Research ArticleWaterfalls Partial Aggregation in Wireless Sensor Networks

Wuyungerile Li1 Bing Jia1 Shunsuke Saruwatari2 and Takashi Watanabe3

1College of Computer Science Inner Mongolia University Hohhot 010021 China2Faculty of Information Shizuoka University Hamamatsu 4328011 Japan3Graduate School of Information Science and Technology Osaka University Osaka 5650871 Japan

Correspondence should be addressed to Bing Jia jiabingimueducn

Received 8 December 2015 Revised 6 March 2016 Accepted 21 March 2016

Academic Editor Miguel A Zamora

Copyright copy 2016 Wuyungerile Li et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In wireless sensor networks (WSNs) energy saving is a critical issue Many research works have been undertaken to save energyData aggregation is one of the schemes that save energy by reducing the amount of data transmission Normally researchers focuson saving energy by aggregating multiple data or turning to achieving short transmission delay in data aggregation few of themare concerned with network lifetime This work achieves an optimum network lifetime by balancing energy consumption amongnodes in network Here we propose a waterfalls partial aggregation controlled by a set of waterfalls pushing rate vectors The firstcontribution of this paper is to propose awaterfalls partial aggregation and tomodel it with queuing theoryThe second contributionis that the optimumnetwork lifetime is achievedmathematically and a near optimumalgorithm is proposed for a given transmissiondelayThe results are compared with existing energy efficient algorithms and the evaluation results show the efficiency of proposedalgorithm

1 Introduction

In the recent development of wireless technology wirelesssensor networks (WSNs) [1] have attracted researchersrsquo atten-tion because of the applicability in many fields for effectivecollection of sensing data with low cost [2] WSN consists ofa large number of sensor nodes where sensor nodes senseevents and generate event data then transmitting the datato a sink node via intermediate sensor nodes in a multihopmanner [3] Sensor nodes are battery-powered with limitedenergy supply moreover in many of the applications sensornodes are deployed in harsh nature environment or vast spaceso that the continuous energy supplement is impossible Inthis case if one of the nodes in the network exhausted energythe network would break down and perform reorganizationwhere the reorganization of the network also consumesmuchenergy and time For these reasons WSNs should be energyefficient For energy saving many researchers are workingon Medium Access Control (MAC) protocols [4] routingprotocols [5] topology control [6] and data aggregation [7]in WSNs

In WSNs data generated from neighboring sensor nodesare often redundant and highly correlated Sensor nodesspend considerable energy for sending or relaying a largenumber of redundant data Moreover a large number of datatransmissions cause data collisions and data congestion Allthese lead to the turning up of data aggregation techniqueData aggregation is defined as the process of aggregating datafrommultiple sensor nodes to eliminate redundant transmis-sion and provide fused information to a sink node In dataaggregation there aremainly three kinds of aggregationwaysThe first is clustering data aggregation where data are col-lected and aggregated at a cluster node and then trans-mitted to a sink node [8 9] The second is hop by hopaggregation which means that data are aggregated at eachintermediate node [10] The third is partial aggregationwhere data aggregation satisfies a time or energy threshold[11] However all these three kinds of data aggregations havetheir drawbacks In clustering data aggregation a cluster headalways consumes much energy than others WSNs alwaysperform a cluster head selecting algorithm to decide a newcluster head which wastes considerable time and energy [12]

Hindawi Publishing CorporationInternational Journal of Distributed Sensor NetworksVolume 2016 Article ID 2392149 11 pageshttpdxdoiorg10115520162392149

2 International Journal of Distributed Sensor Networks

In hop by hop aggregation data suffer long transmissiondelay and nodes on the transmission way suffer unbalancedenergy consumption [10] Compared to these two partialaggregation appears to bemore flexible In partial aggregationwith scheduling technique always set a time period to collectdata for nodes that have aggregating function Here timeperiod is always decided by a given time threshold orrequested data accuracy but always does not consider theenergy consumption of nodes in the network [11 13 14] Aseveryone knows network lifetime is always decided by nodesrsquolifetime when a nodersquos energy is exhausted the network hasto perform reorganization Hence even though the partialaggregation can shorten the transmission delay it still hasthe shortcoming of unbalanced nodes lifetime which leadsto short network lifetime

In this paper to achieve optimum network lifetime atfirst we propose a waterfalls partial aggregation (WPA)which is a kind of scheduling data aggregation scheme whereoptimum network lifetime is achieved by controlling thedata transmitting period Secondly by analyzing the datatransmission process of proposed scheme we determined theformulations of energy consumption and transmission delayof the network which causes advantage for future scientificstudies Then in Section 4 optimum network lifetime is ana-lyzed mathematically and achieved by a heuristics algorithmWe present evaluation in Section 5 Finally we show theconclusion in Section 6

2 Related Work

WSNs have various applications Some applications arerequired to send data as soon as possible while in otherapplications energy saving is much more important Forexample in disaster monitoring or emergency rescue imme-diate data transmission ismore important than others hencedata are always transmitted to neighboring nodes withoutaggregation we call this nonaggregation However in naturemonitoring energy is more significant than transmissiondelay because the replacement of battery for sensor nodesis supposed to be impossible In respect to energy savingdata aggregation technique is widely applied inWSNs that arecomposed of large number of sensor nodes and meanwhileit is not so easy to supply continuous energy Hence we callit full aggregation that processes data aggregation only forenergy saving in the network In some applications of WSNsit is required to trade off node energy and transmissiondelay or some other properties In this case data aggregationshould consider corresponding delay or some other requiredthresholds we call this kind of data aggregation a partialaggregation In this section we introduce nonaggregationfull aggregation and partial aggregation and then discusstheir advantages and disadvantages

21 Nonaggregation Definition of nonaggregation is that anode transmits received data to an adjacent lower nodeimmediately after receiving data from a neighboring nodewhich means that all data are sent to a sink node one by oneas shown in Figure 1

Sink

120582998400n 120582998400nminus1 1205829984003 1205829984002 1205829984001

120582n 120582nminus1 1205822 1205821

nn nnminus1 n2 n1middot middot middot

middot middot middot

Figure 1 Nonaggregation

From the definition of nonaggregation we find that datatransmission rate at a node becomes larger near a sink nodebecause of data relaying properties of a node Hence energyconsumption of nodes near a sink node is much more thanthat in nodes far from the sink As a result nodes near thesink node exhaust energy sooner than others which results inshort network lifetimeMoreover when event data generationrate is large data congestions occur at nodes around a sinknode and they prolong transmission delay as well as let thenetwork lifetime become worse [4ndash6 15ndash18] Sensor MAC[4] (S-MAC) is proposed for energy saving in wireless sensornetworks based on IEEE 80211 MAC protocols In S-MACthe network is always assumed to be less in amount ofdata transmission data process and data aggregation canbe performed in the networks where the transmission delayis considerably tolerant S-MAC uses a listensleep modeland divides the time into frames each frame has the modelof listensleep Besides topology control algorithms arestudied for improving network lifetime and the minimumspanning tree is the typical one LMST [16] is a minimumspanning tree (MST) based topology control algorithm formultihop wireless networks called local minimum spanningtree (LMST) The topology is constructed from each nodewhere a node builds its local MST independently (with theuse of information locally collected) and keeps only one-hopon-tree nodes as its neighbors

22 Full Aggregation Full aggregation is that a node pro-cesses data aggregation when new data generated at itself Inmore detail the node does not transmit any received datafrom neighboring node immediately all arrival data fromneighboring nodes wait for new generated data When thereare new data at a node the node aggregates all arrival datawith its own data and then transmits them to its lower nodeFigure 2 illustrates the full aggregation in detail It is clear tosee from the definition that full aggregation can save muchenergy by aggregating data at every relay node The onlycondition for data transmission is that a new event dataoccurs at a node which means that the data transmissionrate is the same with data generation rate If we assume datageneration rates at all nodes are the same then the numberof data transmissions for each node is the same so thatfull aggregation can achieve balanced energy consumptionamong nodes

However full aggregation has its inevitable defect thattransmission delay is too long for data occurring at nodes farfrom a sink node because of the waiting time for generatingdata at every intermediate node And what is more thetransmission delay is longer when data generation rate islow at nodes PEGASIS (Power-Efficient Gathering in SensorInformation Systems) [10] is one of the energy efficiency

International Journal of Distributed Sensor Networks 3

Sink120582n

120582n

120582nminus1

120582nminus11205822

1205822

1205823 1205821

1205821

nn nnminus1 n2 n1middot middot middot

middot middot middot

Figure 2 Full aggregation

Sink

1Δt 1Δt 1Δt 1Δt 1Δt

120582n 120582nminus1 1205822 1205821

nn nnminus1 n2 n1middot middot middot

middot middot middot

Figure 3 Partial aggregation

chain based data aggregation protocols that employs a greedyalgorithm There are two assumptions in this scheme oneis that all nodes are far from a sink node the other is thatnodes except the end nodes fuse the received data and theirown data and then aggregate them into one packet beforetransmitting them to another neighboring node The mainidea of PEGASIS is forming a chain among the sensor nodesso that each node will receive from and transmit to fused dataonly with a close neighborThe fused data are sent from nodeby node and all the nodes take turns to be the leader fortransmission to the sink nodeThe disadvantage of PEGASISis that transmission delay is too long for nodes that are at theend of a chain

23 Partial Aggregation In partial aggregation data alwayswait a period of time to be transmitted for collecting moredata at a node all collected data are aggregated into one orseveral representative data and then transmitted to an upperneighboring node The waiting time at nodes is adjustableand is always decided by corresponding applications [11 1319ndash21] Figure 3 illustrates a most simple class of partialaggregation in which data wait Δ119905 time for aggregation at anode before they are transmitted (1Δ119905 is data transmissionrate at a node)Whenwaiting timeΔ119905 is out a node aggregatesall generated data and received data into one type of represen-tative data and then sent it out In partial aggregation energyis traded off for improving transmission delay data accuracynetwork lifetime and so forth Since full aggregation leads tolong transmission delay and nonaggregation consumesmuchenergy partial aggregation can balance energy with otherperformances inWSN As the waiting time at nodes is alwaysdecided by applications one can set the waiting time veryshort when they want data with short transmission delay andotherwise the opposite

In in-network cascading timeout data aggregation [13]and ATS-DA (Adaptive Timeout Scheduling for Data Aggre-gation) [11] a sink initially broadcasts a request to all nodesEach node waits for a certain time period to receive datafrom their child nodes The timeout period of each node isset based on the position of the node in the data aggregationtree However these studies are not applicable when real timedata or short delay data are required Because all data at anode have to wait at least SHDavg time no matter whether

the data are important or not Moreover when the data arriv-ing rate is irregular at nodes the node energy consumptionis also irregular among nodes which result in unbalancednetwork In particular in large scaleWSN it is very difficult toachieve fairness for all nodes from the points of transmissiondelay and energy consumption because of data collision andcongestion as well as retransmission

3 Waterfalls Partial Aggregation

31 Sensor NetworkModel In this research we apply queuingtheory to analyze and model wireless sensor network As inmathematical analysis too complex network model makes ittoo sophisticated to get the analytical result and formulationtherefore we define the network model to be the most basicand simplest model of tandem sensor network however theresults can be extensible to other more complex topologiesThe structure and transmission principle of tandem networkare shown in Figure 4

In tandem networks all the nodes deployed in a flat areallocated omnidirectional antennas with the same transmis-sion range Data generated at the nodes are transmitted toa sink node in multihop manner The distance between twoneighboring nodes is the same and all the nodes are withinthe transmission range of their neighboring nodes

(i) 119899119894denotes the 119894th node from the sink

(ii) 119899 is a set of all nodes119873 is natural number set(iii) 119864

119894denotes the energy at node 119899

119894

(iv) 119871max denotes maximum network lifetime(v) 119863119899denotes the total transmission delay

(vi) 120583119894is waterfalls pushing rate at node 119899

119894

(vii) 119905119894is data transmission delay for each node

32 Definition of Waterfalls Partial Aggregation To shortenthe data transmission delay and to lengthen the networklifetime of the network as well as to suppress data congestionaround a sink node we propose a waterfalls partial aggrega-tion (WPA) In WPA data are transmitted to a neighboringnode in two conditions (a) if there are new local generateddata at a node or (b) after waiting a holding time Δ119905 at anode The inverse of the holding time Δ119905

119894at node 119899

119894we

call waterfalls pushing rate and it is denoted as 120583119894where 120583

119894

becomes smaller to nodes nearer a sink node it means 120583119894gt

120583119895 119894 gt 119895 as shown in Figure 5In otherwords data tends to be aggregated rarely at nodes

far from a sink node to suppress delay When there are newgenerated data at a node or the holding time at this nodeis over all data at this node are aggregated and wait forfurther transmission In more detail the holding time fornodes nearer a sink node is longer than the ones far from thesink so that it results in the decrease of the amount of datatransmission near a sink node Finally it can achieve an equalamount of data transmission at each node by controllingthe waterfalls pushing rate vectors The flow chart of dataaggregation and transmission is shown in Figure 6

4 International Journal of Distributed Sensor Networks

SinknN n2n3

n4 n1

middot middot middot

Figure 4 Tandem sensor network

Sink120583nminus1 12058311205832

120583n gt 120583nminus1 gt middot middot middot gt 1205832 gt 1205831

120582n 120582nminus1 1205822 1205821

nn nnminus1 n2 n1middot middot middot

middot middot middot

Figure 5 Waterfalls partial aggregation

Start

Aggregate all data

Are new generated

data

Is server idle

Arrival data

Is channel idle

Send data

Has the holding time

reached

Wait

Wait

No

No

Yes

No

No

Yes

Yes

Yes

(1) Arrival process to server

(2) Service process

(3) Data transmitting process

Figure 6 Flow chart of waterfalls partial aggregation

In the flow chart after data arriving from neighboringnode node 119899

119894starts to check if there are new generated data

at itself If yes node 119899119894aggregates all arrival data with the local

generated data and then waits for service process if no node119899119894checks the holding time if time is run up it aggregates

all arrival data and then goes to service process otherwiseit checks new generated dataThis process we call data arrivalprocess to server After aggregating data node 119899

119894checks if

the server is idle or not If there are no other data waitingfor transmitting we say the server is idle otherwise serveris busy data queue up in the server and wait for furthertransmitting When server is idle node checks if the channelis idle or notWhen channel is idle data transmission process

starts otherwise node 119899119894checks the channel until it turns to

idle Data are sent to a neighboring node if the channel is idle

33 Problem Definition In WPA the waterfalls setting ofholding time is for balancing energy consumption amongnodes as well as suppressing load As we mentioned inSection 2 that network lifetime is always decided by theshortest lifetime node in the network therefore to lengthenthe shortest node lifetime is to extend network lifetimeWhen tolerable maximum transmission delay is given bycorresponding application we set the holding time Δ119905

119894for

nodes in the network according to WPA algorithm Thereason that set a maximum tolerable transmission delay isbecause in a finite network there would have several sets ofwaterfalls pushing rates vectors corresponding to differenttransmission delay According to our definition waterfallspushing rate 120583

119894is inverse of holding time With larger

waterfalls pushing rate 120583119894 a node transmits more data and

results in much energy consumption Thus the problembecomes how to set the waterfalls pushing rates among nodesso that all nodes keep having the same energy consumption

In other words how long should the holding time ofarrival data for data aggregation at a node be to meet theconditions of balanced energy consumption among nodesand given transmission delay For a node if required trans-mission delay is given as119863

119899 then what our algorithm should

do is let

119863119899= 1199051+ 1199052+ sdot sdot sdot + 119905

119894+ sdot sdot sdot + 119905

119899 (119899 isin 119873) (1)

Here 119905119894is transmission delay for node 119899

119894and we deter-

mine the formulation of transmission delay for each node aswell as total transmission delay of the network in Section 34In WPA data have to wait for Δ119905 time at a node for gatheringmore data hence the transmission delay for a node ismainly decided by this Δ119905 On the other hand to balancethe energy consumption among nodes we first obtain totalenergy consumption 119864

119894for node 119899

119894which is composed of

energy of data reception data transmission and overhearingWe show the solution of energy consumption for node 119899

119894

and determine the formulations in Section 34 For node 119899119894

data transmission rate is decided by itself and data receivingrate is decided by its upper neighboring node 119899

119894+1and the

overhearing rate is decided by its lower neighboring node119899119894minus1

For node 119899119894 data transmission rate 1205821015840

119894is decided by data

generation rate 120582119894and waterfalls pushing rate 120583

119894 If we set

the maximum network lifetime as 119871max then we obtain (2)

International Journal of Distributed Sensor Networks 5

G Server

120583i 120582i

Queuerx Queuetx

120582998400998400i+1 120582998400i 120582998400998400i

Figure 7 Analytical model of waterfalls partial aggregation at node119899119894

which means 119871max is decided by nodes lifetime 119871119894 or 119895 when

two arbitrary nodes 119894 and 119895 are balanced on energy 119864119894= 119864119895

119871max = 119871 119894 or 119895 (119864119894 = 119864119895) (1 le 119894 119895 lt 119899) (2)

Therefore for achieving balanced energy among nodes wefirst determine the optimum set of waterfalls pushing rates 120583

119894

that correspond to given 119871max in the network

120583119894= argmax (119871 (120583

119894)) (3)

From the above analysis it is clear that we are trying to obtaina set of waterfalls pushing rates

120583119894| 120583119894lt 120583119894+1 120583119894isin |119873| 1 lt 119894 lt 119899 (4)

For obtaining optimum holding time for each node it isnecessary to know total delay of the network and energyconsumption for a node In the following sections wedetermine the total delay and node energy consumption

34 Analytical Model of WPA

341 Queuing Theory Analysis For determining the trans-mission delay we model the data arrival process and datatransmission process of WPA by queuing theory The analyt-ical model of node 119899

119894is shown in Figure 7

Before explaining the model we introduce QueuerxQueuetx G and Server Queuerx denotes the arrival dataqueue in which arrival data are waiting for data aggregationat node 119899

119894 Queuetx denotes the data queue after aggregation

in which aggregated data are waiting for transmission toa neighbor node G is assumed as a virtual gate betweenQueuerx and Queuetx Data transmitting process of data inQueuetx is accomplished via Server

In the analytical model 12058210158401015840119894+1

is arrival data rate fromupper node toQueuerx of node 119899119894 and is approximately abidedby Poisson distribution At node 119899

119894 event generation rate 120582

119894

is assumed to be Poisson distributionWaterfalls pushing rate120583119894is assumed to be exponential distribution In Queuerx all

arrival data and generated data are aggregated and then joinQueuetx via gate G 120582

1015840

119894is data arrival rate to Queuetx in which

data are waiting for further transmitting 12058210158401015840119894is the data rate

upon exiting from the Server Next we analyze the abovemodel start from data receiving at a node to data that aretransmitted to a neighboring node

342 Event Waiting Time In this subsection we decide thetime interval that datawait inQueuerx for aggregating at node

120582998400998400i+1120582998400998400i+1120582998400998400i+1120582998400998400i+1120582998400998400i+1

0 1 k minus 1 k

2120582i + 120583i

2120582i + 120583i 2120582i + 120583i2120582i + 120583i

middot middot middot middot middot middot

Figure 8 State transition of Queuerx

119899119894 we call this time interval as event waiting time and denote

it by 120591119890(119894) and it is counted from when the first data arrive

to Queuerx until the data are aggregated and join QueuetxTo determine 120591

119890(119894) we first determine the amount of data

waiting for event data in Queuerx at node 119899119894 The amount

of data is denoted as 119876rx(119894) and we describe state transitionrate diagram of Queuerx to determine it in Figure 8 Notethat at node 119899

119894 the arrival data rate 12058210158401015840

119894+1is approximately

Poisson distribution and 120582119894is event data generation rate at

node 119899119894and is approximately Poisson distribution In data

aggregation service time in queuing theory corresponds todata aggregating time which is time interval from last dataaggregation until next data aggregation The basic idea of theanalysis is that data waits in Queuerx for the duration accord-ing to the exponential distribution of average 12120582

119894 since

data aggregation rate is inverse of the mean aggregating timehence data aggregation rate is 2120582

119894 andwhen consideringwith

waterfalls pushing rate data aggregating rate is

120582agg = 2120582119894 + 120583119894 (5)

In the diagram the state variable is the number of datawaiting for an event Assume that a number of data arewaiting for an event data at node 119899

119894in Queuerx If we let 1198751198940

and 119875119894119896

be the probability when the number of data waitingfor an event is 0 and 119896 in Queuerx at node 119899

119894 at state 0

according to Figure 8 we obtain

119875119894012058210158401015840

119894+1=

infin

sum

119895=1

119875119894119895120582agg (6)

At state 119896 we obtain

11987511989411989612058210158401015840

119894+1= 119875119894119896minus1

12058210158401015840

119894+1minus 119875119894119896120582agg (7)

Hence we obtain

119875119894119896=

12058210158401015840

119894+1

12058210158401015840

119894+1+ 120582agg

119875119894119896minus1

119875119894119896minus1

=12058210158401015840

119894+1

12058210158401015840

119894+1+ 120582agg

119875119894119896minus2

1198751198941=

12058210158401015840

119894+1

12058210158401015840

119894+1+ 120582agg

1198751198940

(8)

6 International Journal of Distributed Sensor Networks

Bringing these equations into (6) we determine the relation-ship between 119875

119894119896and 119875

1198940that

119875119894119896= (

12058210158401015840

119894+1

12058210158401015840

119894+1+ 120582agg

)

119896

1198751198940 (9)

As we knowinfin

sum

119896=1

119875119894119896= 1 (10)

Let

120572 =12058210158401015840

119894+1

12058210158401015840

119894+1+ 120582agg

(11)

Then we determineinfin

sum

119896=1

1205721198961198751198940= 1 (12)

Asinfin

sum

119896=1

120572119896=

1

1 minus 120572 (13)

henceinfin

sum

119896=1

1205721198961198751198940=

1

1 minus 1205721198751198940= 1 (14)

Accordingly we obtain

1198751198940= 1 minus 120572 (15)

Substituting 120572 we determine

1198751198940=

2120582119894+ 120583119894

12058210158401015840

119894+1+ 120582agg

(16)

119875119894119896=

(12058210158401015840

119894+1)119896

+ 120582agg

(12058210158401015840

119894+1+ 120582agg)

119896+1 (17)

Now we determine the amount of data that are received inQueuerx(119894) as

119876rx (119894) =infin

sum

119896=0

119896119875119894119896 (18)

Taking (17) into the above equation we obtain

119876rx (119894) =infin

sum

119896=0

119896

(12058210158401015840

119894+1)119896

+ 120582agg

(12058210158401015840

119894+1+ 120582agg)

119896+1 (19)

We express the above equation as

119876rx (119894) =infin

sum

119896=0

119896(12058210158401015840

119894+1

12058210158401015840

119894+1+ 120582agg

)

119896+1120582agg

12058210158401015840

119894+1

(20)

0

Y

X

Figure 9 Property distribution of119883 119884

Let 120573 = 12058210158401015840

119894+1(12058210158401015840

119894+1+ 120582agg) and 120574 = 120582agg120582

10158401015840

119894+1 then we

simplify (20) as

119876rx (119894) =infin

sum

119896=0

119896120573119896+1

120574 =

infin

sum

119896=0

119896120573119896minus1

1205732120574 = 1205732120574

infin

sum

119896=0

119896120573119896minus1

= 1205732120574120597

120597120573

infin

sum

119896=0

120573119896= 1205732120574120597

120597120573

1

1 minus 120573=

1205732120574

(1 minus 120573)2

(21)

As 120573 = 12058210158401015840119894+1(12058210158401015840

119894+1+ 120582agg) and 120582agg = 2120582119894 + 120583119894 so we get that

1 minus 120573 =2120582119894+ 120583119894

12058210158401015840

119894+1+ 2120582119894+ 120583119894

(22)

Therefore we determine that

1205732120574

(1 minus 120573)2

=

(12058210158401015840

119894+1 (12058210158401015840

119894+1+ 2120582119894+ 120583119894))2

(2120582119894+ 120583119894) 12058210158401015840

119894+1

((2120582119894+ 120583119894) (12058210158401015840

119894+1+ 2120582119894+ 120583119894))2

(23)

By calculating the above equations we determine that

119876rx (119894) =12058210158401015840

119894+1

2120582119894+ 120583119894

(24)

According to Littlersquos formula we determine the event waitingtime as follows

120591119890 (119894) =

119876rx (119894)

2120582119894+ 120583119894

=12058210158401015840

119894+1

(2120582119894+ 120583119894)2 (25)

343 Arrival Process to Queue119905119909 From the analytical process

of Queuerx(119894) we find that arrival data rate 1205821015840119894to Queuerx(119894)

is decided by the waterfalls pushing rate 120583119894and event data

generation rate 120582119894at node 119899

119894 The event generation rate 120582

119894

and waterfalls pushing rate 120583119894abide by discrete distribution

and they are independent of one another To determine theformulation of 1205821015840

119894 we calculate the property distribution of

120582119894and 120583

119894 At first we define that 120582

119894and 120583

119894are independent

distribution 119883 and 119884 as shown in Figure 9 We define thevalues of 119883 as 1 2 119864

119883and the values of 119884 as 1 2 119864

119884

Here we prove that the property 119884 is bigger than119883

International Journal of Distributed Sensor Networks 7

We determine the property as follows 119875119883and 119875

119884denote

the properties when119883 = 119894 and 119884 = 119894

119875 [119884 gt 119883]

= 119875119883 (119883 = 1) 119875119884 (119884 = 2) + 119875119884 (119884 = 3) + sdot sdot sdot + 119875119884 (119884

= 119864119884)

+ 119875119883 (119883

= 2) +119875119884 (119884 = 3) + sdot sdot sdot + 119875119884 (119884 = 119864

119884)

+ 119875119883(119883

= 119864119883) + sdot sdot sdot + 119875

119884(119884 = 119864

119883+ 1)

=

119864119883

sum

119894=1

119875119883 (119883 = 119894)

119864119884

sum

119895=119894+1

119875119884(119884 = 119895)

(26)

When applied to continuous distribution we obtain

119875 [119884 gt 119883] = int

infin

0

119875119883 (119905) [int

infin

0

119875119884 (119911) 119889119911] 119889119905 (27)

Here if 119875119883(119905) and 119875

119884(119905) are exponential distribution then

119875119883 (119905) = 120582119883119890

minus120582119883119905

119875119884 (119905) = 120582119884119890

minus120582119884119905

119875 [119884 gt 119883] = int

infin

0

119875119883 (119905) [int

infin

0

120582119884119890minus120582119884119911119889119911] 119889119905

= int

infin

0

119875119883 (119905) [minus119890

minus120582119884119911]infin

119905119889119905

= int

infin

0

119875119883 (119905) (119890

minus120582119884119905) 119889119905

= int

infin

0

120582119883119890minus120582119883119905119890minus120582119884119905119889119905

= int

infin

0

120582119883119890minus(120582119883+120582119884)119905119889119905

=120582119883

120582119883+ 120582119884

[minus119890minus(120582119883+120582119884)119905]

infin

0=

120582119883

120582119883+ 120582119884

(28)

Accordingly we determine the data arrival rate of Queuetx

1205821015840

119894= 120582119894+

120582119894

120582119894+ 120583119894

120583119894 (29)

344 Service Process Since the data generation rate is Pois-son distribution and the waterfalls pushing rate abides byexponential distribution the data arrival rate to Queuetx(119894)approximates to Poisson distribution and each node has oneserver The ACK packet transmission time is not consideredData aggregating time is very short and negligibleThereforethe service time is one-hop data transmission time 120591 In our

work data transmission rate is 119881119888and generated data size is

119878119894 Hence the service time for generated data is

120591 =119878119894

119881119888

(30)

Since 119881119888and 119878

119894are constant hence the service time

for data is fixed and constant According to the aboveanalysis we can determine that the queuing system on serverapproximates to 1198721198631 model As data transmission timeis determined and server utilization ratio is 120578 = 120582

1015840

119894times 120591tx(119894)

hence the average data transmission time 120591tx(119894) is

120591tx (119894) =120591

1 minus 1205821015840

119894120591 (31)

We figure up the probability density function of the servertime by means of Laplace transform in queuing theory andevaluate server waiting time 120591

119904(119894) that data wait for further

transmitting in server

120591119904 (119894) =

1205821015840

119894(120591tx (119894))

2

2 (1 minus 1205821015840

119894120591tx (119894))

(32)

345 Channel Waiting Time In general queuing systemcustomers can be served if there are no other customerswaiting in front of them However when we apply queuingtheory to model wireless communication it is necessary toconsider the impact of wireless channel caused by its proper-ties In wireless sensor networks because of the overhearingof omnidirectional antenna node has to wait a period oftime if its upper or lower neighbors are transmitting dataHere we call this time as channel waiting time 120591

119888(119894) and it

is determined as follows

120591119888 (119894) = 2 times 120578 times 120591tx (119894) = 2120582

1015840

119894times 1205912

tx (119894) (33)

346 Total Delay Total delay119863119899inWPA is the time interval

from when an event data is generated at a node until thedata are received by a sink node in 119873 hops network Theformulation is as follows

119863119899=

119873

sum

119894=1

(120591119890 (119894) + 120591119904 (119894) + 120591119888 (119894) + 120591tx (119894)) (34)

347 Energy Consumption The energy consumption 119864119894for

node 119899119894is the sum of transmission energy consumption

reception energy consumption and overhearing energy con-sumption Now we determine the three kinds of energyconsumption in the following paragraphs Note that 119875tx and119875rx are energy consumption of transmitting or receiving foreach data packet and 119864tx(119894) 119864rx(119894) and 119864oh(119894) are energyconsumption of data transmitting data receiving and over-hearing of node 119899

119894in 119863119899time As we are trying to obtain

the balanced energy consumption among nodes hence wecalculate energy consumption per 119863

119899time for each node

Therefore we get the data transmission energy consumptionaccording to Littlersquos formula as follows

119864tx (119894) = 1205821015840

119894119863119899119875tx (35)

8 International Journal of Distributed Sensor Networks

The reception energy consumption is

119864rx (119894) = 12058210158401015840

119894+1119863119899119875rx (36)

Here 12058210158401015840119894+1

is data arrival rate from upper neighboring node119899119894+1

as the service time is one data packet transmission timehence the data rate getting out from server is same with datatransmission rate which means

12058210158401015840

119894+1= 1205821015840

119894+1 (37)

The overhearing energy consumption is

119864oh (119894) = 1205821015840

119894minus1119863119899119875rx (38)

Here 1205821015840119894minus1

is data transmission rate of node 119899119894minus1

Hence wedetermine total energy consumption119864

119894for a node in time 119863

119899

as

119864119894= 119864tx (119894) + 119864rx (119894) + 119864oh (119894) (39)

4 Optimum Network Lifetime

41 Mathematical Solution According to the analysis in Sec-tion 32 the longest network lifetime is decided by the totaltransmission delay and waterfalls pushing rate vectors Herewe illustrate the relationship between transmission delayenergy consumption and network lifetime so that at lastwe determine the formulations of a set of waterfalls pushingrates Accordingly energy consumption can be written asfollows

119864119894= (1205821015840

119894119875tx + 120582

10158401015840

119894+1119875rx + 120582

1015840

119894minus1119875rx)

= 1205821015840

119894119875tx + (120582

10158401015840

119894+1+ 1205821015840

119894minus1) 119875rx

(40)

To balance energy among nodes we have

1198641= 1198642= sdot sdot sdot = 119864

119894= sdot sdot sdot = 119864

119899 (41)

As the service time is one-hop data transmitting time hence

12058210158401015840

119894= 1205821015840

119894 (42)

Here 119864119894is the energy consumption for node in time 119863

119899

Hence we assume the energy consumption for each unit oftime is 119864 We assume that 120582

119894= 120582 and the network in a finite

network Hence there are no arrival data to the last node inthe network so that it does not need a waterfalls pushing rateand moreover there is no overhearing for the first node thatcounted from the sink node Therefore for node 119899

119899 we have

120583119899= 0

1205821015840

119899= 120582

119864rx (119899) = 0

119864oh (1) = 0

(43)

Accordingly we obtain equations as follows

(120582 +1205821205831

120582 + 1205831

)119875tx + (120582 +1205821205832

120582 + 1205832

)119875rx = 119864

(120582 +1205821205832

120582 + 1205832

)119875tx + (2120582 +1205821205831

120582 + 1205831

+1205821205833

120582 + 1205833

)119875rx = 119864

(120582 +120582120583119894

120582 + 120583119894

)119875tx

+ (2120582 +120582120583119894minus1

120582 + 120583119894minus1

+120582120583119894+1

120582 + 120583119894+1

)119875rx = 119864

120582119875tx + (120582 +120582120583119899minus1

120582 + 120583119899minus1

)119875rx = 119864

(44)

By calculating the above equations we have

120583119899minus1

=120582119875rx (120582 + 120582120583119899minus2 (120582 + 120583119899minus2))

(120582120583119899minus2

(120582 + 120583119899minus2

)) 119875rx minus 120582119875tx(45)

120583119899minus2

=120582119875rx (120582120583119899minus3 (120582 + 120583119899minus3)) + 120582 (119875rx minus 119875tx) (120582120583119899minus1 (120582 + 120583119899minus1))

(119875rx minus 119875tx) (120582 minus 120582120583119899minus1 (120582 + 120583119899minus1)) minus 119875rx (120582120583119899minus3 (120582 + 120583119899minus3))

(46)

120583119894=

1205822120583119894+2

(119875rx minus 119875tx) (120582 + 120583119894+2) minus 1205822120583119894+1

(119875rx minus 119875tx) (120582 + 120583119894+1) minus 1205822120583119894+3119875rx (120582 + 120583119894+3)

120582120583119894+3119875rx (120582 + 120583119894+3) minus 120582120583119894+2 (119875rx minus 119875tx) (120582 + 120583119894+2) + 120582120583119894+1 (119875rx minus 119875tx) (120582 + 120583119894+1) minus 120582119875rx

1205831=

12058221205832(119875rx minus 119875tx) (120582 + 1205832) minus 120582

21205833119875rx (120582 + 1205833) minus 120582

2119875rx

1205821205832(119875tx minus 119875rx) (120582 + 1205832) minus 1205821205833119875rx (120582 + 1205833) + 2120582119875rx minus 120582119875tx

(47)

International Journal of Distributed Sensor Networks 9

Input 120582 120583119899minus1

120578119863appOutput [120583

1 120583119899minus1

]120583max = 2120582

120583119899minus1

= 120583max[1205831 120583119899minus1] = 119891 (120583

119899minus119894)

119863heu = 119892 (1205831 120583119899minus1)

While |119863app minus 119863heu| gt 120578 do120583119899minus2

= 120583119899minus1

minus Δ

[1205831 120583119899minus1] = 119891(120583

119899minus119894)

119863heu = 119892(1205831 120583119899minus1)

EndReturn [120583

1 120583119899minus1]

Algorithm 1 Heuristics algorithm

From the above equations we obtained the relationshipamong waterfalls pushing rate Equation (45) shows that if120583119899minus1

is given then 120583119899minus2

can be determined In (46) if 120583119899minus1

and120583119899minus2

are given then 120583119899minus3

can be determined In this way all 120583119894

can be determined if 120583119899minus1

is known To determine the valuesof 120583119894 we designed a heuristics algorithm as follows In light of

real applications ofWSNwhere the threshold of transmissiondelay is always given in the heuristics algorithm we assumethat the transmission delay is predefined by applicationHence we set the value of120583

119899minus1and then determine the others

However the value of 120583119899minus1

being too large or too little makesthe algorithm too complex to process Therefore it is criticalto determine the maximum value of 120583

119899minus1

42 Heuristics Algorithm In a given application we assumethat data generation rates at all nodes are known and are 120582Thus at node 119899

119899minus1 if the arrival data from node 119899

119899just does

not keep pace with the generated data then the node needswaterfalls pushing rate to transmit the arrival data Howeverthe processing period of waterfalls pushing rate is decided bynode 119899

119899minus1 hence there is again the case that the arrival data

just does not catch up the pace of the waterfalls pushing rate120583119899minus1

Therefore we determine simply that for the arrival datathe average catching up rate of waterfalls pushing rate at node119899119899minus1

is 1205822 Accordingly for transmitting all arrival data themaximum value of 120583

119899minus1at node 119899

119899minus1is

120583max = 2120582 (48)

Hence a heuristics algorithm is described as shown inAlgorithm 1

5 Evaluation

51 Validation of the Queuing Theory Model We implementWPA in the original network simulator written in C++language Analytic results in the previous section are shownas well as simulated result The parameters are shown inTable 1

In the simulation data occurs at each node randomly andindependently Buffer size of each node is infinite Althoughanalytic model assumes that transmission error is negligible

Table 1 Simulation parameters of WPA

Node distance 10 (m)Transmission range 11 (m)Transmission rate 250 (kbps)Data size 4096 (bit)Current consumption for transmission 174 (mA)Current consumption for reception 197 (mA)MAC CSMACARouting DSR

001

01

1

10

100

1000

01 1 10Data generation rate (datasec)

WPA-analysisWPA-simulation

Tota

l del

ay (120583

sec)

Figure 10 Total delay of WPA

transmission errors and retransmission may occur in thesimulation We set data generation rate 120582 = 5 and waterfallspushing rate as 120583

119894= 1 2 3 4 5 for 5-hop tandem network

From Figures 10 and 11 we find that our analytical modelmeets with simulation results so that we basically confirm thecorrectness of queuing theory model of WPA

52 Evaluation of OptimumNetwork Lifetime In this sectionwe evaluate the effectiveness of the proposedwaterfalls partialaggregation Here we consider network environment asnature monitoring network so that we obtain the maximumnetwork lifetime which is the same with network lifetime offull aggregation We set 119864 in (44) which is equal to 119864ful-largestwhere 119864ful-largest denotes the energy consumption of the nodethat consumes most energy in the network The optimalresults can be achieved by numerical packets however themathematical calculation results are too complex For moreunderstanding of our algorithm we obtain the pushing ratevectors applying Excel when the event generation rate is120582 = 1for 5-hop tandem network Nodes are counted from the sinknode which means that the nearest node with the sink isnode 119899

1and its waterfalls pushing rate is 120583

1 The optimum

waterfalls pushing rates are shown in Table 2 when WPAhas the same network lifetime with full aggregation Theanalytical model of nonaggregation and full aggregation isrealized by queuing theory however for the brevity of thepaper we omitted the derivation process

10 International Journal of Distributed Sensor Networks

0

1000

2000

3000

4000

5000

6000

7000

1 2 3 4 5

Ener

gy co

nsum

ptio

n (m

A)

Node ID

WPA-analysisWPA-simulation

Figure 11 Node energy consumption of WPA

0

05

1

15

2

25

3

35

4

1 2 3 4 5Node ID

NonaggregationPartial aggregationFull aggregation

Ener

gy co

nsum

ptio

n (m

A)

Figure 12 Lifetime of nodes

Table 2 Waterfalls pushing rate for optimum network lifetime

Waterfalls pushing rate Values1205831

0021205832

0041205833

0081205834

011205835

05

Figure 12 shows the lifetime of all nodes in the networkwhere we set the waterfalls pushing rate of nodes as the samewith Table 2 From the figure we find that all nodes almosthave the same lifetime in WPA In this case we can saythat the node energy is utilized optimally in WPA In fullaggregation the second and the third node consumed mostenergy and the other nodes have less energy consumptionwhich results in the suboptimum utilization of energy Thenonaggregation has very large energy consumption at thesecond node which results in the short lifetime of the entirenetwork We find from the figure that nonaggregation and

001

01

1

10

100

01 1 10 100

NonaggregationPartial aggregationFull aggregation

Data generation rate (datasec)

Tota

l del

ay (s

ec)

Figure 13 Transmission delay of network

full aggregation have the largest energy consumption attheir second node The reason is that the first node has noadjacent upper node which means that it does not consumeoverhearing energy compared with others

Figure 13 shows the corresponding total delay of thenetwork where we set the same waterfalls pushing rates withTable 1 From the figure we find that when it has the samenetwork lifetime with full aggregation WPA can shorten thetransmission delay by considerable amount When data gen-eration rate becomes larger (larger than 6 types of datasec)transmission delay of nonaggregation gets longer rapidly dueto data congestion at nodes However in WPA the rapidincrease of transmission delay arises later (at data generationrate of 30 types of datasec) than nonaggregation and fullaggregation on data generation line which indicates thatWPA can relieve data congestion at nodes This is becausewith waterfalls pushing rates proposed WPA can adjust thenumber of data transmissions Moreover we find from thefigure that with the given waterfalls pushing rates there isminimum transmission delay and we can determine it inWPA

53 Summary of Evaluation Firstly from the presentationof Section 51 we conclude that our analytical model ofWPA meets with simulation results the diversity of energyconsumption is that we did not consider retransmissionin analytical model and considered it in simulation Sec-ondly in Section 52 compared to nonaggregation and fullaggregation proposed WPA has the superiorities as followsnonaggregation consumes much energy and data congestionoccurs easily when data generation rate is larger In WPAafter data are sent to a neighboring node the data haveto be aggregated together or aggregated with local data forfurther transmitting hence WPA is more energy efficientthan nonaggregationWhen data generation rate is larger at anode data congestion occurs at nonaggregation which leadsto long transmission delay however proposed WPA canrelieve data congestion via reducing the number of data trans-missions by aggregating received data at nodes On the other

International Journal of Distributed Sensor Networks 11

hand full aggregation can save much energy consumptionand suffer long transmission delay Compared to full aggre-gation proposed WPA has similar energy consumption andis superior in balancing energy consumption among nodesMoreover with waterfalls pushing rates WPA can controldata waiting time more reasonably and efficiently than fullaggregation at nodes so that transmission delay of WPA issuperior to full aggregation

6 Conclusion

In this paper we proposed a waterfalls partial aggregationthat can achieve optimum network lifetime in WSN viaapplying data aggregation At first we modeled WPA inqueuing theory and then determined transmission delay ofthe network and energy consumption of a node Then wecontinued to model balanced energy consumption amongnodes The evaluation parts show the correctness of our ana-lytical model and demonstrate the superiorities of proposedWPA

Competing Interests

The authors declare no competing financial interests

Acknowledgments

Thiswork is supported in part by theNatural Science Founda-tion of Inner Mongolia Autonomous Region of China underGrant no 2014BS0603 and no 2014BS0601 and the Programof Higher-Level Talents of Inner Mongolia University underGrant no 30105-135134

References

[1] C-C Shen C Srisathapornphat and C Jaikaeo ldquoSensor infor-mation networking architecture and applicationsrdquo IEEE Per-sonal Communications vol 8 no 4 pp 52ndash59 2001

[2] S D Glaser ldquoSome real-world applications of wireless sen-sor nodesrdquo in Smart Structures and Materials 2004 Sensorsand Smart Structures Technologies for Civil Mechanical andAerospace Systems vol 5391 of Proceedings of SPIE pp 344ndash355San Diego Calif USA March 2004

[3] I F Akyildiz W Su Y Sankarasubramaniam and E CayircildquoWireless sensor networks a surveyrdquo Computer Networks vol38 no 4 pp 393ndash422 2002

[4] W Ye J Heidemann and D Estrin ldquoSensor-MAC (S-MAC)medium access control for wireless sensor networksrdquo in Pro-ceedings of the 21st International Annual Joint Conference of theIEEEComputer and Communications Societies (INFOCOM rsquo02)New York NY USA June 2002

[5] L J G Villalba A L S Orozco A T Cabrera andC J B AbbasldquoRouting protocols in wireless sensor networksrdquo Sensors vol 9no 11 pp 8399ndash8421 2009

[6] P Santi ldquoTopology control in wireless ad hoc and sensornetworksrdquo ACM Computing Surveys vol 37 no 2 pp 164ndash1942005

[7] R Rajagopalan and P K Varshney ldquoData-aggregation tech-niques in sensor networks a surveyrdquo IEEE CommunicationsSurveys and Tutorials vol 8 no 4 pp 48ndash63 2006

[8] W-S Jung K-W Lim Y-B Ko and S-J Park ldquoA hybridapproach for clustering-based data aggregation in wireless sen-sor networksrdquo in Proceedings of the 3rd International ConferenceonDigital Society (ICDS rsquo09) pp 112ndash117 IEEECancunMexicoFebruary 2009

[9] M Kumar S Verma and P S Pratap ldquoClustering approach todata aggregation in wireless sensor networksrdquo in Proceedingsof the 16th IEEE International Conference on Networks pp 1ndash6IEEE New Delhi India December 2008

[10] S Lindsey C Raghavendra and K M Sivalingam ldquoDatagathering algorithms in sensor networks using energy metricsrdquoIEEE Transactions on Parallel and Distributed Systems vol 13no 9 pp 924ndash935 2002

[11] JW Baek Y J Nam andD-W Seo ldquoATS-DA adaptive timeoutscheduling for data aggregation in wireless sensor networksrdquoin Proceedings of the International Conference on InformationNetworking pp 375ndash384 Estoril Portugal 2008

[12] H Chen H Mineno and T Mizuno ldquoAdaptive data aggrega-tion scheme in clustered wireless sensor networksrdquo ComputerCommunications vol 31 no 15 pp 3579ndash3585 2008

[13] I Solis and K Obraczka ldquoIn-network aggregation trade-offsfor data collection in wireless sensor networksrdquo InternationalJournal of Sensor Networks vol 1 no 3-4 pp 200ndash212 2006

[14] F Mirian and M Sabaei ldquoA delay and accuracy sensitive dataaggregation structure in wireless sensor networksrdquo in Proceed-ings of the International Conference on InformationManagementand Engineering (ICIME rsquo09) pp 231ndash235 IEEE Kuala LumpurMalaysia April 2009

[15] P Sun X Zhang Z Dong et al ldquoA novel energy efficientwireless sensor MAC protocolrdquo in Proceedings of the 4th Inter-national Conference on Networked Computing and AdvancedInformation Management pp 68ndash72 Gyeongju South KoreaSeptember 2008

[16] N Li J C Hou and L Sha ldquoDesign and analysis of an mst-based topology control algorithmrdquo in Proceedings of the IEEESocieties 22nd Annual Joint Conference of the IEEE Computerand Communications (INFOCOM rsquo03) vol 3 pp 1702ndash1712 SanFrancisco Calif USA April 2003

[17] B Yin H Shi and Y Shang ldquoA two-level topology controlstrategy for energy efficiency in wireless sensor networksrdquoInternational Journal of Wireless and Mobile Computing vol 4no 1 pp 41ndash49 2010

[18] A Sharif V M Potdar and A J D Rathnayaka ldquoLCARTlightweight congestion aware reliable transport protocol forWSN targeting hetero-geneous trafficrdquo in Proceedings of the17th International Conference on Neural Information Processing(ICONIP rsquo10) Sydney Australia November 2010

[19] B Yu J Li and Y Li ldquoDistributed data aggregation schedulingin wireless sensor networksrdquo in Proceedings of the 28th Confer-ence on Computer Communications (IEEE INFOCOM rsquo09) pp2159ndash2167 Rio de Janeiro Brazil April 2009

[20] M X Cheng and L Yin ldquoEnergy-efficient data gatheringalgorithm in sensor networks with partial aggregationrdquo Inter-national Journal of Sensor Networks vol 4 no 1-2 pp 48ndash542008

[21] A A Allen Probability Statistics and Queueing Theory withComputer Science Applications Gulf Professional Publishing1990

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DistributedSensor Networks

International Journal of

2 International Journal of Distributed Sensor Networks

In hop by hop aggregation data suffer long transmissiondelay and nodes on the transmission way suffer unbalancedenergy consumption [10] Compared to these two partialaggregation appears to bemore flexible In partial aggregationwith scheduling technique always set a time period to collectdata for nodes that have aggregating function Here timeperiod is always decided by a given time threshold orrequested data accuracy but always does not consider theenergy consumption of nodes in the network [11 13 14] Aseveryone knows network lifetime is always decided by nodesrsquolifetime when a nodersquos energy is exhausted the network hasto perform reorganization Hence even though the partialaggregation can shorten the transmission delay it still hasthe shortcoming of unbalanced nodes lifetime which leadsto short network lifetime

In this paper to achieve optimum network lifetime atfirst we propose a waterfalls partial aggregation (WPA)which is a kind of scheduling data aggregation scheme whereoptimum network lifetime is achieved by controlling thedata transmitting period Secondly by analyzing the datatransmission process of proposed scheme we determined theformulations of energy consumption and transmission delayof the network which causes advantage for future scientificstudies Then in Section 4 optimum network lifetime is ana-lyzed mathematically and achieved by a heuristics algorithmWe present evaluation in Section 5 Finally we show theconclusion in Section 6

2 Related Work

WSNs have various applications Some applications arerequired to send data as soon as possible while in otherapplications energy saving is much more important Forexample in disaster monitoring or emergency rescue imme-diate data transmission ismore important than others hencedata are always transmitted to neighboring nodes withoutaggregation we call this nonaggregation However in naturemonitoring energy is more significant than transmissiondelay because the replacement of battery for sensor nodesis supposed to be impossible In respect to energy savingdata aggregation technique is widely applied inWSNs that arecomposed of large number of sensor nodes and meanwhileit is not so easy to supply continuous energy Hence we callit full aggregation that processes data aggregation only forenergy saving in the network In some applications of WSNsit is required to trade off node energy and transmissiondelay or some other properties In this case data aggregationshould consider corresponding delay or some other requiredthresholds we call this kind of data aggregation a partialaggregation In this section we introduce nonaggregationfull aggregation and partial aggregation and then discusstheir advantages and disadvantages

21 Nonaggregation Definition of nonaggregation is that anode transmits received data to an adjacent lower nodeimmediately after receiving data from a neighboring nodewhich means that all data are sent to a sink node one by oneas shown in Figure 1

Sink

120582998400n 120582998400nminus1 1205829984003 1205829984002 1205829984001

120582n 120582nminus1 1205822 1205821

nn nnminus1 n2 n1middot middot middot

middot middot middot

Figure 1 Nonaggregation

From the definition of nonaggregation we find that datatransmission rate at a node becomes larger near a sink nodebecause of data relaying properties of a node Hence energyconsumption of nodes near a sink node is much more thanthat in nodes far from the sink As a result nodes near thesink node exhaust energy sooner than others which results inshort network lifetimeMoreover when event data generationrate is large data congestions occur at nodes around a sinknode and they prolong transmission delay as well as let thenetwork lifetime become worse [4ndash6 15ndash18] Sensor MAC[4] (S-MAC) is proposed for energy saving in wireless sensornetworks based on IEEE 80211 MAC protocols In S-MACthe network is always assumed to be less in amount ofdata transmission data process and data aggregation canbe performed in the networks where the transmission delayis considerably tolerant S-MAC uses a listensleep modeland divides the time into frames each frame has the modelof listensleep Besides topology control algorithms arestudied for improving network lifetime and the minimumspanning tree is the typical one LMST [16] is a minimumspanning tree (MST) based topology control algorithm formultihop wireless networks called local minimum spanningtree (LMST) The topology is constructed from each nodewhere a node builds its local MST independently (with theuse of information locally collected) and keeps only one-hopon-tree nodes as its neighbors

22 Full Aggregation Full aggregation is that a node pro-cesses data aggregation when new data generated at itself Inmore detail the node does not transmit any received datafrom neighboring node immediately all arrival data fromneighboring nodes wait for new generated data When thereare new data at a node the node aggregates all arrival datawith its own data and then transmits them to its lower nodeFigure 2 illustrates the full aggregation in detail It is clear tosee from the definition that full aggregation can save muchenergy by aggregating data at every relay node The onlycondition for data transmission is that a new event dataoccurs at a node which means that the data transmissionrate is the same with data generation rate If we assume datageneration rates at all nodes are the same then the numberof data transmissions for each node is the same so thatfull aggregation can achieve balanced energy consumptionamong nodes

However full aggregation has its inevitable defect thattransmission delay is too long for data occurring at nodes farfrom a sink node because of the waiting time for generatingdata at every intermediate node And what is more thetransmission delay is longer when data generation rate islow at nodes PEGASIS (Power-Efficient Gathering in SensorInformation Systems) [10] is one of the energy efficiency

International Journal of Distributed Sensor Networks 3

Sink120582n

120582n

120582nminus1

120582nminus11205822

1205822

1205823 1205821

1205821

nn nnminus1 n2 n1middot middot middot

middot middot middot

Figure 2 Full aggregation

Sink

1Δt 1Δt 1Δt 1Δt 1Δt

120582n 120582nminus1 1205822 1205821

nn nnminus1 n2 n1middot middot middot

middot middot middot

Figure 3 Partial aggregation

chain based data aggregation protocols that employs a greedyalgorithm There are two assumptions in this scheme oneis that all nodes are far from a sink node the other is thatnodes except the end nodes fuse the received data and theirown data and then aggregate them into one packet beforetransmitting them to another neighboring node The mainidea of PEGASIS is forming a chain among the sensor nodesso that each node will receive from and transmit to fused dataonly with a close neighborThe fused data are sent from nodeby node and all the nodes take turns to be the leader fortransmission to the sink nodeThe disadvantage of PEGASISis that transmission delay is too long for nodes that are at theend of a chain

23 Partial Aggregation In partial aggregation data alwayswait a period of time to be transmitted for collecting moredata at a node all collected data are aggregated into one orseveral representative data and then transmitted to an upperneighboring node The waiting time at nodes is adjustableand is always decided by corresponding applications [11 1319ndash21] Figure 3 illustrates a most simple class of partialaggregation in which data wait Δ119905 time for aggregation at anode before they are transmitted (1Δ119905 is data transmissionrate at a node)Whenwaiting timeΔ119905 is out a node aggregatesall generated data and received data into one type of represen-tative data and then sent it out In partial aggregation energyis traded off for improving transmission delay data accuracynetwork lifetime and so forth Since full aggregation leads tolong transmission delay and nonaggregation consumesmuchenergy partial aggregation can balance energy with otherperformances inWSN As the waiting time at nodes is alwaysdecided by applications one can set the waiting time veryshort when they want data with short transmission delay andotherwise the opposite

In in-network cascading timeout data aggregation [13]and ATS-DA (Adaptive Timeout Scheduling for Data Aggre-gation) [11] a sink initially broadcasts a request to all nodesEach node waits for a certain time period to receive datafrom their child nodes The timeout period of each node isset based on the position of the node in the data aggregationtree However these studies are not applicable when real timedata or short delay data are required Because all data at anode have to wait at least SHDavg time no matter whether

the data are important or not Moreover when the data arriv-ing rate is irregular at nodes the node energy consumptionis also irregular among nodes which result in unbalancednetwork In particular in large scaleWSN it is very difficult toachieve fairness for all nodes from the points of transmissiondelay and energy consumption because of data collision andcongestion as well as retransmission

3 Waterfalls Partial Aggregation

31 Sensor NetworkModel In this research we apply queuingtheory to analyze and model wireless sensor network As inmathematical analysis too complex network model makes ittoo sophisticated to get the analytical result and formulationtherefore we define the network model to be the most basicand simplest model of tandem sensor network however theresults can be extensible to other more complex topologiesThe structure and transmission principle of tandem networkare shown in Figure 4

In tandem networks all the nodes deployed in a flat areallocated omnidirectional antennas with the same transmis-sion range Data generated at the nodes are transmitted toa sink node in multihop manner The distance between twoneighboring nodes is the same and all the nodes are withinthe transmission range of their neighboring nodes

(i) 119899119894denotes the 119894th node from the sink

(ii) 119899 is a set of all nodes119873 is natural number set(iii) 119864

119894denotes the energy at node 119899

119894

(iv) 119871max denotes maximum network lifetime(v) 119863119899denotes the total transmission delay

(vi) 120583119894is waterfalls pushing rate at node 119899

119894

(vii) 119905119894is data transmission delay for each node

32 Definition of Waterfalls Partial Aggregation To shortenthe data transmission delay and to lengthen the networklifetime of the network as well as to suppress data congestionaround a sink node we propose a waterfalls partial aggrega-tion (WPA) In WPA data are transmitted to a neighboringnode in two conditions (a) if there are new local generateddata at a node or (b) after waiting a holding time Δ119905 at anode The inverse of the holding time Δ119905

119894at node 119899

119894we

call waterfalls pushing rate and it is denoted as 120583119894where 120583

119894

becomes smaller to nodes nearer a sink node it means 120583119894gt

120583119895 119894 gt 119895 as shown in Figure 5In otherwords data tends to be aggregated rarely at nodes

far from a sink node to suppress delay When there are newgenerated data at a node or the holding time at this nodeis over all data at this node are aggregated and wait forfurther transmission In more detail the holding time fornodes nearer a sink node is longer than the ones far from thesink so that it results in the decrease of the amount of datatransmission near a sink node Finally it can achieve an equalamount of data transmission at each node by controllingthe waterfalls pushing rate vectors The flow chart of dataaggregation and transmission is shown in Figure 6

4 International Journal of Distributed Sensor Networks

SinknN n2n3

n4 n1

middot middot middot

Figure 4 Tandem sensor network

Sink120583nminus1 12058311205832

120583n gt 120583nminus1 gt middot middot middot gt 1205832 gt 1205831

120582n 120582nminus1 1205822 1205821

nn nnminus1 n2 n1middot middot middot

middot middot middot

Figure 5 Waterfalls partial aggregation

Start

Aggregate all data

Are new generated

data

Is server idle

Arrival data

Is channel idle

Send data

Has the holding time

reached

Wait

Wait

No

No

Yes

No

No

Yes

Yes

Yes

(1) Arrival process to server

(2) Service process

(3) Data transmitting process

Figure 6 Flow chart of waterfalls partial aggregation

In the flow chart after data arriving from neighboringnode node 119899

119894starts to check if there are new generated data

at itself If yes node 119899119894aggregates all arrival data with the local

generated data and then waits for service process if no node119899119894checks the holding time if time is run up it aggregates

all arrival data and then goes to service process otherwiseit checks new generated dataThis process we call data arrivalprocess to server After aggregating data node 119899

119894checks if

the server is idle or not If there are no other data waitingfor transmitting we say the server is idle otherwise serveris busy data queue up in the server and wait for furthertransmitting When server is idle node checks if the channelis idle or notWhen channel is idle data transmission process

starts otherwise node 119899119894checks the channel until it turns to

idle Data are sent to a neighboring node if the channel is idle

33 Problem Definition In WPA the waterfalls setting ofholding time is for balancing energy consumption amongnodes as well as suppressing load As we mentioned inSection 2 that network lifetime is always decided by theshortest lifetime node in the network therefore to lengthenthe shortest node lifetime is to extend network lifetimeWhen tolerable maximum transmission delay is given bycorresponding application we set the holding time Δ119905

119894for

nodes in the network according to WPA algorithm Thereason that set a maximum tolerable transmission delay isbecause in a finite network there would have several sets ofwaterfalls pushing rates vectors corresponding to differenttransmission delay According to our definition waterfallspushing rate 120583

119894is inverse of holding time With larger

waterfalls pushing rate 120583119894 a node transmits more data and

results in much energy consumption Thus the problembecomes how to set the waterfalls pushing rates among nodesso that all nodes keep having the same energy consumption

In other words how long should the holding time ofarrival data for data aggregation at a node be to meet theconditions of balanced energy consumption among nodesand given transmission delay For a node if required trans-mission delay is given as119863

119899 then what our algorithm should

do is let

119863119899= 1199051+ 1199052+ sdot sdot sdot + 119905

119894+ sdot sdot sdot + 119905

119899 (119899 isin 119873) (1)

Here 119905119894is transmission delay for node 119899

119894and we deter-

mine the formulation of transmission delay for each node aswell as total transmission delay of the network in Section 34In WPA data have to wait for Δ119905 time at a node for gatheringmore data hence the transmission delay for a node ismainly decided by this Δ119905 On the other hand to balancethe energy consumption among nodes we first obtain totalenergy consumption 119864

119894for node 119899

119894which is composed of

energy of data reception data transmission and overhearingWe show the solution of energy consumption for node 119899

119894

and determine the formulations in Section 34 For node 119899119894

data transmission rate is decided by itself and data receivingrate is decided by its upper neighboring node 119899

119894+1and the

overhearing rate is decided by its lower neighboring node119899119894minus1

For node 119899119894 data transmission rate 1205821015840

119894is decided by data

generation rate 120582119894and waterfalls pushing rate 120583

119894 If we set

the maximum network lifetime as 119871max then we obtain (2)

International Journal of Distributed Sensor Networks 5

G Server

120583i 120582i

Queuerx Queuetx

120582998400998400i+1 120582998400i 120582998400998400i

Figure 7 Analytical model of waterfalls partial aggregation at node119899119894

which means 119871max is decided by nodes lifetime 119871119894 or 119895 when

two arbitrary nodes 119894 and 119895 are balanced on energy 119864119894= 119864119895

119871max = 119871 119894 or 119895 (119864119894 = 119864119895) (1 le 119894 119895 lt 119899) (2)

Therefore for achieving balanced energy among nodes wefirst determine the optimum set of waterfalls pushing rates 120583

119894

that correspond to given 119871max in the network

120583119894= argmax (119871 (120583

119894)) (3)

From the above analysis it is clear that we are trying to obtaina set of waterfalls pushing rates

120583119894| 120583119894lt 120583119894+1 120583119894isin |119873| 1 lt 119894 lt 119899 (4)

For obtaining optimum holding time for each node it isnecessary to know total delay of the network and energyconsumption for a node In the following sections wedetermine the total delay and node energy consumption

34 Analytical Model of WPA

341 Queuing Theory Analysis For determining the trans-mission delay we model the data arrival process and datatransmission process of WPA by queuing theory The analyt-ical model of node 119899

119894is shown in Figure 7

Before explaining the model we introduce QueuerxQueuetx G and Server Queuerx denotes the arrival dataqueue in which arrival data are waiting for data aggregationat node 119899

119894 Queuetx denotes the data queue after aggregation

in which aggregated data are waiting for transmission toa neighbor node G is assumed as a virtual gate betweenQueuerx and Queuetx Data transmitting process of data inQueuetx is accomplished via Server

In the analytical model 12058210158401015840119894+1

is arrival data rate fromupper node toQueuerx of node 119899119894 and is approximately abidedby Poisson distribution At node 119899

119894 event generation rate 120582

119894

is assumed to be Poisson distributionWaterfalls pushing rate120583119894is assumed to be exponential distribution In Queuerx all

arrival data and generated data are aggregated and then joinQueuetx via gate G 120582

1015840

119894is data arrival rate to Queuetx in which

data are waiting for further transmitting 12058210158401015840119894is the data rate

upon exiting from the Server Next we analyze the abovemodel start from data receiving at a node to data that aretransmitted to a neighboring node

342 Event Waiting Time In this subsection we decide thetime interval that datawait inQueuerx for aggregating at node

120582998400998400i+1120582998400998400i+1120582998400998400i+1120582998400998400i+1120582998400998400i+1

0 1 k minus 1 k

2120582i + 120583i

2120582i + 120583i 2120582i + 120583i2120582i + 120583i

middot middot middot middot middot middot

Figure 8 State transition of Queuerx

119899119894 we call this time interval as event waiting time and denote

it by 120591119890(119894) and it is counted from when the first data arrive

to Queuerx until the data are aggregated and join QueuetxTo determine 120591

119890(119894) we first determine the amount of data

waiting for event data in Queuerx at node 119899119894 The amount

of data is denoted as 119876rx(119894) and we describe state transitionrate diagram of Queuerx to determine it in Figure 8 Notethat at node 119899

119894 the arrival data rate 12058210158401015840

119894+1is approximately

Poisson distribution and 120582119894is event data generation rate at

node 119899119894and is approximately Poisson distribution In data

aggregation service time in queuing theory corresponds todata aggregating time which is time interval from last dataaggregation until next data aggregation The basic idea of theanalysis is that data waits in Queuerx for the duration accord-ing to the exponential distribution of average 12120582

119894 since

data aggregation rate is inverse of the mean aggregating timehence data aggregation rate is 2120582

119894 andwhen consideringwith

waterfalls pushing rate data aggregating rate is

120582agg = 2120582119894 + 120583119894 (5)

In the diagram the state variable is the number of datawaiting for an event Assume that a number of data arewaiting for an event data at node 119899

119894in Queuerx If we let 1198751198940

and 119875119894119896

be the probability when the number of data waitingfor an event is 0 and 119896 in Queuerx at node 119899

119894 at state 0

according to Figure 8 we obtain

119875119894012058210158401015840

119894+1=

infin

sum

119895=1

119875119894119895120582agg (6)

At state 119896 we obtain

11987511989411989612058210158401015840

119894+1= 119875119894119896minus1

12058210158401015840

119894+1minus 119875119894119896120582agg (7)

Hence we obtain

119875119894119896=

12058210158401015840

119894+1

12058210158401015840

119894+1+ 120582agg

119875119894119896minus1

119875119894119896minus1

=12058210158401015840

119894+1

12058210158401015840

119894+1+ 120582agg

119875119894119896minus2

1198751198941=

12058210158401015840

119894+1

12058210158401015840

119894+1+ 120582agg

1198751198940

(8)

6 International Journal of Distributed Sensor Networks

Bringing these equations into (6) we determine the relation-ship between 119875

119894119896and 119875

1198940that

119875119894119896= (

12058210158401015840

119894+1

12058210158401015840

119894+1+ 120582agg

)

119896

1198751198940 (9)

As we knowinfin

sum

119896=1

119875119894119896= 1 (10)

Let

120572 =12058210158401015840

119894+1

12058210158401015840

119894+1+ 120582agg

(11)

Then we determineinfin

sum

119896=1

1205721198961198751198940= 1 (12)

Asinfin

sum

119896=1

120572119896=

1

1 minus 120572 (13)

henceinfin

sum

119896=1

1205721198961198751198940=

1

1 minus 1205721198751198940= 1 (14)

Accordingly we obtain

1198751198940= 1 minus 120572 (15)

Substituting 120572 we determine

1198751198940=

2120582119894+ 120583119894

12058210158401015840

119894+1+ 120582agg

(16)

119875119894119896=

(12058210158401015840

119894+1)119896

+ 120582agg

(12058210158401015840

119894+1+ 120582agg)

119896+1 (17)

Now we determine the amount of data that are received inQueuerx(119894) as

119876rx (119894) =infin

sum

119896=0

119896119875119894119896 (18)

Taking (17) into the above equation we obtain

119876rx (119894) =infin

sum

119896=0

119896

(12058210158401015840

119894+1)119896

+ 120582agg

(12058210158401015840

119894+1+ 120582agg)

119896+1 (19)

We express the above equation as

119876rx (119894) =infin

sum

119896=0

119896(12058210158401015840

119894+1

12058210158401015840

119894+1+ 120582agg

)

119896+1120582agg

12058210158401015840

119894+1

(20)

0

Y

X

Figure 9 Property distribution of119883 119884

Let 120573 = 12058210158401015840

119894+1(12058210158401015840

119894+1+ 120582agg) and 120574 = 120582agg120582

10158401015840

119894+1 then we

simplify (20) as

119876rx (119894) =infin

sum

119896=0

119896120573119896+1

120574 =

infin

sum

119896=0

119896120573119896minus1

1205732120574 = 1205732120574

infin

sum

119896=0

119896120573119896minus1

= 1205732120574120597

120597120573

infin

sum

119896=0

120573119896= 1205732120574120597

120597120573

1

1 minus 120573=

1205732120574

(1 minus 120573)2

(21)

As 120573 = 12058210158401015840119894+1(12058210158401015840

119894+1+ 120582agg) and 120582agg = 2120582119894 + 120583119894 so we get that

1 minus 120573 =2120582119894+ 120583119894

12058210158401015840

119894+1+ 2120582119894+ 120583119894

(22)

Therefore we determine that

1205732120574

(1 minus 120573)2

=

(12058210158401015840

119894+1 (12058210158401015840

119894+1+ 2120582119894+ 120583119894))2

(2120582119894+ 120583119894) 12058210158401015840

119894+1

((2120582119894+ 120583119894) (12058210158401015840

119894+1+ 2120582119894+ 120583119894))2

(23)

By calculating the above equations we determine that

119876rx (119894) =12058210158401015840

119894+1

2120582119894+ 120583119894

(24)

According to Littlersquos formula we determine the event waitingtime as follows

120591119890 (119894) =

119876rx (119894)

2120582119894+ 120583119894

=12058210158401015840

119894+1

(2120582119894+ 120583119894)2 (25)

343 Arrival Process to Queue119905119909 From the analytical process

of Queuerx(119894) we find that arrival data rate 1205821015840119894to Queuerx(119894)

is decided by the waterfalls pushing rate 120583119894and event data

generation rate 120582119894at node 119899

119894 The event generation rate 120582

119894

and waterfalls pushing rate 120583119894abide by discrete distribution

and they are independent of one another To determine theformulation of 1205821015840

119894 we calculate the property distribution of

120582119894and 120583

119894 At first we define that 120582

119894and 120583

119894are independent

distribution 119883 and 119884 as shown in Figure 9 We define thevalues of 119883 as 1 2 119864

119883and the values of 119884 as 1 2 119864

119884

Here we prove that the property 119884 is bigger than119883

International Journal of Distributed Sensor Networks 7

We determine the property as follows 119875119883and 119875

119884denote

the properties when119883 = 119894 and 119884 = 119894

119875 [119884 gt 119883]

= 119875119883 (119883 = 1) 119875119884 (119884 = 2) + 119875119884 (119884 = 3) + sdot sdot sdot + 119875119884 (119884

= 119864119884)

+ 119875119883 (119883

= 2) +119875119884 (119884 = 3) + sdot sdot sdot + 119875119884 (119884 = 119864

119884)

+ 119875119883(119883

= 119864119883) + sdot sdot sdot + 119875

119884(119884 = 119864

119883+ 1)

=

119864119883

sum

119894=1

119875119883 (119883 = 119894)

119864119884

sum

119895=119894+1

119875119884(119884 = 119895)

(26)

When applied to continuous distribution we obtain

119875 [119884 gt 119883] = int

infin

0

119875119883 (119905) [int

infin

0

119875119884 (119911) 119889119911] 119889119905 (27)

Here if 119875119883(119905) and 119875

119884(119905) are exponential distribution then

119875119883 (119905) = 120582119883119890

minus120582119883119905

119875119884 (119905) = 120582119884119890

minus120582119884119905

119875 [119884 gt 119883] = int

infin

0

119875119883 (119905) [int

infin

0

120582119884119890minus120582119884119911119889119911] 119889119905

= int

infin

0

119875119883 (119905) [minus119890

minus120582119884119911]infin

119905119889119905

= int

infin

0

119875119883 (119905) (119890

minus120582119884119905) 119889119905

= int

infin

0

120582119883119890minus120582119883119905119890minus120582119884119905119889119905

= int

infin

0

120582119883119890minus(120582119883+120582119884)119905119889119905

=120582119883

120582119883+ 120582119884

[minus119890minus(120582119883+120582119884)119905]

infin

0=

120582119883

120582119883+ 120582119884

(28)

Accordingly we determine the data arrival rate of Queuetx

1205821015840

119894= 120582119894+

120582119894

120582119894+ 120583119894

120583119894 (29)

344 Service Process Since the data generation rate is Pois-son distribution and the waterfalls pushing rate abides byexponential distribution the data arrival rate to Queuetx(119894)approximates to Poisson distribution and each node has oneserver The ACK packet transmission time is not consideredData aggregating time is very short and negligibleThereforethe service time is one-hop data transmission time 120591 In our

work data transmission rate is 119881119888and generated data size is

119878119894 Hence the service time for generated data is

120591 =119878119894

119881119888

(30)

Since 119881119888and 119878

119894are constant hence the service time

for data is fixed and constant According to the aboveanalysis we can determine that the queuing system on serverapproximates to 1198721198631 model As data transmission timeis determined and server utilization ratio is 120578 = 120582

1015840

119894times 120591tx(119894)

hence the average data transmission time 120591tx(119894) is

120591tx (119894) =120591

1 minus 1205821015840

119894120591 (31)

We figure up the probability density function of the servertime by means of Laplace transform in queuing theory andevaluate server waiting time 120591

119904(119894) that data wait for further

transmitting in server

120591119904 (119894) =

1205821015840

119894(120591tx (119894))

2

2 (1 minus 1205821015840

119894120591tx (119894))

(32)

345 Channel Waiting Time In general queuing systemcustomers can be served if there are no other customerswaiting in front of them However when we apply queuingtheory to model wireless communication it is necessary toconsider the impact of wireless channel caused by its proper-ties In wireless sensor networks because of the overhearingof omnidirectional antenna node has to wait a period oftime if its upper or lower neighbors are transmitting dataHere we call this time as channel waiting time 120591

119888(119894) and it

is determined as follows

120591119888 (119894) = 2 times 120578 times 120591tx (119894) = 2120582

1015840

119894times 1205912

tx (119894) (33)

346 Total Delay Total delay119863119899inWPA is the time interval

from when an event data is generated at a node until thedata are received by a sink node in 119873 hops network Theformulation is as follows

119863119899=

119873

sum

119894=1

(120591119890 (119894) + 120591119904 (119894) + 120591119888 (119894) + 120591tx (119894)) (34)

347 Energy Consumption The energy consumption 119864119894for

node 119899119894is the sum of transmission energy consumption

reception energy consumption and overhearing energy con-sumption Now we determine the three kinds of energyconsumption in the following paragraphs Note that 119875tx and119875rx are energy consumption of transmitting or receiving foreach data packet and 119864tx(119894) 119864rx(119894) and 119864oh(119894) are energyconsumption of data transmitting data receiving and over-hearing of node 119899

119894in 119863119899time As we are trying to obtain

the balanced energy consumption among nodes hence wecalculate energy consumption per 119863

119899time for each node

Therefore we get the data transmission energy consumptionaccording to Littlersquos formula as follows

119864tx (119894) = 1205821015840

119894119863119899119875tx (35)

8 International Journal of Distributed Sensor Networks

The reception energy consumption is

119864rx (119894) = 12058210158401015840

119894+1119863119899119875rx (36)

Here 12058210158401015840119894+1

is data arrival rate from upper neighboring node119899119894+1

as the service time is one data packet transmission timehence the data rate getting out from server is same with datatransmission rate which means

12058210158401015840

119894+1= 1205821015840

119894+1 (37)

The overhearing energy consumption is

119864oh (119894) = 1205821015840

119894minus1119863119899119875rx (38)

Here 1205821015840119894minus1

is data transmission rate of node 119899119894minus1

Hence wedetermine total energy consumption119864

119894for a node in time 119863

119899

as

119864119894= 119864tx (119894) + 119864rx (119894) + 119864oh (119894) (39)

4 Optimum Network Lifetime

41 Mathematical Solution According to the analysis in Sec-tion 32 the longest network lifetime is decided by the totaltransmission delay and waterfalls pushing rate vectors Herewe illustrate the relationship between transmission delayenergy consumption and network lifetime so that at lastwe determine the formulations of a set of waterfalls pushingrates Accordingly energy consumption can be written asfollows

119864119894= (1205821015840

119894119875tx + 120582

10158401015840

119894+1119875rx + 120582

1015840

119894minus1119875rx)

= 1205821015840

119894119875tx + (120582

10158401015840

119894+1+ 1205821015840

119894minus1) 119875rx

(40)

To balance energy among nodes we have

1198641= 1198642= sdot sdot sdot = 119864

119894= sdot sdot sdot = 119864

119899 (41)

As the service time is one-hop data transmitting time hence

12058210158401015840

119894= 1205821015840

119894 (42)

Here 119864119894is the energy consumption for node in time 119863

119899

Hence we assume the energy consumption for each unit oftime is 119864 We assume that 120582

119894= 120582 and the network in a finite

network Hence there are no arrival data to the last node inthe network so that it does not need a waterfalls pushing rateand moreover there is no overhearing for the first node thatcounted from the sink node Therefore for node 119899

119899 we have

120583119899= 0

1205821015840

119899= 120582

119864rx (119899) = 0

119864oh (1) = 0

(43)

Accordingly we obtain equations as follows

(120582 +1205821205831

120582 + 1205831

)119875tx + (120582 +1205821205832

120582 + 1205832

)119875rx = 119864

(120582 +1205821205832

120582 + 1205832

)119875tx + (2120582 +1205821205831

120582 + 1205831

+1205821205833

120582 + 1205833

)119875rx = 119864

(120582 +120582120583119894

120582 + 120583119894

)119875tx

+ (2120582 +120582120583119894minus1

120582 + 120583119894minus1

+120582120583119894+1

120582 + 120583119894+1

)119875rx = 119864

120582119875tx + (120582 +120582120583119899minus1

120582 + 120583119899minus1

)119875rx = 119864

(44)

By calculating the above equations we have

120583119899minus1

=120582119875rx (120582 + 120582120583119899minus2 (120582 + 120583119899minus2))

(120582120583119899minus2

(120582 + 120583119899minus2

)) 119875rx minus 120582119875tx(45)

120583119899minus2

=120582119875rx (120582120583119899minus3 (120582 + 120583119899minus3)) + 120582 (119875rx minus 119875tx) (120582120583119899minus1 (120582 + 120583119899minus1))

(119875rx minus 119875tx) (120582 minus 120582120583119899minus1 (120582 + 120583119899minus1)) minus 119875rx (120582120583119899minus3 (120582 + 120583119899minus3))

(46)

120583119894=

1205822120583119894+2

(119875rx minus 119875tx) (120582 + 120583119894+2) minus 1205822120583119894+1

(119875rx minus 119875tx) (120582 + 120583119894+1) minus 1205822120583119894+3119875rx (120582 + 120583119894+3)

120582120583119894+3119875rx (120582 + 120583119894+3) minus 120582120583119894+2 (119875rx minus 119875tx) (120582 + 120583119894+2) + 120582120583119894+1 (119875rx minus 119875tx) (120582 + 120583119894+1) minus 120582119875rx

1205831=

12058221205832(119875rx minus 119875tx) (120582 + 1205832) minus 120582

21205833119875rx (120582 + 1205833) minus 120582

2119875rx

1205821205832(119875tx minus 119875rx) (120582 + 1205832) minus 1205821205833119875rx (120582 + 1205833) + 2120582119875rx minus 120582119875tx

(47)

International Journal of Distributed Sensor Networks 9

Input 120582 120583119899minus1

120578119863appOutput [120583

1 120583119899minus1

]120583max = 2120582

120583119899minus1

= 120583max[1205831 120583119899minus1] = 119891 (120583

119899minus119894)

119863heu = 119892 (1205831 120583119899minus1)

While |119863app minus 119863heu| gt 120578 do120583119899minus2

= 120583119899minus1

minus Δ

[1205831 120583119899minus1] = 119891(120583

119899minus119894)

119863heu = 119892(1205831 120583119899minus1)

EndReturn [120583

1 120583119899minus1]

Algorithm 1 Heuristics algorithm

From the above equations we obtained the relationshipamong waterfalls pushing rate Equation (45) shows that if120583119899minus1

is given then 120583119899minus2

can be determined In (46) if 120583119899minus1

and120583119899minus2

are given then 120583119899minus3

can be determined In this way all 120583119894

can be determined if 120583119899minus1

is known To determine the valuesof 120583119894 we designed a heuristics algorithm as follows In light of

real applications ofWSNwhere the threshold of transmissiondelay is always given in the heuristics algorithm we assumethat the transmission delay is predefined by applicationHence we set the value of120583

119899minus1and then determine the others

However the value of 120583119899minus1

being too large or too little makesthe algorithm too complex to process Therefore it is criticalto determine the maximum value of 120583

119899minus1

42 Heuristics Algorithm In a given application we assumethat data generation rates at all nodes are known and are 120582Thus at node 119899

119899minus1 if the arrival data from node 119899

119899just does

not keep pace with the generated data then the node needswaterfalls pushing rate to transmit the arrival data Howeverthe processing period of waterfalls pushing rate is decided bynode 119899

119899minus1 hence there is again the case that the arrival data

just does not catch up the pace of the waterfalls pushing rate120583119899minus1

Therefore we determine simply that for the arrival datathe average catching up rate of waterfalls pushing rate at node119899119899minus1

is 1205822 Accordingly for transmitting all arrival data themaximum value of 120583

119899minus1at node 119899

119899minus1is

120583max = 2120582 (48)

Hence a heuristics algorithm is described as shown inAlgorithm 1

5 Evaluation

51 Validation of the Queuing Theory Model We implementWPA in the original network simulator written in C++language Analytic results in the previous section are shownas well as simulated result The parameters are shown inTable 1

In the simulation data occurs at each node randomly andindependently Buffer size of each node is infinite Althoughanalytic model assumes that transmission error is negligible

Table 1 Simulation parameters of WPA

Node distance 10 (m)Transmission range 11 (m)Transmission rate 250 (kbps)Data size 4096 (bit)Current consumption for transmission 174 (mA)Current consumption for reception 197 (mA)MAC CSMACARouting DSR

001

01

1

10

100

1000

01 1 10Data generation rate (datasec)

WPA-analysisWPA-simulation

Tota

l del

ay (120583

sec)

Figure 10 Total delay of WPA

transmission errors and retransmission may occur in thesimulation We set data generation rate 120582 = 5 and waterfallspushing rate as 120583

119894= 1 2 3 4 5 for 5-hop tandem network

From Figures 10 and 11 we find that our analytical modelmeets with simulation results so that we basically confirm thecorrectness of queuing theory model of WPA

52 Evaluation of OptimumNetwork Lifetime In this sectionwe evaluate the effectiveness of the proposedwaterfalls partialaggregation Here we consider network environment asnature monitoring network so that we obtain the maximumnetwork lifetime which is the same with network lifetime offull aggregation We set 119864 in (44) which is equal to 119864ful-largestwhere 119864ful-largest denotes the energy consumption of the nodethat consumes most energy in the network The optimalresults can be achieved by numerical packets however themathematical calculation results are too complex For moreunderstanding of our algorithm we obtain the pushing ratevectors applying Excel when the event generation rate is120582 = 1for 5-hop tandem network Nodes are counted from the sinknode which means that the nearest node with the sink isnode 119899

1and its waterfalls pushing rate is 120583

1 The optimum

waterfalls pushing rates are shown in Table 2 when WPAhas the same network lifetime with full aggregation Theanalytical model of nonaggregation and full aggregation isrealized by queuing theory however for the brevity of thepaper we omitted the derivation process

10 International Journal of Distributed Sensor Networks

0

1000

2000

3000

4000

5000

6000

7000

1 2 3 4 5

Ener

gy co

nsum

ptio

n (m

A)

Node ID

WPA-analysisWPA-simulation

Figure 11 Node energy consumption of WPA

0

05

1

15

2

25

3

35

4

1 2 3 4 5Node ID

NonaggregationPartial aggregationFull aggregation

Ener

gy co

nsum

ptio

n (m

A)

Figure 12 Lifetime of nodes

Table 2 Waterfalls pushing rate for optimum network lifetime

Waterfalls pushing rate Values1205831

0021205832

0041205833

0081205834

011205835

05

Figure 12 shows the lifetime of all nodes in the networkwhere we set the waterfalls pushing rate of nodes as the samewith Table 2 From the figure we find that all nodes almosthave the same lifetime in WPA In this case we can saythat the node energy is utilized optimally in WPA In fullaggregation the second and the third node consumed mostenergy and the other nodes have less energy consumptionwhich results in the suboptimum utilization of energy Thenonaggregation has very large energy consumption at thesecond node which results in the short lifetime of the entirenetwork We find from the figure that nonaggregation and

001

01

1

10

100

01 1 10 100

NonaggregationPartial aggregationFull aggregation

Data generation rate (datasec)

Tota

l del

ay (s

ec)

Figure 13 Transmission delay of network

full aggregation have the largest energy consumption attheir second node The reason is that the first node has noadjacent upper node which means that it does not consumeoverhearing energy compared with others

Figure 13 shows the corresponding total delay of thenetwork where we set the same waterfalls pushing rates withTable 1 From the figure we find that when it has the samenetwork lifetime with full aggregation WPA can shorten thetransmission delay by considerable amount When data gen-eration rate becomes larger (larger than 6 types of datasec)transmission delay of nonaggregation gets longer rapidly dueto data congestion at nodes However in WPA the rapidincrease of transmission delay arises later (at data generationrate of 30 types of datasec) than nonaggregation and fullaggregation on data generation line which indicates thatWPA can relieve data congestion at nodes This is becausewith waterfalls pushing rates proposed WPA can adjust thenumber of data transmissions Moreover we find from thefigure that with the given waterfalls pushing rates there isminimum transmission delay and we can determine it inWPA

53 Summary of Evaluation Firstly from the presentationof Section 51 we conclude that our analytical model ofWPA meets with simulation results the diversity of energyconsumption is that we did not consider retransmissionin analytical model and considered it in simulation Sec-ondly in Section 52 compared to nonaggregation and fullaggregation proposed WPA has the superiorities as followsnonaggregation consumes much energy and data congestionoccurs easily when data generation rate is larger In WPAafter data are sent to a neighboring node the data haveto be aggregated together or aggregated with local data forfurther transmitting hence WPA is more energy efficientthan nonaggregationWhen data generation rate is larger at anode data congestion occurs at nonaggregation which leadsto long transmission delay however proposed WPA canrelieve data congestion via reducing the number of data trans-missions by aggregating received data at nodes On the other

International Journal of Distributed Sensor Networks 11

hand full aggregation can save much energy consumptionand suffer long transmission delay Compared to full aggre-gation proposed WPA has similar energy consumption andis superior in balancing energy consumption among nodesMoreover with waterfalls pushing rates WPA can controldata waiting time more reasonably and efficiently than fullaggregation at nodes so that transmission delay of WPA issuperior to full aggregation

6 Conclusion

In this paper we proposed a waterfalls partial aggregationthat can achieve optimum network lifetime in WSN viaapplying data aggregation At first we modeled WPA inqueuing theory and then determined transmission delay ofthe network and energy consumption of a node Then wecontinued to model balanced energy consumption amongnodes The evaluation parts show the correctness of our ana-lytical model and demonstrate the superiorities of proposedWPA

Competing Interests

The authors declare no competing financial interests

Acknowledgments

Thiswork is supported in part by theNatural Science Founda-tion of Inner Mongolia Autonomous Region of China underGrant no 2014BS0603 and no 2014BS0601 and the Programof Higher-Level Talents of Inner Mongolia University underGrant no 30105-135134

References

[1] C-C Shen C Srisathapornphat and C Jaikaeo ldquoSensor infor-mation networking architecture and applicationsrdquo IEEE Per-sonal Communications vol 8 no 4 pp 52ndash59 2001

[2] S D Glaser ldquoSome real-world applications of wireless sen-sor nodesrdquo in Smart Structures and Materials 2004 Sensorsand Smart Structures Technologies for Civil Mechanical andAerospace Systems vol 5391 of Proceedings of SPIE pp 344ndash355San Diego Calif USA March 2004

[3] I F Akyildiz W Su Y Sankarasubramaniam and E CayircildquoWireless sensor networks a surveyrdquo Computer Networks vol38 no 4 pp 393ndash422 2002

[4] W Ye J Heidemann and D Estrin ldquoSensor-MAC (S-MAC)medium access control for wireless sensor networksrdquo in Pro-ceedings of the 21st International Annual Joint Conference of theIEEEComputer and Communications Societies (INFOCOM rsquo02)New York NY USA June 2002

[5] L J G Villalba A L S Orozco A T Cabrera andC J B AbbasldquoRouting protocols in wireless sensor networksrdquo Sensors vol 9no 11 pp 8399ndash8421 2009

[6] P Santi ldquoTopology control in wireless ad hoc and sensornetworksrdquo ACM Computing Surveys vol 37 no 2 pp 164ndash1942005

[7] R Rajagopalan and P K Varshney ldquoData-aggregation tech-niques in sensor networks a surveyrdquo IEEE CommunicationsSurveys and Tutorials vol 8 no 4 pp 48ndash63 2006

[8] W-S Jung K-W Lim Y-B Ko and S-J Park ldquoA hybridapproach for clustering-based data aggregation in wireless sen-sor networksrdquo in Proceedings of the 3rd International ConferenceonDigital Society (ICDS rsquo09) pp 112ndash117 IEEECancunMexicoFebruary 2009

[9] M Kumar S Verma and P S Pratap ldquoClustering approach todata aggregation in wireless sensor networksrdquo in Proceedingsof the 16th IEEE International Conference on Networks pp 1ndash6IEEE New Delhi India December 2008

[10] S Lindsey C Raghavendra and K M Sivalingam ldquoDatagathering algorithms in sensor networks using energy metricsrdquoIEEE Transactions on Parallel and Distributed Systems vol 13no 9 pp 924ndash935 2002

[11] JW Baek Y J Nam andD-W Seo ldquoATS-DA adaptive timeoutscheduling for data aggregation in wireless sensor networksrdquoin Proceedings of the International Conference on InformationNetworking pp 375ndash384 Estoril Portugal 2008

[12] H Chen H Mineno and T Mizuno ldquoAdaptive data aggrega-tion scheme in clustered wireless sensor networksrdquo ComputerCommunications vol 31 no 15 pp 3579ndash3585 2008

[13] I Solis and K Obraczka ldquoIn-network aggregation trade-offsfor data collection in wireless sensor networksrdquo InternationalJournal of Sensor Networks vol 1 no 3-4 pp 200ndash212 2006

[14] F Mirian and M Sabaei ldquoA delay and accuracy sensitive dataaggregation structure in wireless sensor networksrdquo in Proceed-ings of the International Conference on InformationManagementand Engineering (ICIME rsquo09) pp 231ndash235 IEEE Kuala LumpurMalaysia April 2009

[15] P Sun X Zhang Z Dong et al ldquoA novel energy efficientwireless sensor MAC protocolrdquo in Proceedings of the 4th Inter-national Conference on Networked Computing and AdvancedInformation Management pp 68ndash72 Gyeongju South KoreaSeptember 2008

[16] N Li J C Hou and L Sha ldquoDesign and analysis of an mst-based topology control algorithmrdquo in Proceedings of the IEEESocieties 22nd Annual Joint Conference of the IEEE Computerand Communications (INFOCOM rsquo03) vol 3 pp 1702ndash1712 SanFrancisco Calif USA April 2003

[17] B Yin H Shi and Y Shang ldquoA two-level topology controlstrategy for energy efficiency in wireless sensor networksrdquoInternational Journal of Wireless and Mobile Computing vol 4no 1 pp 41ndash49 2010

[18] A Sharif V M Potdar and A J D Rathnayaka ldquoLCARTlightweight congestion aware reliable transport protocol forWSN targeting hetero-geneous trafficrdquo in Proceedings of the17th International Conference on Neural Information Processing(ICONIP rsquo10) Sydney Australia November 2010

[19] B Yu J Li and Y Li ldquoDistributed data aggregation schedulingin wireless sensor networksrdquo in Proceedings of the 28th Confer-ence on Computer Communications (IEEE INFOCOM rsquo09) pp2159ndash2167 Rio de Janeiro Brazil April 2009

[20] M X Cheng and L Yin ldquoEnergy-efficient data gatheringalgorithm in sensor networks with partial aggregationrdquo Inter-national Journal of Sensor Networks vol 4 no 1-2 pp 48ndash542008

[21] A A Allen Probability Statistics and Queueing Theory withComputer Science Applications Gulf Professional Publishing1990

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Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

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Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

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Navigation and Observation

International Journal of

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DistributedSensor Networks

International Journal of

International Journal of Distributed Sensor Networks 3

Sink120582n

120582n

120582nminus1

120582nminus11205822

1205822

1205823 1205821

1205821

nn nnminus1 n2 n1middot middot middot

middot middot middot

Figure 2 Full aggregation

Sink

1Δt 1Δt 1Δt 1Δt 1Δt

120582n 120582nminus1 1205822 1205821

nn nnminus1 n2 n1middot middot middot

middot middot middot

Figure 3 Partial aggregation

chain based data aggregation protocols that employs a greedyalgorithm There are two assumptions in this scheme oneis that all nodes are far from a sink node the other is thatnodes except the end nodes fuse the received data and theirown data and then aggregate them into one packet beforetransmitting them to another neighboring node The mainidea of PEGASIS is forming a chain among the sensor nodesso that each node will receive from and transmit to fused dataonly with a close neighborThe fused data are sent from nodeby node and all the nodes take turns to be the leader fortransmission to the sink nodeThe disadvantage of PEGASISis that transmission delay is too long for nodes that are at theend of a chain

23 Partial Aggregation In partial aggregation data alwayswait a period of time to be transmitted for collecting moredata at a node all collected data are aggregated into one orseveral representative data and then transmitted to an upperneighboring node The waiting time at nodes is adjustableand is always decided by corresponding applications [11 1319ndash21] Figure 3 illustrates a most simple class of partialaggregation in which data wait Δ119905 time for aggregation at anode before they are transmitted (1Δ119905 is data transmissionrate at a node)Whenwaiting timeΔ119905 is out a node aggregatesall generated data and received data into one type of represen-tative data and then sent it out In partial aggregation energyis traded off for improving transmission delay data accuracynetwork lifetime and so forth Since full aggregation leads tolong transmission delay and nonaggregation consumesmuchenergy partial aggregation can balance energy with otherperformances inWSN As the waiting time at nodes is alwaysdecided by applications one can set the waiting time veryshort when they want data with short transmission delay andotherwise the opposite

In in-network cascading timeout data aggregation [13]and ATS-DA (Adaptive Timeout Scheduling for Data Aggre-gation) [11] a sink initially broadcasts a request to all nodesEach node waits for a certain time period to receive datafrom their child nodes The timeout period of each node isset based on the position of the node in the data aggregationtree However these studies are not applicable when real timedata or short delay data are required Because all data at anode have to wait at least SHDavg time no matter whether

the data are important or not Moreover when the data arriv-ing rate is irregular at nodes the node energy consumptionis also irregular among nodes which result in unbalancednetwork In particular in large scaleWSN it is very difficult toachieve fairness for all nodes from the points of transmissiondelay and energy consumption because of data collision andcongestion as well as retransmission

3 Waterfalls Partial Aggregation

31 Sensor NetworkModel In this research we apply queuingtheory to analyze and model wireless sensor network As inmathematical analysis too complex network model makes ittoo sophisticated to get the analytical result and formulationtherefore we define the network model to be the most basicand simplest model of tandem sensor network however theresults can be extensible to other more complex topologiesThe structure and transmission principle of tandem networkare shown in Figure 4

In tandem networks all the nodes deployed in a flat areallocated omnidirectional antennas with the same transmis-sion range Data generated at the nodes are transmitted toa sink node in multihop manner The distance between twoneighboring nodes is the same and all the nodes are withinthe transmission range of their neighboring nodes

(i) 119899119894denotes the 119894th node from the sink

(ii) 119899 is a set of all nodes119873 is natural number set(iii) 119864

119894denotes the energy at node 119899

119894

(iv) 119871max denotes maximum network lifetime(v) 119863119899denotes the total transmission delay

(vi) 120583119894is waterfalls pushing rate at node 119899

119894

(vii) 119905119894is data transmission delay for each node

32 Definition of Waterfalls Partial Aggregation To shortenthe data transmission delay and to lengthen the networklifetime of the network as well as to suppress data congestionaround a sink node we propose a waterfalls partial aggrega-tion (WPA) In WPA data are transmitted to a neighboringnode in two conditions (a) if there are new local generateddata at a node or (b) after waiting a holding time Δ119905 at anode The inverse of the holding time Δ119905

119894at node 119899

119894we

call waterfalls pushing rate and it is denoted as 120583119894where 120583

119894

becomes smaller to nodes nearer a sink node it means 120583119894gt

120583119895 119894 gt 119895 as shown in Figure 5In otherwords data tends to be aggregated rarely at nodes

far from a sink node to suppress delay When there are newgenerated data at a node or the holding time at this nodeis over all data at this node are aggregated and wait forfurther transmission In more detail the holding time fornodes nearer a sink node is longer than the ones far from thesink so that it results in the decrease of the amount of datatransmission near a sink node Finally it can achieve an equalamount of data transmission at each node by controllingthe waterfalls pushing rate vectors The flow chart of dataaggregation and transmission is shown in Figure 6

4 International Journal of Distributed Sensor Networks

SinknN n2n3

n4 n1

middot middot middot

Figure 4 Tandem sensor network

Sink120583nminus1 12058311205832

120583n gt 120583nminus1 gt middot middot middot gt 1205832 gt 1205831

120582n 120582nminus1 1205822 1205821

nn nnminus1 n2 n1middot middot middot

middot middot middot

Figure 5 Waterfalls partial aggregation

Start

Aggregate all data

Are new generated

data

Is server idle

Arrival data

Is channel idle

Send data

Has the holding time

reached

Wait

Wait

No

No

Yes

No

No

Yes

Yes

Yes

(1) Arrival process to server

(2) Service process

(3) Data transmitting process

Figure 6 Flow chart of waterfalls partial aggregation

In the flow chart after data arriving from neighboringnode node 119899

119894starts to check if there are new generated data

at itself If yes node 119899119894aggregates all arrival data with the local

generated data and then waits for service process if no node119899119894checks the holding time if time is run up it aggregates

all arrival data and then goes to service process otherwiseit checks new generated dataThis process we call data arrivalprocess to server After aggregating data node 119899

119894checks if

the server is idle or not If there are no other data waitingfor transmitting we say the server is idle otherwise serveris busy data queue up in the server and wait for furthertransmitting When server is idle node checks if the channelis idle or notWhen channel is idle data transmission process

starts otherwise node 119899119894checks the channel until it turns to

idle Data are sent to a neighboring node if the channel is idle

33 Problem Definition In WPA the waterfalls setting ofholding time is for balancing energy consumption amongnodes as well as suppressing load As we mentioned inSection 2 that network lifetime is always decided by theshortest lifetime node in the network therefore to lengthenthe shortest node lifetime is to extend network lifetimeWhen tolerable maximum transmission delay is given bycorresponding application we set the holding time Δ119905

119894for

nodes in the network according to WPA algorithm Thereason that set a maximum tolerable transmission delay isbecause in a finite network there would have several sets ofwaterfalls pushing rates vectors corresponding to differenttransmission delay According to our definition waterfallspushing rate 120583

119894is inverse of holding time With larger

waterfalls pushing rate 120583119894 a node transmits more data and

results in much energy consumption Thus the problembecomes how to set the waterfalls pushing rates among nodesso that all nodes keep having the same energy consumption

In other words how long should the holding time ofarrival data for data aggregation at a node be to meet theconditions of balanced energy consumption among nodesand given transmission delay For a node if required trans-mission delay is given as119863

119899 then what our algorithm should

do is let

119863119899= 1199051+ 1199052+ sdot sdot sdot + 119905

119894+ sdot sdot sdot + 119905

119899 (119899 isin 119873) (1)

Here 119905119894is transmission delay for node 119899

119894and we deter-

mine the formulation of transmission delay for each node aswell as total transmission delay of the network in Section 34In WPA data have to wait for Δ119905 time at a node for gatheringmore data hence the transmission delay for a node ismainly decided by this Δ119905 On the other hand to balancethe energy consumption among nodes we first obtain totalenergy consumption 119864

119894for node 119899

119894which is composed of

energy of data reception data transmission and overhearingWe show the solution of energy consumption for node 119899

119894

and determine the formulations in Section 34 For node 119899119894

data transmission rate is decided by itself and data receivingrate is decided by its upper neighboring node 119899

119894+1and the

overhearing rate is decided by its lower neighboring node119899119894minus1

For node 119899119894 data transmission rate 1205821015840

119894is decided by data

generation rate 120582119894and waterfalls pushing rate 120583

119894 If we set

the maximum network lifetime as 119871max then we obtain (2)

International Journal of Distributed Sensor Networks 5

G Server

120583i 120582i

Queuerx Queuetx

120582998400998400i+1 120582998400i 120582998400998400i

Figure 7 Analytical model of waterfalls partial aggregation at node119899119894

which means 119871max is decided by nodes lifetime 119871119894 or 119895 when

two arbitrary nodes 119894 and 119895 are balanced on energy 119864119894= 119864119895

119871max = 119871 119894 or 119895 (119864119894 = 119864119895) (1 le 119894 119895 lt 119899) (2)

Therefore for achieving balanced energy among nodes wefirst determine the optimum set of waterfalls pushing rates 120583

119894

that correspond to given 119871max in the network

120583119894= argmax (119871 (120583

119894)) (3)

From the above analysis it is clear that we are trying to obtaina set of waterfalls pushing rates

120583119894| 120583119894lt 120583119894+1 120583119894isin |119873| 1 lt 119894 lt 119899 (4)

For obtaining optimum holding time for each node it isnecessary to know total delay of the network and energyconsumption for a node In the following sections wedetermine the total delay and node energy consumption

34 Analytical Model of WPA

341 Queuing Theory Analysis For determining the trans-mission delay we model the data arrival process and datatransmission process of WPA by queuing theory The analyt-ical model of node 119899

119894is shown in Figure 7

Before explaining the model we introduce QueuerxQueuetx G and Server Queuerx denotes the arrival dataqueue in which arrival data are waiting for data aggregationat node 119899

119894 Queuetx denotes the data queue after aggregation

in which aggregated data are waiting for transmission toa neighbor node G is assumed as a virtual gate betweenQueuerx and Queuetx Data transmitting process of data inQueuetx is accomplished via Server

In the analytical model 12058210158401015840119894+1

is arrival data rate fromupper node toQueuerx of node 119899119894 and is approximately abidedby Poisson distribution At node 119899

119894 event generation rate 120582

119894

is assumed to be Poisson distributionWaterfalls pushing rate120583119894is assumed to be exponential distribution In Queuerx all

arrival data and generated data are aggregated and then joinQueuetx via gate G 120582

1015840

119894is data arrival rate to Queuetx in which

data are waiting for further transmitting 12058210158401015840119894is the data rate

upon exiting from the Server Next we analyze the abovemodel start from data receiving at a node to data that aretransmitted to a neighboring node

342 Event Waiting Time In this subsection we decide thetime interval that datawait inQueuerx for aggregating at node

120582998400998400i+1120582998400998400i+1120582998400998400i+1120582998400998400i+1120582998400998400i+1

0 1 k minus 1 k

2120582i + 120583i

2120582i + 120583i 2120582i + 120583i2120582i + 120583i

middot middot middot middot middot middot

Figure 8 State transition of Queuerx

119899119894 we call this time interval as event waiting time and denote

it by 120591119890(119894) and it is counted from when the first data arrive

to Queuerx until the data are aggregated and join QueuetxTo determine 120591

119890(119894) we first determine the amount of data

waiting for event data in Queuerx at node 119899119894 The amount

of data is denoted as 119876rx(119894) and we describe state transitionrate diagram of Queuerx to determine it in Figure 8 Notethat at node 119899

119894 the arrival data rate 12058210158401015840

119894+1is approximately

Poisson distribution and 120582119894is event data generation rate at

node 119899119894and is approximately Poisson distribution In data

aggregation service time in queuing theory corresponds todata aggregating time which is time interval from last dataaggregation until next data aggregation The basic idea of theanalysis is that data waits in Queuerx for the duration accord-ing to the exponential distribution of average 12120582

119894 since

data aggregation rate is inverse of the mean aggregating timehence data aggregation rate is 2120582

119894 andwhen consideringwith

waterfalls pushing rate data aggregating rate is

120582agg = 2120582119894 + 120583119894 (5)

In the diagram the state variable is the number of datawaiting for an event Assume that a number of data arewaiting for an event data at node 119899

119894in Queuerx If we let 1198751198940

and 119875119894119896

be the probability when the number of data waitingfor an event is 0 and 119896 in Queuerx at node 119899

119894 at state 0

according to Figure 8 we obtain

119875119894012058210158401015840

119894+1=

infin

sum

119895=1

119875119894119895120582agg (6)

At state 119896 we obtain

11987511989411989612058210158401015840

119894+1= 119875119894119896minus1

12058210158401015840

119894+1minus 119875119894119896120582agg (7)

Hence we obtain

119875119894119896=

12058210158401015840

119894+1

12058210158401015840

119894+1+ 120582agg

119875119894119896minus1

119875119894119896minus1

=12058210158401015840

119894+1

12058210158401015840

119894+1+ 120582agg

119875119894119896minus2

1198751198941=

12058210158401015840

119894+1

12058210158401015840

119894+1+ 120582agg

1198751198940

(8)

6 International Journal of Distributed Sensor Networks

Bringing these equations into (6) we determine the relation-ship between 119875

119894119896and 119875

1198940that

119875119894119896= (

12058210158401015840

119894+1

12058210158401015840

119894+1+ 120582agg

)

119896

1198751198940 (9)

As we knowinfin

sum

119896=1

119875119894119896= 1 (10)

Let

120572 =12058210158401015840

119894+1

12058210158401015840

119894+1+ 120582agg

(11)

Then we determineinfin

sum

119896=1

1205721198961198751198940= 1 (12)

Asinfin

sum

119896=1

120572119896=

1

1 minus 120572 (13)

henceinfin

sum

119896=1

1205721198961198751198940=

1

1 minus 1205721198751198940= 1 (14)

Accordingly we obtain

1198751198940= 1 minus 120572 (15)

Substituting 120572 we determine

1198751198940=

2120582119894+ 120583119894

12058210158401015840

119894+1+ 120582agg

(16)

119875119894119896=

(12058210158401015840

119894+1)119896

+ 120582agg

(12058210158401015840

119894+1+ 120582agg)

119896+1 (17)

Now we determine the amount of data that are received inQueuerx(119894) as

119876rx (119894) =infin

sum

119896=0

119896119875119894119896 (18)

Taking (17) into the above equation we obtain

119876rx (119894) =infin

sum

119896=0

119896

(12058210158401015840

119894+1)119896

+ 120582agg

(12058210158401015840

119894+1+ 120582agg)

119896+1 (19)

We express the above equation as

119876rx (119894) =infin

sum

119896=0

119896(12058210158401015840

119894+1

12058210158401015840

119894+1+ 120582agg

)

119896+1120582agg

12058210158401015840

119894+1

(20)

0

Y

X

Figure 9 Property distribution of119883 119884

Let 120573 = 12058210158401015840

119894+1(12058210158401015840

119894+1+ 120582agg) and 120574 = 120582agg120582

10158401015840

119894+1 then we

simplify (20) as

119876rx (119894) =infin

sum

119896=0

119896120573119896+1

120574 =

infin

sum

119896=0

119896120573119896minus1

1205732120574 = 1205732120574

infin

sum

119896=0

119896120573119896minus1

= 1205732120574120597

120597120573

infin

sum

119896=0

120573119896= 1205732120574120597

120597120573

1

1 minus 120573=

1205732120574

(1 minus 120573)2

(21)

As 120573 = 12058210158401015840119894+1(12058210158401015840

119894+1+ 120582agg) and 120582agg = 2120582119894 + 120583119894 so we get that

1 minus 120573 =2120582119894+ 120583119894

12058210158401015840

119894+1+ 2120582119894+ 120583119894

(22)

Therefore we determine that

1205732120574

(1 minus 120573)2

=

(12058210158401015840

119894+1 (12058210158401015840

119894+1+ 2120582119894+ 120583119894))2

(2120582119894+ 120583119894) 12058210158401015840

119894+1

((2120582119894+ 120583119894) (12058210158401015840

119894+1+ 2120582119894+ 120583119894))2

(23)

By calculating the above equations we determine that

119876rx (119894) =12058210158401015840

119894+1

2120582119894+ 120583119894

(24)

According to Littlersquos formula we determine the event waitingtime as follows

120591119890 (119894) =

119876rx (119894)

2120582119894+ 120583119894

=12058210158401015840

119894+1

(2120582119894+ 120583119894)2 (25)

343 Arrival Process to Queue119905119909 From the analytical process

of Queuerx(119894) we find that arrival data rate 1205821015840119894to Queuerx(119894)

is decided by the waterfalls pushing rate 120583119894and event data

generation rate 120582119894at node 119899

119894 The event generation rate 120582

119894

and waterfalls pushing rate 120583119894abide by discrete distribution

and they are independent of one another To determine theformulation of 1205821015840

119894 we calculate the property distribution of

120582119894and 120583

119894 At first we define that 120582

119894and 120583

119894are independent

distribution 119883 and 119884 as shown in Figure 9 We define thevalues of 119883 as 1 2 119864

119883and the values of 119884 as 1 2 119864

119884

Here we prove that the property 119884 is bigger than119883

International Journal of Distributed Sensor Networks 7

We determine the property as follows 119875119883and 119875

119884denote

the properties when119883 = 119894 and 119884 = 119894

119875 [119884 gt 119883]

= 119875119883 (119883 = 1) 119875119884 (119884 = 2) + 119875119884 (119884 = 3) + sdot sdot sdot + 119875119884 (119884

= 119864119884)

+ 119875119883 (119883

= 2) +119875119884 (119884 = 3) + sdot sdot sdot + 119875119884 (119884 = 119864

119884)

+ 119875119883(119883

= 119864119883) + sdot sdot sdot + 119875

119884(119884 = 119864

119883+ 1)

=

119864119883

sum

119894=1

119875119883 (119883 = 119894)

119864119884

sum

119895=119894+1

119875119884(119884 = 119895)

(26)

When applied to continuous distribution we obtain

119875 [119884 gt 119883] = int

infin

0

119875119883 (119905) [int

infin

0

119875119884 (119911) 119889119911] 119889119905 (27)

Here if 119875119883(119905) and 119875

119884(119905) are exponential distribution then

119875119883 (119905) = 120582119883119890

minus120582119883119905

119875119884 (119905) = 120582119884119890

minus120582119884119905

119875 [119884 gt 119883] = int

infin

0

119875119883 (119905) [int

infin

0

120582119884119890minus120582119884119911119889119911] 119889119905

= int

infin

0

119875119883 (119905) [minus119890

minus120582119884119911]infin

119905119889119905

= int

infin

0

119875119883 (119905) (119890

minus120582119884119905) 119889119905

= int

infin

0

120582119883119890minus120582119883119905119890minus120582119884119905119889119905

= int

infin

0

120582119883119890minus(120582119883+120582119884)119905119889119905

=120582119883

120582119883+ 120582119884

[minus119890minus(120582119883+120582119884)119905]

infin

0=

120582119883

120582119883+ 120582119884

(28)

Accordingly we determine the data arrival rate of Queuetx

1205821015840

119894= 120582119894+

120582119894

120582119894+ 120583119894

120583119894 (29)

344 Service Process Since the data generation rate is Pois-son distribution and the waterfalls pushing rate abides byexponential distribution the data arrival rate to Queuetx(119894)approximates to Poisson distribution and each node has oneserver The ACK packet transmission time is not consideredData aggregating time is very short and negligibleThereforethe service time is one-hop data transmission time 120591 In our

work data transmission rate is 119881119888and generated data size is

119878119894 Hence the service time for generated data is

120591 =119878119894

119881119888

(30)

Since 119881119888and 119878

119894are constant hence the service time

for data is fixed and constant According to the aboveanalysis we can determine that the queuing system on serverapproximates to 1198721198631 model As data transmission timeis determined and server utilization ratio is 120578 = 120582

1015840

119894times 120591tx(119894)

hence the average data transmission time 120591tx(119894) is

120591tx (119894) =120591

1 minus 1205821015840

119894120591 (31)

We figure up the probability density function of the servertime by means of Laplace transform in queuing theory andevaluate server waiting time 120591

119904(119894) that data wait for further

transmitting in server

120591119904 (119894) =

1205821015840

119894(120591tx (119894))

2

2 (1 minus 1205821015840

119894120591tx (119894))

(32)

345 Channel Waiting Time In general queuing systemcustomers can be served if there are no other customerswaiting in front of them However when we apply queuingtheory to model wireless communication it is necessary toconsider the impact of wireless channel caused by its proper-ties In wireless sensor networks because of the overhearingof omnidirectional antenna node has to wait a period oftime if its upper or lower neighbors are transmitting dataHere we call this time as channel waiting time 120591

119888(119894) and it

is determined as follows

120591119888 (119894) = 2 times 120578 times 120591tx (119894) = 2120582

1015840

119894times 1205912

tx (119894) (33)

346 Total Delay Total delay119863119899inWPA is the time interval

from when an event data is generated at a node until thedata are received by a sink node in 119873 hops network Theformulation is as follows

119863119899=

119873

sum

119894=1

(120591119890 (119894) + 120591119904 (119894) + 120591119888 (119894) + 120591tx (119894)) (34)

347 Energy Consumption The energy consumption 119864119894for

node 119899119894is the sum of transmission energy consumption

reception energy consumption and overhearing energy con-sumption Now we determine the three kinds of energyconsumption in the following paragraphs Note that 119875tx and119875rx are energy consumption of transmitting or receiving foreach data packet and 119864tx(119894) 119864rx(119894) and 119864oh(119894) are energyconsumption of data transmitting data receiving and over-hearing of node 119899

119894in 119863119899time As we are trying to obtain

the balanced energy consumption among nodes hence wecalculate energy consumption per 119863

119899time for each node

Therefore we get the data transmission energy consumptionaccording to Littlersquos formula as follows

119864tx (119894) = 1205821015840

119894119863119899119875tx (35)

8 International Journal of Distributed Sensor Networks

The reception energy consumption is

119864rx (119894) = 12058210158401015840

119894+1119863119899119875rx (36)

Here 12058210158401015840119894+1

is data arrival rate from upper neighboring node119899119894+1

as the service time is one data packet transmission timehence the data rate getting out from server is same with datatransmission rate which means

12058210158401015840

119894+1= 1205821015840

119894+1 (37)

The overhearing energy consumption is

119864oh (119894) = 1205821015840

119894minus1119863119899119875rx (38)

Here 1205821015840119894minus1

is data transmission rate of node 119899119894minus1

Hence wedetermine total energy consumption119864

119894for a node in time 119863

119899

as

119864119894= 119864tx (119894) + 119864rx (119894) + 119864oh (119894) (39)

4 Optimum Network Lifetime

41 Mathematical Solution According to the analysis in Sec-tion 32 the longest network lifetime is decided by the totaltransmission delay and waterfalls pushing rate vectors Herewe illustrate the relationship between transmission delayenergy consumption and network lifetime so that at lastwe determine the formulations of a set of waterfalls pushingrates Accordingly energy consumption can be written asfollows

119864119894= (1205821015840

119894119875tx + 120582

10158401015840

119894+1119875rx + 120582

1015840

119894minus1119875rx)

= 1205821015840

119894119875tx + (120582

10158401015840

119894+1+ 1205821015840

119894minus1) 119875rx

(40)

To balance energy among nodes we have

1198641= 1198642= sdot sdot sdot = 119864

119894= sdot sdot sdot = 119864

119899 (41)

As the service time is one-hop data transmitting time hence

12058210158401015840

119894= 1205821015840

119894 (42)

Here 119864119894is the energy consumption for node in time 119863

119899

Hence we assume the energy consumption for each unit oftime is 119864 We assume that 120582

119894= 120582 and the network in a finite

network Hence there are no arrival data to the last node inthe network so that it does not need a waterfalls pushing rateand moreover there is no overhearing for the first node thatcounted from the sink node Therefore for node 119899

119899 we have

120583119899= 0

1205821015840

119899= 120582

119864rx (119899) = 0

119864oh (1) = 0

(43)

Accordingly we obtain equations as follows

(120582 +1205821205831

120582 + 1205831

)119875tx + (120582 +1205821205832

120582 + 1205832

)119875rx = 119864

(120582 +1205821205832

120582 + 1205832

)119875tx + (2120582 +1205821205831

120582 + 1205831

+1205821205833

120582 + 1205833

)119875rx = 119864

(120582 +120582120583119894

120582 + 120583119894

)119875tx

+ (2120582 +120582120583119894minus1

120582 + 120583119894minus1

+120582120583119894+1

120582 + 120583119894+1

)119875rx = 119864

120582119875tx + (120582 +120582120583119899minus1

120582 + 120583119899minus1

)119875rx = 119864

(44)

By calculating the above equations we have

120583119899minus1

=120582119875rx (120582 + 120582120583119899minus2 (120582 + 120583119899minus2))

(120582120583119899minus2

(120582 + 120583119899minus2

)) 119875rx minus 120582119875tx(45)

120583119899minus2

=120582119875rx (120582120583119899minus3 (120582 + 120583119899minus3)) + 120582 (119875rx minus 119875tx) (120582120583119899minus1 (120582 + 120583119899minus1))

(119875rx minus 119875tx) (120582 minus 120582120583119899minus1 (120582 + 120583119899minus1)) minus 119875rx (120582120583119899minus3 (120582 + 120583119899minus3))

(46)

120583119894=

1205822120583119894+2

(119875rx minus 119875tx) (120582 + 120583119894+2) minus 1205822120583119894+1

(119875rx minus 119875tx) (120582 + 120583119894+1) minus 1205822120583119894+3119875rx (120582 + 120583119894+3)

120582120583119894+3119875rx (120582 + 120583119894+3) minus 120582120583119894+2 (119875rx minus 119875tx) (120582 + 120583119894+2) + 120582120583119894+1 (119875rx minus 119875tx) (120582 + 120583119894+1) minus 120582119875rx

1205831=

12058221205832(119875rx minus 119875tx) (120582 + 1205832) minus 120582

21205833119875rx (120582 + 1205833) minus 120582

2119875rx

1205821205832(119875tx minus 119875rx) (120582 + 1205832) minus 1205821205833119875rx (120582 + 1205833) + 2120582119875rx minus 120582119875tx

(47)

International Journal of Distributed Sensor Networks 9

Input 120582 120583119899minus1

120578119863appOutput [120583

1 120583119899minus1

]120583max = 2120582

120583119899minus1

= 120583max[1205831 120583119899minus1] = 119891 (120583

119899minus119894)

119863heu = 119892 (1205831 120583119899minus1)

While |119863app minus 119863heu| gt 120578 do120583119899minus2

= 120583119899minus1

minus Δ

[1205831 120583119899minus1] = 119891(120583

119899minus119894)

119863heu = 119892(1205831 120583119899minus1)

EndReturn [120583

1 120583119899minus1]

Algorithm 1 Heuristics algorithm

From the above equations we obtained the relationshipamong waterfalls pushing rate Equation (45) shows that if120583119899minus1

is given then 120583119899minus2

can be determined In (46) if 120583119899minus1

and120583119899minus2

are given then 120583119899minus3

can be determined In this way all 120583119894

can be determined if 120583119899minus1

is known To determine the valuesof 120583119894 we designed a heuristics algorithm as follows In light of

real applications ofWSNwhere the threshold of transmissiondelay is always given in the heuristics algorithm we assumethat the transmission delay is predefined by applicationHence we set the value of120583

119899minus1and then determine the others

However the value of 120583119899minus1

being too large or too little makesthe algorithm too complex to process Therefore it is criticalto determine the maximum value of 120583

119899minus1

42 Heuristics Algorithm In a given application we assumethat data generation rates at all nodes are known and are 120582Thus at node 119899

119899minus1 if the arrival data from node 119899

119899just does

not keep pace with the generated data then the node needswaterfalls pushing rate to transmit the arrival data Howeverthe processing period of waterfalls pushing rate is decided bynode 119899

119899minus1 hence there is again the case that the arrival data

just does not catch up the pace of the waterfalls pushing rate120583119899minus1

Therefore we determine simply that for the arrival datathe average catching up rate of waterfalls pushing rate at node119899119899minus1

is 1205822 Accordingly for transmitting all arrival data themaximum value of 120583

119899minus1at node 119899

119899minus1is

120583max = 2120582 (48)

Hence a heuristics algorithm is described as shown inAlgorithm 1

5 Evaluation

51 Validation of the Queuing Theory Model We implementWPA in the original network simulator written in C++language Analytic results in the previous section are shownas well as simulated result The parameters are shown inTable 1

In the simulation data occurs at each node randomly andindependently Buffer size of each node is infinite Althoughanalytic model assumes that transmission error is negligible

Table 1 Simulation parameters of WPA

Node distance 10 (m)Transmission range 11 (m)Transmission rate 250 (kbps)Data size 4096 (bit)Current consumption for transmission 174 (mA)Current consumption for reception 197 (mA)MAC CSMACARouting DSR

001

01

1

10

100

1000

01 1 10Data generation rate (datasec)

WPA-analysisWPA-simulation

Tota

l del

ay (120583

sec)

Figure 10 Total delay of WPA

transmission errors and retransmission may occur in thesimulation We set data generation rate 120582 = 5 and waterfallspushing rate as 120583

119894= 1 2 3 4 5 for 5-hop tandem network

From Figures 10 and 11 we find that our analytical modelmeets with simulation results so that we basically confirm thecorrectness of queuing theory model of WPA

52 Evaluation of OptimumNetwork Lifetime In this sectionwe evaluate the effectiveness of the proposedwaterfalls partialaggregation Here we consider network environment asnature monitoring network so that we obtain the maximumnetwork lifetime which is the same with network lifetime offull aggregation We set 119864 in (44) which is equal to 119864ful-largestwhere 119864ful-largest denotes the energy consumption of the nodethat consumes most energy in the network The optimalresults can be achieved by numerical packets however themathematical calculation results are too complex For moreunderstanding of our algorithm we obtain the pushing ratevectors applying Excel when the event generation rate is120582 = 1for 5-hop tandem network Nodes are counted from the sinknode which means that the nearest node with the sink isnode 119899

1and its waterfalls pushing rate is 120583

1 The optimum

waterfalls pushing rates are shown in Table 2 when WPAhas the same network lifetime with full aggregation Theanalytical model of nonaggregation and full aggregation isrealized by queuing theory however for the brevity of thepaper we omitted the derivation process

10 International Journal of Distributed Sensor Networks

0

1000

2000

3000

4000

5000

6000

7000

1 2 3 4 5

Ener

gy co

nsum

ptio

n (m

A)

Node ID

WPA-analysisWPA-simulation

Figure 11 Node energy consumption of WPA

0

05

1

15

2

25

3

35

4

1 2 3 4 5Node ID

NonaggregationPartial aggregationFull aggregation

Ener

gy co

nsum

ptio

n (m

A)

Figure 12 Lifetime of nodes

Table 2 Waterfalls pushing rate for optimum network lifetime

Waterfalls pushing rate Values1205831

0021205832

0041205833

0081205834

011205835

05

Figure 12 shows the lifetime of all nodes in the networkwhere we set the waterfalls pushing rate of nodes as the samewith Table 2 From the figure we find that all nodes almosthave the same lifetime in WPA In this case we can saythat the node energy is utilized optimally in WPA In fullaggregation the second and the third node consumed mostenergy and the other nodes have less energy consumptionwhich results in the suboptimum utilization of energy Thenonaggregation has very large energy consumption at thesecond node which results in the short lifetime of the entirenetwork We find from the figure that nonaggregation and

001

01

1

10

100

01 1 10 100

NonaggregationPartial aggregationFull aggregation

Data generation rate (datasec)

Tota

l del

ay (s

ec)

Figure 13 Transmission delay of network

full aggregation have the largest energy consumption attheir second node The reason is that the first node has noadjacent upper node which means that it does not consumeoverhearing energy compared with others

Figure 13 shows the corresponding total delay of thenetwork where we set the same waterfalls pushing rates withTable 1 From the figure we find that when it has the samenetwork lifetime with full aggregation WPA can shorten thetransmission delay by considerable amount When data gen-eration rate becomes larger (larger than 6 types of datasec)transmission delay of nonaggregation gets longer rapidly dueto data congestion at nodes However in WPA the rapidincrease of transmission delay arises later (at data generationrate of 30 types of datasec) than nonaggregation and fullaggregation on data generation line which indicates thatWPA can relieve data congestion at nodes This is becausewith waterfalls pushing rates proposed WPA can adjust thenumber of data transmissions Moreover we find from thefigure that with the given waterfalls pushing rates there isminimum transmission delay and we can determine it inWPA

53 Summary of Evaluation Firstly from the presentationof Section 51 we conclude that our analytical model ofWPA meets with simulation results the diversity of energyconsumption is that we did not consider retransmissionin analytical model and considered it in simulation Sec-ondly in Section 52 compared to nonaggregation and fullaggregation proposed WPA has the superiorities as followsnonaggregation consumes much energy and data congestionoccurs easily when data generation rate is larger In WPAafter data are sent to a neighboring node the data haveto be aggregated together or aggregated with local data forfurther transmitting hence WPA is more energy efficientthan nonaggregationWhen data generation rate is larger at anode data congestion occurs at nonaggregation which leadsto long transmission delay however proposed WPA canrelieve data congestion via reducing the number of data trans-missions by aggregating received data at nodes On the other

International Journal of Distributed Sensor Networks 11

hand full aggregation can save much energy consumptionand suffer long transmission delay Compared to full aggre-gation proposed WPA has similar energy consumption andis superior in balancing energy consumption among nodesMoreover with waterfalls pushing rates WPA can controldata waiting time more reasonably and efficiently than fullaggregation at nodes so that transmission delay of WPA issuperior to full aggregation

6 Conclusion

In this paper we proposed a waterfalls partial aggregationthat can achieve optimum network lifetime in WSN viaapplying data aggregation At first we modeled WPA inqueuing theory and then determined transmission delay ofthe network and energy consumption of a node Then wecontinued to model balanced energy consumption amongnodes The evaluation parts show the correctness of our ana-lytical model and demonstrate the superiorities of proposedWPA

Competing Interests

The authors declare no competing financial interests

Acknowledgments

Thiswork is supported in part by theNatural Science Founda-tion of Inner Mongolia Autonomous Region of China underGrant no 2014BS0603 and no 2014BS0601 and the Programof Higher-Level Talents of Inner Mongolia University underGrant no 30105-135134

References

[1] C-C Shen C Srisathapornphat and C Jaikaeo ldquoSensor infor-mation networking architecture and applicationsrdquo IEEE Per-sonal Communications vol 8 no 4 pp 52ndash59 2001

[2] S D Glaser ldquoSome real-world applications of wireless sen-sor nodesrdquo in Smart Structures and Materials 2004 Sensorsand Smart Structures Technologies for Civil Mechanical andAerospace Systems vol 5391 of Proceedings of SPIE pp 344ndash355San Diego Calif USA March 2004

[3] I F Akyildiz W Su Y Sankarasubramaniam and E CayircildquoWireless sensor networks a surveyrdquo Computer Networks vol38 no 4 pp 393ndash422 2002

[4] W Ye J Heidemann and D Estrin ldquoSensor-MAC (S-MAC)medium access control for wireless sensor networksrdquo in Pro-ceedings of the 21st International Annual Joint Conference of theIEEEComputer and Communications Societies (INFOCOM rsquo02)New York NY USA June 2002

[5] L J G Villalba A L S Orozco A T Cabrera andC J B AbbasldquoRouting protocols in wireless sensor networksrdquo Sensors vol 9no 11 pp 8399ndash8421 2009

[6] P Santi ldquoTopology control in wireless ad hoc and sensornetworksrdquo ACM Computing Surveys vol 37 no 2 pp 164ndash1942005

[7] R Rajagopalan and P K Varshney ldquoData-aggregation tech-niques in sensor networks a surveyrdquo IEEE CommunicationsSurveys and Tutorials vol 8 no 4 pp 48ndash63 2006

[8] W-S Jung K-W Lim Y-B Ko and S-J Park ldquoA hybridapproach for clustering-based data aggregation in wireless sen-sor networksrdquo in Proceedings of the 3rd International ConferenceonDigital Society (ICDS rsquo09) pp 112ndash117 IEEECancunMexicoFebruary 2009

[9] M Kumar S Verma and P S Pratap ldquoClustering approach todata aggregation in wireless sensor networksrdquo in Proceedingsof the 16th IEEE International Conference on Networks pp 1ndash6IEEE New Delhi India December 2008

[10] S Lindsey C Raghavendra and K M Sivalingam ldquoDatagathering algorithms in sensor networks using energy metricsrdquoIEEE Transactions on Parallel and Distributed Systems vol 13no 9 pp 924ndash935 2002

[11] JW Baek Y J Nam andD-W Seo ldquoATS-DA adaptive timeoutscheduling for data aggregation in wireless sensor networksrdquoin Proceedings of the International Conference on InformationNetworking pp 375ndash384 Estoril Portugal 2008

[12] H Chen H Mineno and T Mizuno ldquoAdaptive data aggrega-tion scheme in clustered wireless sensor networksrdquo ComputerCommunications vol 31 no 15 pp 3579ndash3585 2008

[13] I Solis and K Obraczka ldquoIn-network aggregation trade-offsfor data collection in wireless sensor networksrdquo InternationalJournal of Sensor Networks vol 1 no 3-4 pp 200ndash212 2006

[14] F Mirian and M Sabaei ldquoA delay and accuracy sensitive dataaggregation structure in wireless sensor networksrdquo in Proceed-ings of the International Conference on InformationManagementand Engineering (ICIME rsquo09) pp 231ndash235 IEEE Kuala LumpurMalaysia April 2009

[15] P Sun X Zhang Z Dong et al ldquoA novel energy efficientwireless sensor MAC protocolrdquo in Proceedings of the 4th Inter-national Conference on Networked Computing and AdvancedInformation Management pp 68ndash72 Gyeongju South KoreaSeptember 2008

[16] N Li J C Hou and L Sha ldquoDesign and analysis of an mst-based topology control algorithmrdquo in Proceedings of the IEEESocieties 22nd Annual Joint Conference of the IEEE Computerand Communications (INFOCOM rsquo03) vol 3 pp 1702ndash1712 SanFrancisco Calif USA April 2003

[17] B Yin H Shi and Y Shang ldquoA two-level topology controlstrategy for energy efficiency in wireless sensor networksrdquoInternational Journal of Wireless and Mobile Computing vol 4no 1 pp 41ndash49 2010

[18] A Sharif V M Potdar and A J D Rathnayaka ldquoLCARTlightweight congestion aware reliable transport protocol forWSN targeting hetero-geneous trafficrdquo in Proceedings of the17th International Conference on Neural Information Processing(ICONIP rsquo10) Sydney Australia November 2010

[19] B Yu J Li and Y Li ldquoDistributed data aggregation schedulingin wireless sensor networksrdquo in Proceedings of the 28th Confer-ence on Computer Communications (IEEE INFOCOM rsquo09) pp2159ndash2167 Rio de Janeiro Brazil April 2009

[20] M X Cheng and L Yin ldquoEnergy-efficient data gatheringalgorithm in sensor networks with partial aggregationrdquo Inter-national Journal of Sensor Networks vol 4 no 1-2 pp 48ndash542008

[21] A A Allen Probability Statistics and Queueing Theory withComputer Science Applications Gulf Professional Publishing1990

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DistributedSensor Networks

International Journal of

4 International Journal of Distributed Sensor Networks

SinknN n2n3

n4 n1

middot middot middot

Figure 4 Tandem sensor network

Sink120583nminus1 12058311205832

120583n gt 120583nminus1 gt middot middot middot gt 1205832 gt 1205831

120582n 120582nminus1 1205822 1205821

nn nnminus1 n2 n1middot middot middot

middot middot middot

Figure 5 Waterfalls partial aggregation

Start

Aggregate all data

Are new generated

data

Is server idle

Arrival data

Is channel idle

Send data

Has the holding time

reached

Wait

Wait

No

No

Yes

No

No

Yes

Yes

Yes

(1) Arrival process to server

(2) Service process

(3) Data transmitting process

Figure 6 Flow chart of waterfalls partial aggregation

In the flow chart after data arriving from neighboringnode node 119899

119894starts to check if there are new generated data

at itself If yes node 119899119894aggregates all arrival data with the local

generated data and then waits for service process if no node119899119894checks the holding time if time is run up it aggregates

all arrival data and then goes to service process otherwiseit checks new generated dataThis process we call data arrivalprocess to server After aggregating data node 119899

119894checks if

the server is idle or not If there are no other data waitingfor transmitting we say the server is idle otherwise serveris busy data queue up in the server and wait for furthertransmitting When server is idle node checks if the channelis idle or notWhen channel is idle data transmission process

starts otherwise node 119899119894checks the channel until it turns to

idle Data are sent to a neighboring node if the channel is idle

33 Problem Definition In WPA the waterfalls setting ofholding time is for balancing energy consumption amongnodes as well as suppressing load As we mentioned inSection 2 that network lifetime is always decided by theshortest lifetime node in the network therefore to lengthenthe shortest node lifetime is to extend network lifetimeWhen tolerable maximum transmission delay is given bycorresponding application we set the holding time Δ119905

119894for

nodes in the network according to WPA algorithm Thereason that set a maximum tolerable transmission delay isbecause in a finite network there would have several sets ofwaterfalls pushing rates vectors corresponding to differenttransmission delay According to our definition waterfallspushing rate 120583

119894is inverse of holding time With larger

waterfalls pushing rate 120583119894 a node transmits more data and

results in much energy consumption Thus the problembecomes how to set the waterfalls pushing rates among nodesso that all nodes keep having the same energy consumption

In other words how long should the holding time ofarrival data for data aggregation at a node be to meet theconditions of balanced energy consumption among nodesand given transmission delay For a node if required trans-mission delay is given as119863

119899 then what our algorithm should

do is let

119863119899= 1199051+ 1199052+ sdot sdot sdot + 119905

119894+ sdot sdot sdot + 119905

119899 (119899 isin 119873) (1)

Here 119905119894is transmission delay for node 119899

119894and we deter-

mine the formulation of transmission delay for each node aswell as total transmission delay of the network in Section 34In WPA data have to wait for Δ119905 time at a node for gatheringmore data hence the transmission delay for a node ismainly decided by this Δ119905 On the other hand to balancethe energy consumption among nodes we first obtain totalenergy consumption 119864

119894for node 119899

119894which is composed of

energy of data reception data transmission and overhearingWe show the solution of energy consumption for node 119899

119894

and determine the formulations in Section 34 For node 119899119894

data transmission rate is decided by itself and data receivingrate is decided by its upper neighboring node 119899

119894+1and the

overhearing rate is decided by its lower neighboring node119899119894minus1

For node 119899119894 data transmission rate 1205821015840

119894is decided by data

generation rate 120582119894and waterfalls pushing rate 120583

119894 If we set

the maximum network lifetime as 119871max then we obtain (2)

International Journal of Distributed Sensor Networks 5

G Server

120583i 120582i

Queuerx Queuetx

120582998400998400i+1 120582998400i 120582998400998400i

Figure 7 Analytical model of waterfalls partial aggregation at node119899119894

which means 119871max is decided by nodes lifetime 119871119894 or 119895 when

two arbitrary nodes 119894 and 119895 are balanced on energy 119864119894= 119864119895

119871max = 119871 119894 or 119895 (119864119894 = 119864119895) (1 le 119894 119895 lt 119899) (2)

Therefore for achieving balanced energy among nodes wefirst determine the optimum set of waterfalls pushing rates 120583

119894

that correspond to given 119871max in the network

120583119894= argmax (119871 (120583

119894)) (3)

From the above analysis it is clear that we are trying to obtaina set of waterfalls pushing rates

120583119894| 120583119894lt 120583119894+1 120583119894isin |119873| 1 lt 119894 lt 119899 (4)

For obtaining optimum holding time for each node it isnecessary to know total delay of the network and energyconsumption for a node In the following sections wedetermine the total delay and node energy consumption

34 Analytical Model of WPA

341 Queuing Theory Analysis For determining the trans-mission delay we model the data arrival process and datatransmission process of WPA by queuing theory The analyt-ical model of node 119899

119894is shown in Figure 7

Before explaining the model we introduce QueuerxQueuetx G and Server Queuerx denotes the arrival dataqueue in which arrival data are waiting for data aggregationat node 119899

119894 Queuetx denotes the data queue after aggregation

in which aggregated data are waiting for transmission toa neighbor node G is assumed as a virtual gate betweenQueuerx and Queuetx Data transmitting process of data inQueuetx is accomplished via Server

In the analytical model 12058210158401015840119894+1

is arrival data rate fromupper node toQueuerx of node 119899119894 and is approximately abidedby Poisson distribution At node 119899

119894 event generation rate 120582

119894

is assumed to be Poisson distributionWaterfalls pushing rate120583119894is assumed to be exponential distribution In Queuerx all

arrival data and generated data are aggregated and then joinQueuetx via gate G 120582

1015840

119894is data arrival rate to Queuetx in which

data are waiting for further transmitting 12058210158401015840119894is the data rate

upon exiting from the Server Next we analyze the abovemodel start from data receiving at a node to data that aretransmitted to a neighboring node

342 Event Waiting Time In this subsection we decide thetime interval that datawait inQueuerx for aggregating at node

120582998400998400i+1120582998400998400i+1120582998400998400i+1120582998400998400i+1120582998400998400i+1

0 1 k minus 1 k

2120582i + 120583i

2120582i + 120583i 2120582i + 120583i2120582i + 120583i

middot middot middot middot middot middot

Figure 8 State transition of Queuerx

119899119894 we call this time interval as event waiting time and denote

it by 120591119890(119894) and it is counted from when the first data arrive

to Queuerx until the data are aggregated and join QueuetxTo determine 120591

119890(119894) we first determine the amount of data

waiting for event data in Queuerx at node 119899119894 The amount

of data is denoted as 119876rx(119894) and we describe state transitionrate diagram of Queuerx to determine it in Figure 8 Notethat at node 119899

119894 the arrival data rate 12058210158401015840

119894+1is approximately

Poisson distribution and 120582119894is event data generation rate at

node 119899119894and is approximately Poisson distribution In data

aggregation service time in queuing theory corresponds todata aggregating time which is time interval from last dataaggregation until next data aggregation The basic idea of theanalysis is that data waits in Queuerx for the duration accord-ing to the exponential distribution of average 12120582

119894 since

data aggregation rate is inverse of the mean aggregating timehence data aggregation rate is 2120582

119894 andwhen consideringwith

waterfalls pushing rate data aggregating rate is

120582agg = 2120582119894 + 120583119894 (5)

In the diagram the state variable is the number of datawaiting for an event Assume that a number of data arewaiting for an event data at node 119899

119894in Queuerx If we let 1198751198940

and 119875119894119896

be the probability when the number of data waitingfor an event is 0 and 119896 in Queuerx at node 119899

119894 at state 0

according to Figure 8 we obtain

119875119894012058210158401015840

119894+1=

infin

sum

119895=1

119875119894119895120582agg (6)

At state 119896 we obtain

11987511989411989612058210158401015840

119894+1= 119875119894119896minus1

12058210158401015840

119894+1minus 119875119894119896120582agg (7)

Hence we obtain

119875119894119896=

12058210158401015840

119894+1

12058210158401015840

119894+1+ 120582agg

119875119894119896minus1

119875119894119896minus1

=12058210158401015840

119894+1

12058210158401015840

119894+1+ 120582agg

119875119894119896minus2

1198751198941=

12058210158401015840

119894+1

12058210158401015840

119894+1+ 120582agg

1198751198940

(8)

6 International Journal of Distributed Sensor Networks

Bringing these equations into (6) we determine the relation-ship between 119875

119894119896and 119875

1198940that

119875119894119896= (

12058210158401015840

119894+1

12058210158401015840

119894+1+ 120582agg

)

119896

1198751198940 (9)

As we knowinfin

sum

119896=1

119875119894119896= 1 (10)

Let

120572 =12058210158401015840

119894+1

12058210158401015840

119894+1+ 120582agg

(11)

Then we determineinfin

sum

119896=1

1205721198961198751198940= 1 (12)

Asinfin

sum

119896=1

120572119896=

1

1 minus 120572 (13)

henceinfin

sum

119896=1

1205721198961198751198940=

1

1 minus 1205721198751198940= 1 (14)

Accordingly we obtain

1198751198940= 1 minus 120572 (15)

Substituting 120572 we determine

1198751198940=

2120582119894+ 120583119894

12058210158401015840

119894+1+ 120582agg

(16)

119875119894119896=

(12058210158401015840

119894+1)119896

+ 120582agg

(12058210158401015840

119894+1+ 120582agg)

119896+1 (17)

Now we determine the amount of data that are received inQueuerx(119894) as

119876rx (119894) =infin

sum

119896=0

119896119875119894119896 (18)

Taking (17) into the above equation we obtain

119876rx (119894) =infin

sum

119896=0

119896

(12058210158401015840

119894+1)119896

+ 120582agg

(12058210158401015840

119894+1+ 120582agg)

119896+1 (19)

We express the above equation as

119876rx (119894) =infin

sum

119896=0

119896(12058210158401015840

119894+1

12058210158401015840

119894+1+ 120582agg

)

119896+1120582agg

12058210158401015840

119894+1

(20)

0

Y

X

Figure 9 Property distribution of119883 119884

Let 120573 = 12058210158401015840

119894+1(12058210158401015840

119894+1+ 120582agg) and 120574 = 120582agg120582

10158401015840

119894+1 then we

simplify (20) as

119876rx (119894) =infin

sum

119896=0

119896120573119896+1

120574 =

infin

sum

119896=0

119896120573119896minus1

1205732120574 = 1205732120574

infin

sum

119896=0

119896120573119896minus1

= 1205732120574120597

120597120573

infin

sum

119896=0

120573119896= 1205732120574120597

120597120573

1

1 minus 120573=

1205732120574

(1 minus 120573)2

(21)

As 120573 = 12058210158401015840119894+1(12058210158401015840

119894+1+ 120582agg) and 120582agg = 2120582119894 + 120583119894 so we get that

1 minus 120573 =2120582119894+ 120583119894

12058210158401015840

119894+1+ 2120582119894+ 120583119894

(22)

Therefore we determine that

1205732120574

(1 minus 120573)2

=

(12058210158401015840

119894+1 (12058210158401015840

119894+1+ 2120582119894+ 120583119894))2

(2120582119894+ 120583119894) 12058210158401015840

119894+1

((2120582119894+ 120583119894) (12058210158401015840

119894+1+ 2120582119894+ 120583119894))2

(23)

By calculating the above equations we determine that

119876rx (119894) =12058210158401015840

119894+1

2120582119894+ 120583119894

(24)

According to Littlersquos formula we determine the event waitingtime as follows

120591119890 (119894) =

119876rx (119894)

2120582119894+ 120583119894

=12058210158401015840

119894+1

(2120582119894+ 120583119894)2 (25)

343 Arrival Process to Queue119905119909 From the analytical process

of Queuerx(119894) we find that arrival data rate 1205821015840119894to Queuerx(119894)

is decided by the waterfalls pushing rate 120583119894and event data

generation rate 120582119894at node 119899

119894 The event generation rate 120582

119894

and waterfalls pushing rate 120583119894abide by discrete distribution

and they are independent of one another To determine theformulation of 1205821015840

119894 we calculate the property distribution of

120582119894and 120583

119894 At first we define that 120582

119894and 120583

119894are independent

distribution 119883 and 119884 as shown in Figure 9 We define thevalues of 119883 as 1 2 119864

119883and the values of 119884 as 1 2 119864

119884

Here we prove that the property 119884 is bigger than119883

International Journal of Distributed Sensor Networks 7

We determine the property as follows 119875119883and 119875

119884denote

the properties when119883 = 119894 and 119884 = 119894

119875 [119884 gt 119883]

= 119875119883 (119883 = 1) 119875119884 (119884 = 2) + 119875119884 (119884 = 3) + sdot sdot sdot + 119875119884 (119884

= 119864119884)

+ 119875119883 (119883

= 2) +119875119884 (119884 = 3) + sdot sdot sdot + 119875119884 (119884 = 119864

119884)

+ 119875119883(119883

= 119864119883) + sdot sdot sdot + 119875

119884(119884 = 119864

119883+ 1)

=

119864119883

sum

119894=1

119875119883 (119883 = 119894)

119864119884

sum

119895=119894+1

119875119884(119884 = 119895)

(26)

When applied to continuous distribution we obtain

119875 [119884 gt 119883] = int

infin

0

119875119883 (119905) [int

infin

0

119875119884 (119911) 119889119911] 119889119905 (27)

Here if 119875119883(119905) and 119875

119884(119905) are exponential distribution then

119875119883 (119905) = 120582119883119890

minus120582119883119905

119875119884 (119905) = 120582119884119890

minus120582119884119905

119875 [119884 gt 119883] = int

infin

0

119875119883 (119905) [int

infin

0

120582119884119890minus120582119884119911119889119911] 119889119905

= int

infin

0

119875119883 (119905) [minus119890

minus120582119884119911]infin

119905119889119905

= int

infin

0

119875119883 (119905) (119890

minus120582119884119905) 119889119905

= int

infin

0

120582119883119890minus120582119883119905119890minus120582119884119905119889119905

= int

infin

0

120582119883119890minus(120582119883+120582119884)119905119889119905

=120582119883

120582119883+ 120582119884

[minus119890minus(120582119883+120582119884)119905]

infin

0=

120582119883

120582119883+ 120582119884

(28)

Accordingly we determine the data arrival rate of Queuetx

1205821015840

119894= 120582119894+

120582119894

120582119894+ 120583119894

120583119894 (29)

344 Service Process Since the data generation rate is Pois-son distribution and the waterfalls pushing rate abides byexponential distribution the data arrival rate to Queuetx(119894)approximates to Poisson distribution and each node has oneserver The ACK packet transmission time is not consideredData aggregating time is very short and negligibleThereforethe service time is one-hop data transmission time 120591 In our

work data transmission rate is 119881119888and generated data size is

119878119894 Hence the service time for generated data is

120591 =119878119894

119881119888

(30)

Since 119881119888and 119878

119894are constant hence the service time

for data is fixed and constant According to the aboveanalysis we can determine that the queuing system on serverapproximates to 1198721198631 model As data transmission timeis determined and server utilization ratio is 120578 = 120582

1015840

119894times 120591tx(119894)

hence the average data transmission time 120591tx(119894) is

120591tx (119894) =120591

1 minus 1205821015840

119894120591 (31)

We figure up the probability density function of the servertime by means of Laplace transform in queuing theory andevaluate server waiting time 120591

119904(119894) that data wait for further

transmitting in server

120591119904 (119894) =

1205821015840

119894(120591tx (119894))

2

2 (1 minus 1205821015840

119894120591tx (119894))

(32)

345 Channel Waiting Time In general queuing systemcustomers can be served if there are no other customerswaiting in front of them However when we apply queuingtheory to model wireless communication it is necessary toconsider the impact of wireless channel caused by its proper-ties In wireless sensor networks because of the overhearingof omnidirectional antenna node has to wait a period oftime if its upper or lower neighbors are transmitting dataHere we call this time as channel waiting time 120591

119888(119894) and it

is determined as follows

120591119888 (119894) = 2 times 120578 times 120591tx (119894) = 2120582

1015840

119894times 1205912

tx (119894) (33)

346 Total Delay Total delay119863119899inWPA is the time interval

from when an event data is generated at a node until thedata are received by a sink node in 119873 hops network Theformulation is as follows

119863119899=

119873

sum

119894=1

(120591119890 (119894) + 120591119904 (119894) + 120591119888 (119894) + 120591tx (119894)) (34)

347 Energy Consumption The energy consumption 119864119894for

node 119899119894is the sum of transmission energy consumption

reception energy consumption and overhearing energy con-sumption Now we determine the three kinds of energyconsumption in the following paragraphs Note that 119875tx and119875rx are energy consumption of transmitting or receiving foreach data packet and 119864tx(119894) 119864rx(119894) and 119864oh(119894) are energyconsumption of data transmitting data receiving and over-hearing of node 119899

119894in 119863119899time As we are trying to obtain

the balanced energy consumption among nodes hence wecalculate energy consumption per 119863

119899time for each node

Therefore we get the data transmission energy consumptionaccording to Littlersquos formula as follows

119864tx (119894) = 1205821015840

119894119863119899119875tx (35)

8 International Journal of Distributed Sensor Networks

The reception energy consumption is

119864rx (119894) = 12058210158401015840

119894+1119863119899119875rx (36)

Here 12058210158401015840119894+1

is data arrival rate from upper neighboring node119899119894+1

as the service time is one data packet transmission timehence the data rate getting out from server is same with datatransmission rate which means

12058210158401015840

119894+1= 1205821015840

119894+1 (37)

The overhearing energy consumption is

119864oh (119894) = 1205821015840

119894minus1119863119899119875rx (38)

Here 1205821015840119894minus1

is data transmission rate of node 119899119894minus1

Hence wedetermine total energy consumption119864

119894for a node in time 119863

119899

as

119864119894= 119864tx (119894) + 119864rx (119894) + 119864oh (119894) (39)

4 Optimum Network Lifetime

41 Mathematical Solution According to the analysis in Sec-tion 32 the longest network lifetime is decided by the totaltransmission delay and waterfalls pushing rate vectors Herewe illustrate the relationship between transmission delayenergy consumption and network lifetime so that at lastwe determine the formulations of a set of waterfalls pushingrates Accordingly energy consumption can be written asfollows

119864119894= (1205821015840

119894119875tx + 120582

10158401015840

119894+1119875rx + 120582

1015840

119894minus1119875rx)

= 1205821015840

119894119875tx + (120582

10158401015840

119894+1+ 1205821015840

119894minus1) 119875rx

(40)

To balance energy among nodes we have

1198641= 1198642= sdot sdot sdot = 119864

119894= sdot sdot sdot = 119864

119899 (41)

As the service time is one-hop data transmitting time hence

12058210158401015840

119894= 1205821015840

119894 (42)

Here 119864119894is the energy consumption for node in time 119863

119899

Hence we assume the energy consumption for each unit oftime is 119864 We assume that 120582

119894= 120582 and the network in a finite

network Hence there are no arrival data to the last node inthe network so that it does not need a waterfalls pushing rateand moreover there is no overhearing for the first node thatcounted from the sink node Therefore for node 119899

119899 we have

120583119899= 0

1205821015840

119899= 120582

119864rx (119899) = 0

119864oh (1) = 0

(43)

Accordingly we obtain equations as follows

(120582 +1205821205831

120582 + 1205831

)119875tx + (120582 +1205821205832

120582 + 1205832

)119875rx = 119864

(120582 +1205821205832

120582 + 1205832

)119875tx + (2120582 +1205821205831

120582 + 1205831

+1205821205833

120582 + 1205833

)119875rx = 119864

(120582 +120582120583119894

120582 + 120583119894

)119875tx

+ (2120582 +120582120583119894minus1

120582 + 120583119894minus1

+120582120583119894+1

120582 + 120583119894+1

)119875rx = 119864

120582119875tx + (120582 +120582120583119899minus1

120582 + 120583119899minus1

)119875rx = 119864

(44)

By calculating the above equations we have

120583119899minus1

=120582119875rx (120582 + 120582120583119899minus2 (120582 + 120583119899minus2))

(120582120583119899minus2

(120582 + 120583119899minus2

)) 119875rx minus 120582119875tx(45)

120583119899minus2

=120582119875rx (120582120583119899minus3 (120582 + 120583119899minus3)) + 120582 (119875rx minus 119875tx) (120582120583119899minus1 (120582 + 120583119899minus1))

(119875rx minus 119875tx) (120582 minus 120582120583119899minus1 (120582 + 120583119899minus1)) minus 119875rx (120582120583119899minus3 (120582 + 120583119899minus3))

(46)

120583119894=

1205822120583119894+2

(119875rx minus 119875tx) (120582 + 120583119894+2) minus 1205822120583119894+1

(119875rx minus 119875tx) (120582 + 120583119894+1) minus 1205822120583119894+3119875rx (120582 + 120583119894+3)

120582120583119894+3119875rx (120582 + 120583119894+3) minus 120582120583119894+2 (119875rx minus 119875tx) (120582 + 120583119894+2) + 120582120583119894+1 (119875rx minus 119875tx) (120582 + 120583119894+1) minus 120582119875rx

1205831=

12058221205832(119875rx minus 119875tx) (120582 + 1205832) minus 120582

21205833119875rx (120582 + 1205833) minus 120582

2119875rx

1205821205832(119875tx minus 119875rx) (120582 + 1205832) minus 1205821205833119875rx (120582 + 1205833) + 2120582119875rx minus 120582119875tx

(47)

International Journal of Distributed Sensor Networks 9

Input 120582 120583119899minus1

120578119863appOutput [120583

1 120583119899minus1

]120583max = 2120582

120583119899minus1

= 120583max[1205831 120583119899minus1] = 119891 (120583

119899minus119894)

119863heu = 119892 (1205831 120583119899minus1)

While |119863app minus 119863heu| gt 120578 do120583119899minus2

= 120583119899minus1

minus Δ

[1205831 120583119899minus1] = 119891(120583

119899minus119894)

119863heu = 119892(1205831 120583119899minus1)

EndReturn [120583

1 120583119899minus1]

Algorithm 1 Heuristics algorithm

From the above equations we obtained the relationshipamong waterfalls pushing rate Equation (45) shows that if120583119899minus1

is given then 120583119899minus2

can be determined In (46) if 120583119899minus1

and120583119899minus2

are given then 120583119899minus3

can be determined In this way all 120583119894

can be determined if 120583119899minus1

is known To determine the valuesof 120583119894 we designed a heuristics algorithm as follows In light of

real applications ofWSNwhere the threshold of transmissiondelay is always given in the heuristics algorithm we assumethat the transmission delay is predefined by applicationHence we set the value of120583

119899minus1and then determine the others

However the value of 120583119899minus1

being too large or too little makesthe algorithm too complex to process Therefore it is criticalto determine the maximum value of 120583

119899minus1

42 Heuristics Algorithm In a given application we assumethat data generation rates at all nodes are known and are 120582Thus at node 119899

119899minus1 if the arrival data from node 119899

119899just does

not keep pace with the generated data then the node needswaterfalls pushing rate to transmit the arrival data Howeverthe processing period of waterfalls pushing rate is decided bynode 119899

119899minus1 hence there is again the case that the arrival data

just does not catch up the pace of the waterfalls pushing rate120583119899minus1

Therefore we determine simply that for the arrival datathe average catching up rate of waterfalls pushing rate at node119899119899minus1

is 1205822 Accordingly for transmitting all arrival data themaximum value of 120583

119899minus1at node 119899

119899minus1is

120583max = 2120582 (48)

Hence a heuristics algorithm is described as shown inAlgorithm 1

5 Evaluation

51 Validation of the Queuing Theory Model We implementWPA in the original network simulator written in C++language Analytic results in the previous section are shownas well as simulated result The parameters are shown inTable 1

In the simulation data occurs at each node randomly andindependently Buffer size of each node is infinite Althoughanalytic model assumes that transmission error is negligible

Table 1 Simulation parameters of WPA

Node distance 10 (m)Transmission range 11 (m)Transmission rate 250 (kbps)Data size 4096 (bit)Current consumption for transmission 174 (mA)Current consumption for reception 197 (mA)MAC CSMACARouting DSR

001

01

1

10

100

1000

01 1 10Data generation rate (datasec)

WPA-analysisWPA-simulation

Tota

l del

ay (120583

sec)

Figure 10 Total delay of WPA

transmission errors and retransmission may occur in thesimulation We set data generation rate 120582 = 5 and waterfallspushing rate as 120583

119894= 1 2 3 4 5 for 5-hop tandem network

From Figures 10 and 11 we find that our analytical modelmeets with simulation results so that we basically confirm thecorrectness of queuing theory model of WPA

52 Evaluation of OptimumNetwork Lifetime In this sectionwe evaluate the effectiveness of the proposedwaterfalls partialaggregation Here we consider network environment asnature monitoring network so that we obtain the maximumnetwork lifetime which is the same with network lifetime offull aggregation We set 119864 in (44) which is equal to 119864ful-largestwhere 119864ful-largest denotes the energy consumption of the nodethat consumes most energy in the network The optimalresults can be achieved by numerical packets however themathematical calculation results are too complex For moreunderstanding of our algorithm we obtain the pushing ratevectors applying Excel when the event generation rate is120582 = 1for 5-hop tandem network Nodes are counted from the sinknode which means that the nearest node with the sink isnode 119899

1and its waterfalls pushing rate is 120583

1 The optimum

waterfalls pushing rates are shown in Table 2 when WPAhas the same network lifetime with full aggregation Theanalytical model of nonaggregation and full aggregation isrealized by queuing theory however for the brevity of thepaper we omitted the derivation process

10 International Journal of Distributed Sensor Networks

0

1000

2000

3000

4000

5000

6000

7000

1 2 3 4 5

Ener

gy co

nsum

ptio

n (m

A)

Node ID

WPA-analysisWPA-simulation

Figure 11 Node energy consumption of WPA

0

05

1

15

2

25

3

35

4

1 2 3 4 5Node ID

NonaggregationPartial aggregationFull aggregation

Ener

gy co

nsum

ptio

n (m

A)

Figure 12 Lifetime of nodes

Table 2 Waterfalls pushing rate for optimum network lifetime

Waterfalls pushing rate Values1205831

0021205832

0041205833

0081205834

011205835

05

Figure 12 shows the lifetime of all nodes in the networkwhere we set the waterfalls pushing rate of nodes as the samewith Table 2 From the figure we find that all nodes almosthave the same lifetime in WPA In this case we can saythat the node energy is utilized optimally in WPA In fullaggregation the second and the third node consumed mostenergy and the other nodes have less energy consumptionwhich results in the suboptimum utilization of energy Thenonaggregation has very large energy consumption at thesecond node which results in the short lifetime of the entirenetwork We find from the figure that nonaggregation and

001

01

1

10

100

01 1 10 100

NonaggregationPartial aggregationFull aggregation

Data generation rate (datasec)

Tota

l del

ay (s

ec)

Figure 13 Transmission delay of network

full aggregation have the largest energy consumption attheir second node The reason is that the first node has noadjacent upper node which means that it does not consumeoverhearing energy compared with others

Figure 13 shows the corresponding total delay of thenetwork where we set the same waterfalls pushing rates withTable 1 From the figure we find that when it has the samenetwork lifetime with full aggregation WPA can shorten thetransmission delay by considerable amount When data gen-eration rate becomes larger (larger than 6 types of datasec)transmission delay of nonaggregation gets longer rapidly dueto data congestion at nodes However in WPA the rapidincrease of transmission delay arises later (at data generationrate of 30 types of datasec) than nonaggregation and fullaggregation on data generation line which indicates thatWPA can relieve data congestion at nodes This is becausewith waterfalls pushing rates proposed WPA can adjust thenumber of data transmissions Moreover we find from thefigure that with the given waterfalls pushing rates there isminimum transmission delay and we can determine it inWPA

53 Summary of Evaluation Firstly from the presentationof Section 51 we conclude that our analytical model ofWPA meets with simulation results the diversity of energyconsumption is that we did not consider retransmissionin analytical model and considered it in simulation Sec-ondly in Section 52 compared to nonaggregation and fullaggregation proposed WPA has the superiorities as followsnonaggregation consumes much energy and data congestionoccurs easily when data generation rate is larger In WPAafter data are sent to a neighboring node the data haveto be aggregated together or aggregated with local data forfurther transmitting hence WPA is more energy efficientthan nonaggregationWhen data generation rate is larger at anode data congestion occurs at nonaggregation which leadsto long transmission delay however proposed WPA canrelieve data congestion via reducing the number of data trans-missions by aggregating received data at nodes On the other

International Journal of Distributed Sensor Networks 11

hand full aggregation can save much energy consumptionand suffer long transmission delay Compared to full aggre-gation proposed WPA has similar energy consumption andis superior in balancing energy consumption among nodesMoreover with waterfalls pushing rates WPA can controldata waiting time more reasonably and efficiently than fullaggregation at nodes so that transmission delay of WPA issuperior to full aggregation

6 Conclusion

In this paper we proposed a waterfalls partial aggregationthat can achieve optimum network lifetime in WSN viaapplying data aggregation At first we modeled WPA inqueuing theory and then determined transmission delay ofthe network and energy consumption of a node Then wecontinued to model balanced energy consumption amongnodes The evaluation parts show the correctness of our ana-lytical model and demonstrate the superiorities of proposedWPA

Competing Interests

The authors declare no competing financial interests

Acknowledgments

Thiswork is supported in part by theNatural Science Founda-tion of Inner Mongolia Autonomous Region of China underGrant no 2014BS0603 and no 2014BS0601 and the Programof Higher-Level Talents of Inner Mongolia University underGrant no 30105-135134

References

[1] C-C Shen C Srisathapornphat and C Jaikaeo ldquoSensor infor-mation networking architecture and applicationsrdquo IEEE Per-sonal Communications vol 8 no 4 pp 52ndash59 2001

[2] S D Glaser ldquoSome real-world applications of wireless sen-sor nodesrdquo in Smart Structures and Materials 2004 Sensorsand Smart Structures Technologies for Civil Mechanical andAerospace Systems vol 5391 of Proceedings of SPIE pp 344ndash355San Diego Calif USA March 2004

[3] I F Akyildiz W Su Y Sankarasubramaniam and E CayircildquoWireless sensor networks a surveyrdquo Computer Networks vol38 no 4 pp 393ndash422 2002

[4] W Ye J Heidemann and D Estrin ldquoSensor-MAC (S-MAC)medium access control for wireless sensor networksrdquo in Pro-ceedings of the 21st International Annual Joint Conference of theIEEEComputer and Communications Societies (INFOCOM rsquo02)New York NY USA June 2002

[5] L J G Villalba A L S Orozco A T Cabrera andC J B AbbasldquoRouting protocols in wireless sensor networksrdquo Sensors vol 9no 11 pp 8399ndash8421 2009

[6] P Santi ldquoTopology control in wireless ad hoc and sensornetworksrdquo ACM Computing Surveys vol 37 no 2 pp 164ndash1942005

[7] R Rajagopalan and P K Varshney ldquoData-aggregation tech-niques in sensor networks a surveyrdquo IEEE CommunicationsSurveys and Tutorials vol 8 no 4 pp 48ndash63 2006

[8] W-S Jung K-W Lim Y-B Ko and S-J Park ldquoA hybridapproach for clustering-based data aggregation in wireless sen-sor networksrdquo in Proceedings of the 3rd International ConferenceonDigital Society (ICDS rsquo09) pp 112ndash117 IEEECancunMexicoFebruary 2009

[9] M Kumar S Verma and P S Pratap ldquoClustering approach todata aggregation in wireless sensor networksrdquo in Proceedingsof the 16th IEEE International Conference on Networks pp 1ndash6IEEE New Delhi India December 2008

[10] S Lindsey C Raghavendra and K M Sivalingam ldquoDatagathering algorithms in sensor networks using energy metricsrdquoIEEE Transactions on Parallel and Distributed Systems vol 13no 9 pp 924ndash935 2002

[11] JW Baek Y J Nam andD-W Seo ldquoATS-DA adaptive timeoutscheduling for data aggregation in wireless sensor networksrdquoin Proceedings of the International Conference on InformationNetworking pp 375ndash384 Estoril Portugal 2008

[12] H Chen H Mineno and T Mizuno ldquoAdaptive data aggrega-tion scheme in clustered wireless sensor networksrdquo ComputerCommunications vol 31 no 15 pp 3579ndash3585 2008

[13] I Solis and K Obraczka ldquoIn-network aggregation trade-offsfor data collection in wireless sensor networksrdquo InternationalJournal of Sensor Networks vol 1 no 3-4 pp 200ndash212 2006

[14] F Mirian and M Sabaei ldquoA delay and accuracy sensitive dataaggregation structure in wireless sensor networksrdquo in Proceed-ings of the International Conference on InformationManagementand Engineering (ICIME rsquo09) pp 231ndash235 IEEE Kuala LumpurMalaysia April 2009

[15] P Sun X Zhang Z Dong et al ldquoA novel energy efficientwireless sensor MAC protocolrdquo in Proceedings of the 4th Inter-national Conference on Networked Computing and AdvancedInformation Management pp 68ndash72 Gyeongju South KoreaSeptember 2008

[16] N Li J C Hou and L Sha ldquoDesign and analysis of an mst-based topology control algorithmrdquo in Proceedings of the IEEESocieties 22nd Annual Joint Conference of the IEEE Computerand Communications (INFOCOM rsquo03) vol 3 pp 1702ndash1712 SanFrancisco Calif USA April 2003

[17] B Yin H Shi and Y Shang ldquoA two-level topology controlstrategy for energy efficiency in wireless sensor networksrdquoInternational Journal of Wireless and Mobile Computing vol 4no 1 pp 41ndash49 2010

[18] A Sharif V M Potdar and A J D Rathnayaka ldquoLCARTlightweight congestion aware reliable transport protocol forWSN targeting hetero-geneous trafficrdquo in Proceedings of the17th International Conference on Neural Information Processing(ICONIP rsquo10) Sydney Australia November 2010

[19] B Yu J Li and Y Li ldquoDistributed data aggregation schedulingin wireless sensor networksrdquo in Proceedings of the 28th Confer-ence on Computer Communications (IEEE INFOCOM rsquo09) pp2159ndash2167 Rio de Janeiro Brazil April 2009

[20] M X Cheng and L Yin ldquoEnergy-efficient data gatheringalgorithm in sensor networks with partial aggregationrdquo Inter-national Journal of Sensor Networks vol 4 no 1-2 pp 48ndash542008

[21] A A Allen Probability Statistics and Queueing Theory withComputer Science Applications Gulf Professional Publishing1990

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Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

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Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

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Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of Distributed Sensor Networks 5

G Server

120583i 120582i

Queuerx Queuetx

120582998400998400i+1 120582998400i 120582998400998400i

Figure 7 Analytical model of waterfalls partial aggregation at node119899119894

which means 119871max is decided by nodes lifetime 119871119894 or 119895 when

two arbitrary nodes 119894 and 119895 are balanced on energy 119864119894= 119864119895

119871max = 119871 119894 or 119895 (119864119894 = 119864119895) (1 le 119894 119895 lt 119899) (2)

Therefore for achieving balanced energy among nodes wefirst determine the optimum set of waterfalls pushing rates 120583

119894

that correspond to given 119871max in the network

120583119894= argmax (119871 (120583

119894)) (3)

From the above analysis it is clear that we are trying to obtaina set of waterfalls pushing rates

120583119894| 120583119894lt 120583119894+1 120583119894isin |119873| 1 lt 119894 lt 119899 (4)

For obtaining optimum holding time for each node it isnecessary to know total delay of the network and energyconsumption for a node In the following sections wedetermine the total delay and node energy consumption

34 Analytical Model of WPA

341 Queuing Theory Analysis For determining the trans-mission delay we model the data arrival process and datatransmission process of WPA by queuing theory The analyt-ical model of node 119899

119894is shown in Figure 7

Before explaining the model we introduce QueuerxQueuetx G and Server Queuerx denotes the arrival dataqueue in which arrival data are waiting for data aggregationat node 119899

119894 Queuetx denotes the data queue after aggregation

in which aggregated data are waiting for transmission toa neighbor node G is assumed as a virtual gate betweenQueuerx and Queuetx Data transmitting process of data inQueuetx is accomplished via Server

In the analytical model 12058210158401015840119894+1

is arrival data rate fromupper node toQueuerx of node 119899119894 and is approximately abidedby Poisson distribution At node 119899

119894 event generation rate 120582

119894

is assumed to be Poisson distributionWaterfalls pushing rate120583119894is assumed to be exponential distribution In Queuerx all

arrival data and generated data are aggregated and then joinQueuetx via gate G 120582

1015840

119894is data arrival rate to Queuetx in which

data are waiting for further transmitting 12058210158401015840119894is the data rate

upon exiting from the Server Next we analyze the abovemodel start from data receiving at a node to data that aretransmitted to a neighboring node

342 Event Waiting Time In this subsection we decide thetime interval that datawait inQueuerx for aggregating at node

120582998400998400i+1120582998400998400i+1120582998400998400i+1120582998400998400i+1120582998400998400i+1

0 1 k minus 1 k

2120582i + 120583i

2120582i + 120583i 2120582i + 120583i2120582i + 120583i

middot middot middot middot middot middot

Figure 8 State transition of Queuerx

119899119894 we call this time interval as event waiting time and denote

it by 120591119890(119894) and it is counted from when the first data arrive

to Queuerx until the data are aggregated and join QueuetxTo determine 120591

119890(119894) we first determine the amount of data

waiting for event data in Queuerx at node 119899119894 The amount

of data is denoted as 119876rx(119894) and we describe state transitionrate diagram of Queuerx to determine it in Figure 8 Notethat at node 119899

119894 the arrival data rate 12058210158401015840

119894+1is approximately

Poisson distribution and 120582119894is event data generation rate at

node 119899119894and is approximately Poisson distribution In data

aggregation service time in queuing theory corresponds todata aggregating time which is time interval from last dataaggregation until next data aggregation The basic idea of theanalysis is that data waits in Queuerx for the duration accord-ing to the exponential distribution of average 12120582

119894 since

data aggregation rate is inverse of the mean aggregating timehence data aggregation rate is 2120582

119894 andwhen consideringwith

waterfalls pushing rate data aggregating rate is

120582agg = 2120582119894 + 120583119894 (5)

In the diagram the state variable is the number of datawaiting for an event Assume that a number of data arewaiting for an event data at node 119899

119894in Queuerx If we let 1198751198940

and 119875119894119896

be the probability when the number of data waitingfor an event is 0 and 119896 in Queuerx at node 119899

119894 at state 0

according to Figure 8 we obtain

119875119894012058210158401015840

119894+1=

infin

sum

119895=1

119875119894119895120582agg (6)

At state 119896 we obtain

11987511989411989612058210158401015840

119894+1= 119875119894119896minus1

12058210158401015840

119894+1minus 119875119894119896120582agg (7)

Hence we obtain

119875119894119896=

12058210158401015840

119894+1

12058210158401015840

119894+1+ 120582agg

119875119894119896minus1

119875119894119896minus1

=12058210158401015840

119894+1

12058210158401015840

119894+1+ 120582agg

119875119894119896minus2

1198751198941=

12058210158401015840

119894+1

12058210158401015840

119894+1+ 120582agg

1198751198940

(8)

6 International Journal of Distributed Sensor Networks

Bringing these equations into (6) we determine the relation-ship between 119875

119894119896and 119875

1198940that

119875119894119896= (

12058210158401015840

119894+1

12058210158401015840

119894+1+ 120582agg

)

119896

1198751198940 (9)

As we knowinfin

sum

119896=1

119875119894119896= 1 (10)

Let

120572 =12058210158401015840

119894+1

12058210158401015840

119894+1+ 120582agg

(11)

Then we determineinfin

sum

119896=1

1205721198961198751198940= 1 (12)

Asinfin

sum

119896=1

120572119896=

1

1 minus 120572 (13)

henceinfin

sum

119896=1

1205721198961198751198940=

1

1 minus 1205721198751198940= 1 (14)

Accordingly we obtain

1198751198940= 1 minus 120572 (15)

Substituting 120572 we determine

1198751198940=

2120582119894+ 120583119894

12058210158401015840

119894+1+ 120582agg

(16)

119875119894119896=

(12058210158401015840

119894+1)119896

+ 120582agg

(12058210158401015840

119894+1+ 120582agg)

119896+1 (17)

Now we determine the amount of data that are received inQueuerx(119894) as

119876rx (119894) =infin

sum

119896=0

119896119875119894119896 (18)

Taking (17) into the above equation we obtain

119876rx (119894) =infin

sum

119896=0

119896

(12058210158401015840

119894+1)119896

+ 120582agg

(12058210158401015840

119894+1+ 120582agg)

119896+1 (19)

We express the above equation as

119876rx (119894) =infin

sum

119896=0

119896(12058210158401015840

119894+1

12058210158401015840

119894+1+ 120582agg

)

119896+1120582agg

12058210158401015840

119894+1

(20)

0

Y

X

Figure 9 Property distribution of119883 119884

Let 120573 = 12058210158401015840

119894+1(12058210158401015840

119894+1+ 120582agg) and 120574 = 120582agg120582

10158401015840

119894+1 then we

simplify (20) as

119876rx (119894) =infin

sum

119896=0

119896120573119896+1

120574 =

infin

sum

119896=0

119896120573119896minus1

1205732120574 = 1205732120574

infin

sum

119896=0

119896120573119896minus1

= 1205732120574120597

120597120573

infin

sum

119896=0

120573119896= 1205732120574120597

120597120573

1

1 minus 120573=

1205732120574

(1 minus 120573)2

(21)

As 120573 = 12058210158401015840119894+1(12058210158401015840

119894+1+ 120582agg) and 120582agg = 2120582119894 + 120583119894 so we get that

1 minus 120573 =2120582119894+ 120583119894

12058210158401015840

119894+1+ 2120582119894+ 120583119894

(22)

Therefore we determine that

1205732120574

(1 minus 120573)2

=

(12058210158401015840

119894+1 (12058210158401015840

119894+1+ 2120582119894+ 120583119894))2

(2120582119894+ 120583119894) 12058210158401015840

119894+1

((2120582119894+ 120583119894) (12058210158401015840

119894+1+ 2120582119894+ 120583119894))2

(23)

By calculating the above equations we determine that

119876rx (119894) =12058210158401015840

119894+1

2120582119894+ 120583119894

(24)

According to Littlersquos formula we determine the event waitingtime as follows

120591119890 (119894) =

119876rx (119894)

2120582119894+ 120583119894

=12058210158401015840

119894+1

(2120582119894+ 120583119894)2 (25)

343 Arrival Process to Queue119905119909 From the analytical process

of Queuerx(119894) we find that arrival data rate 1205821015840119894to Queuerx(119894)

is decided by the waterfalls pushing rate 120583119894and event data

generation rate 120582119894at node 119899

119894 The event generation rate 120582

119894

and waterfalls pushing rate 120583119894abide by discrete distribution

and they are independent of one another To determine theformulation of 1205821015840

119894 we calculate the property distribution of

120582119894and 120583

119894 At first we define that 120582

119894and 120583

119894are independent

distribution 119883 and 119884 as shown in Figure 9 We define thevalues of 119883 as 1 2 119864

119883and the values of 119884 as 1 2 119864

119884

Here we prove that the property 119884 is bigger than119883

International Journal of Distributed Sensor Networks 7

We determine the property as follows 119875119883and 119875

119884denote

the properties when119883 = 119894 and 119884 = 119894

119875 [119884 gt 119883]

= 119875119883 (119883 = 1) 119875119884 (119884 = 2) + 119875119884 (119884 = 3) + sdot sdot sdot + 119875119884 (119884

= 119864119884)

+ 119875119883 (119883

= 2) +119875119884 (119884 = 3) + sdot sdot sdot + 119875119884 (119884 = 119864

119884)

+ 119875119883(119883

= 119864119883) + sdot sdot sdot + 119875

119884(119884 = 119864

119883+ 1)

=

119864119883

sum

119894=1

119875119883 (119883 = 119894)

119864119884

sum

119895=119894+1

119875119884(119884 = 119895)

(26)

When applied to continuous distribution we obtain

119875 [119884 gt 119883] = int

infin

0

119875119883 (119905) [int

infin

0

119875119884 (119911) 119889119911] 119889119905 (27)

Here if 119875119883(119905) and 119875

119884(119905) are exponential distribution then

119875119883 (119905) = 120582119883119890

minus120582119883119905

119875119884 (119905) = 120582119884119890

minus120582119884119905

119875 [119884 gt 119883] = int

infin

0

119875119883 (119905) [int

infin

0

120582119884119890minus120582119884119911119889119911] 119889119905

= int

infin

0

119875119883 (119905) [minus119890

minus120582119884119911]infin

119905119889119905

= int

infin

0

119875119883 (119905) (119890

minus120582119884119905) 119889119905

= int

infin

0

120582119883119890minus120582119883119905119890minus120582119884119905119889119905

= int

infin

0

120582119883119890minus(120582119883+120582119884)119905119889119905

=120582119883

120582119883+ 120582119884

[minus119890minus(120582119883+120582119884)119905]

infin

0=

120582119883

120582119883+ 120582119884

(28)

Accordingly we determine the data arrival rate of Queuetx

1205821015840

119894= 120582119894+

120582119894

120582119894+ 120583119894

120583119894 (29)

344 Service Process Since the data generation rate is Pois-son distribution and the waterfalls pushing rate abides byexponential distribution the data arrival rate to Queuetx(119894)approximates to Poisson distribution and each node has oneserver The ACK packet transmission time is not consideredData aggregating time is very short and negligibleThereforethe service time is one-hop data transmission time 120591 In our

work data transmission rate is 119881119888and generated data size is

119878119894 Hence the service time for generated data is

120591 =119878119894

119881119888

(30)

Since 119881119888and 119878

119894are constant hence the service time

for data is fixed and constant According to the aboveanalysis we can determine that the queuing system on serverapproximates to 1198721198631 model As data transmission timeis determined and server utilization ratio is 120578 = 120582

1015840

119894times 120591tx(119894)

hence the average data transmission time 120591tx(119894) is

120591tx (119894) =120591

1 minus 1205821015840

119894120591 (31)

We figure up the probability density function of the servertime by means of Laplace transform in queuing theory andevaluate server waiting time 120591

119904(119894) that data wait for further

transmitting in server

120591119904 (119894) =

1205821015840

119894(120591tx (119894))

2

2 (1 minus 1205821015840

119894120591tx (119894))

(32)

345 Channel Waiting Time In general queuing systemcustomers can be served if there are no other customerswaiting in front of them However when we apply queuingtheory to model wireless communication it is necessary toconsider the impact of wireless channel caused by its proper-ties In wireless sensor networks because of the overhearingof omnidirectional antenna node has to wait a period oftime if its upper or lower neighbors are transmitting dataHere we call this time as channel waiting time 120591

119888(119894) and it

is determined as follows

120591119888 (119894) = 2 times 120578 times 120591tx (119894) = 2120582

1015840

119894times 1205912

tx (119894) (33)

346 Total Delay Total delay119863119899inWPA is the time interval

from when an event data is generated at a node until thedata are received by a sink node in 119873 hops network Theformulation is as follows

119863119899=

119873

sum

119894=1

(120591119890 (119894) + 120591119904 (119894) + 120591119888 (119894) + 120591tx (119894)) (34)

347 Energy Consumption The energy consumption 119864119894for

node 119899119894is the sum of transmission energy consumption

reception energy consumption and overhearing energy con-sumption Now we determine the three kinds of energyconsumption in the following paragraphs Note that 119875tx and119875rx are energy consumption of transmitting or receiving foreach data packet and 119864tx(119894) 119864rx(119894) and 119864oh(119894) are energyconsumption of data transmitting data receiving and over-hearing of node 119899

119894in 119863119899time As we are trying to obtain

the balanced energy consumption among nodes hence wecalculate energy consumption per 119863

119899time for each node

Therefore we get the data transmission energy consumptionaccording to Littlersquos formula as follows

119864tx (119894) = 1205821015840

119894119863119899119875tx (35)

8 International Journal of Distributed Sensor Networks

The reception energy consumption is

119864rx (119894) = 12058210158401015840

119894+1119863119899119875rx (36)

Here 12058210158401015840119894+1

is data arrival rate from upper neighboring node119899119894+1

as the service time is one data packet transmission timehence the data rate getting out from server is same with datatransmission rate which means

12058210158401015840

119894+1= 1205821015840

119894+1 (37)

The overhearing energy consumption is

119864oh (119894) = 1205821015840

119894minus1119863119899119875rx (38)

Here 1205821015840119894minus1

is data transmission rate of node 119899119894minus1

Hence wedetermine total energy consumption119864

119894for a node in time 119863

119899

as

119864119894= 119864tx (119894) + 119864rx (119894) + 119864oh (119894) (39)

4 Optimum Network Lifetime

41 Mathematical Solution According to the analysis in Sec-tion 32 the longest network lifetime is decided by the totaltransmission delay and waterfalls pushing rate vectors Herewe illustrate the relationship between transmission delayenergy consumption and network lifetime so that at lastwe determine the formulations of a set of waterfalls pushingrates Accordingly energy consumption can be written asfollows

119864119894= (1205821015840

119894119875tx + 120582

10158401015840

119894+1119875rx + 120582

1015840

119894minus1119875rx)

= 1205821015840

119894119875tx + (120582

10158401015840

119894+1+ 1205821015840

119894minus1) 119875rx

(40)

To balance energy among nodes we have

1198641= 1198642= sdot sdot sdot = 119864

119894= sdot sdot sdot = 119864

119899 (41)

As the service time is one-hop data transmitting time hence

12058210158401015840

119894= 1205821015840

119894 (42)

Here 119864119894is the energy consumption for node in time 119863

119899

Hence we assume the energy consumption for each unit oftime is 119864 We assume that 120582

119894= 120582 and the network in a finite

network Hence there are no arrival data to the last node inthe network so that it does not need a waterfalls pushing rateand moreover there is no overhearing for the first node thatcounted from the sink node Therefore for node 119899

119899 we have

120583119899= 0

1205821015840

119899= 120582

119864rx (119899) = 0

119864oh (1) = 0

(43)

Accordingly we obtain equations as follows

(120582 +1205821205831

120582 + 1205831

)119875tx + (120582 +1205821205832

120582 + 1205832

)119875rx = 119864

(120582 +1205821205832

120582 + 1205832

)119875tx + (2120582 +1205821205831

120582 + 1205831

+1205821205833

120582 + 1205833

)119875rx = 119864

(120582 +120582120583119894

120582 + 120583119894

)119875tx

+ (2120582 +120582120583119894minus1

120582 + 120583119894minus1

+120582120583119894+1

120582 + 120583119894+1

)119875rx = 119864

120582119875tx + (120582 +120582120583119899minus1

120582 + 120583119899minus1

)119875rx = 119864

(44)

By calculating the above equations we have

120583119899minus1

=120582119875rx (120582 + 120582120583119899minus2 (120582 + 120583119899minus2))

(120582120583119899minus2

(120582 + 120583119899minus2

)) 119875rx minus 120582119875tx(45)

120583119899minus2

=120582119875rx (120582120583119899minus3 (120582 + 120583119899minus3)) + 120582 (119875rx minus 119875tx) (120582120583119899minus1 (120582 + 120583119899minus1))

(119875rx minus 119875tx) (120582 minus 120582120583119899minus1 (120582 + 120583119899minus1)) minus 119875rx (120582120583119899minus3 (120582 + 120583119899minus3))

(46)

120583119894=

1205822120583119894+2

(119875rx minus 119875tx) (120582 + 120583119894+2) minus 1205822120583119894+1

(119875rx minus 119875tx) (120582 + 120583119894+1) minus 1205822120583119894+3119875rx (120582 + 120583119894+3)

120582120583119894+3119875rx (120582 + 120583119894+3) minus 120582120583119894+2 (119875rx minus 119875tx) (120582 + 120583119894+2) + 120582120583119894+1 (119875rx minus 119875tx) (120582 + 120583119894+1) minus 120582119875rx

1205831=

12058221205832(119875rx minus 119875tx) (120582 + 1205832) minus 120582

21205833119875rx (120582 + 1205833) minus 120582

2119875rx

1205821205832(119875tx minus 119875rx) (120582 + 1205832) minus 1205821205833119875rx (120582 + 1205833) + 2120582119875rx minus 120582119875tx

(47)

International Journal of Distributed Sensor Networks 9

Input 120582 120583119899minus1

120578119863appOutput [120583

1 120583119899minus1

]120583max = 2120582

120583119899minus1

= 120583max[1205831 120583119899minus1] = 119891 (120583

119899minus119894)

119863heu = 119892 (1205831 120583119899minus1)

While |119863app minus 119863heu| gt 120578 do120583119899minus2

= 120583119899minus1

minus Δ

[1205831 120583119899minus1] = 119891(120583

119899minus119894)

119863heu = 119892(1205831 120583119899minus1)

EndReturn [120583

1 120583119899minus1]

Algorithm 1 Heuristics algorithm

From the above equations we obtained the relationshipamong waterfalls pushing rate Equation (45) shows that if120583119899minus1

is given then 120583119899minus2

can be determined In (46) if 120583119899minus1

and120583119899minus2

are given then 120583119899minus3

can be determined In this way all 120583119894

can be determined if 120583119899minus1

is known To determine the valuesof 120583119894 we designed a heuristics algorithm as follows In light of

real applications ofWSNwhere the threshold of transmissiondelay is always given in the heuristics algorithm we assumethat the transmission delay is predefined by applicationHence we set the value of120583

119899minus1and then determine the others

However the value of 120583119899minus1

being too large or too little makesthe algorithm too complex to process Therefore it is criticalto determine the maximum value of 120583

119899minus1

42 Heuristics Algorithm In a given application we assumethat data generation rates at all nodes are known and are 120582Thus at node 119899

119899minus1 if the arrival data from node 119899

119899just does

not keep pace with the generated data then the node needswaterfalls pushing rate to transmit the arrival data Howeverthe processing period of waterfalls pushing rate is decided bynode 119899

119899minus1 hence there is again the case that the arrival data

just does not catch up the pace of the waterfalls pushing rate120583119899minus1

Therefore we determine simply that for the arrival datathe average catching up rate of waterfalls pushing rate at node119899119899minus1

is 1205822 Accordingly for transmitting all arrival data themaximum value of 120583

119899minus1at node 119899

119899minus1is

120583max = 2120582 (48)

Hence a heuristics algorithm is described as shown inAlgorithm 1

5 Evaluation

51 Validation of the Queuing Theory Model We implementWPA in the original network simulator written in C++language Analytic results in the previous section are shownas well as simulated result The parameters are shown inTable 1

In the simulation data occurs at each node randomly andindependently Buffer size of each node is infinite Althoughanalytic model assumes that transmission error is negligible

Table 1 Simulation parameters of WPA

Node distance 10 (m)Transmission range 11 (m)Transmission rate 250 (kbps)Data size 4096 (bit)Current consumption for transmission 174 (mA)Current consumption for reception 197 (mA)MAC CSMACARouting DSR

001

01

1

10

100

1000

01 1 10Data generation rate (datasec)

WPA-analysisWPA-simulation

Tota

l del

ay (120583

sec)

Figure 10 Total delay of WPA

transmission errors and retransmission may occur in thesimulation We set data generation rate 120582 = 5 and waterfallspushing rate as 120583

119894= 1 2 3 4 5 for 5-hop tandem network

From Figures 10 and 11 we find that our analytical modelmeets with simulation results so that we basically confirm thecorrectness of queuing theory model of WPA

52 Evaluation of OptimumNetwork Lifetime In this sectionwe evaluate the effectiveness of the proposedwaterfalls partialaggregation Here we consider network environment asnature monitoring network so that we obtain the maximumnetwork lifetime which is the same with network lifetime offull aggregation We set 119864 in (44) which is equal to 119864ful-largestwhere 119864ful-largest denotes the energy consumption of the nodethat consumes most energy in the network The optimalresults can be achieved by numerical packets however themathematical calculation results are too complex For moreunderstanding of our algorithm we obtain the pushing ratevectors applying Excel when the event generation rate is120582 = 1for 5-hop tandem network Nodes are counted from the sinknode which means that the nearest node with the sink isnode 119899

1and its waterfalls pushing rate is 120583

1 The optimum

waterfalls pushing rates are shown in Table 2 when WPAhas the same network lifetime with full aggregation Theanalytical model of nonaggregation and full aggregation isrealized by queuing theory however for the brevity of thepaper we omitted the derivation process

10 International Journal of Distributed Sensor Networks

0

1000

2000

3000

4000

5000

6000

7000

1 2 3 4 5

Ener

gy co

nsum

ptio

n (m

A)

Node ID

WPA-analysisWPA-simulation

Figure 11 Node energy consumption of WPA

0

05

1

15

2

25

3

35

4

1 2 3 4 5Node ID

NonaggregationPartial aggregationFull aggregation

Ener

gy co

nsum

ptio

n (m

A)

Figure 12 Lifetime of nodes

Table 2 Waterfalls pushing rate for optimum network lifetime

Waterfalls pushing rate Values1205831

0021205832

0041205833

0081205834

011205835

05

Figure 12 shows the lifetime of all nodes in the networkwhere we set the waterfalls pushing rate of nodes as the samewith Table 2 From the figure we find that all nodes almosthave the same lifetime in WPA In this case we can saythat the node energy is utilized optimally in WPA In fullaggregation the second and the third node consumed mostenergy and the other nodes have less energy consumptionwhich results in the suboptimum utilization of energy Thenonaggregation has very large energy consumption at thesecond node which results in the short lifetime of the entirenetwork We find from the figure that nonaggregation and

001

01

1

10

100

01 1 10 100

NonaggregationPartial aggregationFull aggregation

Data generation rate (datasec)

Tota

l del

ay (s

ec)

Figure 13 Transmission delay of network

full aggregation have the largest energy consumption attheir second node The reason is that the first node has noadjacent upper node which means that it does not consumeoverhearing energy compared with others

Figure 13 shows the corresponding total delay of thenetwork where we set the same waterfalls pushing rates withTable 1 From the figure we find that when it has the samenetwork lifetime with full aggregation WPA can shorten thetransmission delay by considerable amount When data gen-eration rate becomes larger (larger than 6 types of datasec)transmission delay of nonaggregation gets longer rapidly dueto data congestion at nodes However in WPA the rapidincrease of transmission delay arises later (at data generationrate of 30 types of datasec) than nonaggregation and fullaggregation on data generation line which indicates thatWPA can relieve data congestion at nodes This is becausewith waterfalls pushing rates proposed WPA can adjust thenumber of data transmissions Moreover we find from thefigure that with the given waterfalls pushing rates there isminimum transmission delay and we can determine it inWPA

53 Summary of Evaluation Firstly from the presentationof Section 51 we conclude that our analytical model ofWPA meets with simulation results the diversity of energyconsumption is that we did not consider retransmissionin analytical model and considered it in simulation Sec-ondly in Section 52 compared to nonaggregation and fullaggregation proposed WPA has the superiorities as followsnonaggregation consumes much energy and data congestionoccurs easily when data generation rate is larger In WPAafter data are sent to a neighboring node the data haveto be aggregated together or aggregated with local data forfurther transmitting hence WPA is more energy efficientthan nonaggregationWhen data generation rate is larger at anode data congestion occurs at nonaggregation which leadsto long transmission delay however proposed WPA canrelieve data congestion via reducing the number of data trans-missions by aggregating received data at nodes On the other

International Journal of Distributed Sensor Networks 11

hand full aggregation can save much energy consumptionand suffer long transmission delay Compared to full aggre-gation proposed WPA has similar energy consumption andis superior in balancing energy consumption among nodesMoreover with waterfalls pushing rates WPA can controldata waiting time more reasonably and efficiently than fullaggregation at nodes so that transmission delay of WPA issuperior to full aggregation

6 Conclusion

In this paper we proposed a waterfalls partial aggregationthat can achieve optimum network lifetime in WSN viaapplying data aggregation At first we modeled WPA inqueuing theory and then determined transmission delay ofthe network and energy consumption of a node Then wecontinued to model balanced energy consumption amongnodes The evaluation parts show the correctness of our ana-lytical model and demonstrate the superiorities of proposedWPA

Competing Interests

The authors declare no competing financial interests

Acknowledgments

Thiswork is supported in part by theNatural Science Founda-tion of Inner Mongolia Autonomous Region of China underGrant no 2014BS0603 and no 2014BS0601 and the Programof Higher-Level Talents of Inner Mongolia University underGrant no 30105-135134

References

[1] C-C Shen C Srisathapornphat and C Jaikaeo ldquoSensor infor-mation networking architecture and applicationsrdquo IEEE Per-sonal Communications vol 8 no 4 pp 52ndash59 2001

[2] S D Glaser ldquoSome real-world applications of wireless sen-sor nodesrdquo in Smart Structures and Materials 2004 Sensorsand Smart Structures Technologies for Civil Mechanical andAerospace Systems vol 5391 of Proceedings of SPIE pp 344ndash355San Diego Calif USA March 2004

[3] I F Akyildiz W Su Y Sankarasubramaniam and E CayircildquoWireless sensor networks a surveyrdquo Computer Networks vol38 no 4 pp 393ndash422 2002

[4] W Ye J Heidemann and D Estrin ldquoSensor-MAC (S-MAC)medium access control for wireless sensor networksrdquo in Pro-ceedings of the 21st International Annual Joint Conference of theIEEEComputer and Communications Societies (INFOCOM rsquo02)New York NY USA June 2002

[5] L J G Villalba A L S Orozco A T Cabrera andC J B AbbasldquoRouting protocols in wireless sensor networksrdquo Sensors vol 9no 11 pp 8399ndash8421 2009

[6] P Santi ldquoTopology control in wireless ad hoc and sensornetworksrdquo ACM Computing Surveys vol 37 no 2 pp 164ndash1942005

[7] R Rajagopalan and P K Varshney ldquoData-aggregation tech-niques in sensor networks a surveyrdquo IEEE CommunicationsSurveys and Tutorials vol 8 no 4 pp 48ndash63 2006

[8] W-S Jung K-W Lim Y-B Ko and S-J Park ldquoA hybridapproach for clustering-based data aggregation in wireless sen-sor networksrdquo in Proceedings of the 3rd International ConferenceonDigital Society (ICDS rsquo09) pp 112ndash117 IEEECancunMexicoFebruary 2009

[9] M Kumar S Verma and P S Pratap ldquoClustering approach todata aggregation in wireless sensor networksrdquo in Proceedingsof the 16th IEEE International Conference on Networks pp 1ndash6IEEE New Delhi India December 2008

[10] S Lindsey C Raghavendra and K M Sivalingam ldquoDatagathering algorithms in sensor networks using energy metricsrdquoIEEE Transactions on Parallel and Distributed Systems vol 13no 9 pp 924ndash935 2002

[11] JW Baek Y J Nam andD-W Seo ldquoATS-DA adaptive timeoutscheduling for data aggregation in wireless sensor networksrdquoin Proceedings of the International Conference on InformationNetworking pp 375ndash384 Estoril Portugal 2008

[12] H Chen H Mineno and T Mizuno ldquoAdaptive data aggrega-tion scheme in clustered wireless sensor networksrdquo ComputerCommunications vol 31 no 15 pp 3579ndash3585 2008

[13] I Solis and K Obraczka ldquoIn-network aggregation trade-offsfor data collection in wireless sensor networksrdquo InternationalJournal of Sensor Networks vol 1 no 3-4 pp 200ndash212 2006

[14] F Mirian and M Sabaei ldquoA delay and accuracy sensitive dataaggregation structure in wireless sensor networksrdquo in Proceed-ings of the International Conference on InformationManagementand Engineering (ICIME rsquo09) pp 231ndash235 IEEE Kuala LumpurMalaysia April 2009

[15] P Sun X Zhang Z Dong et al ldquoA novel energy efficientwireless sensor MAC protocolrdquo in Proceedings of the 4th Inter-national Conference on Networked Computing and AdvancedInformation Management pp 68ndash72 Gyeongju South KoreaSeptember 2008

[16] N Li J C Hou and L Sha ldquoDesign and analysis of an mst-based topology control algorithmrdquo in Proceedings of the IEEESocieties 22nd Annual Joint Conference of the IEEE Computerand Communications (INFOCOM rsquo03) vol 3 pp 1702ndash1712 SanFrancisco Calif USA April 2003

[17] B Yin H Shi and Y Shang ldquoA two-level topology controlstrategy for energy efficiency in wireless sensor networksrdquoInternational Journal of Wireless and Mobile Computing vol 4no 1 pp 41ndash49 2010

[18] A Sharif V M Potdar and A J D Rathnayaka ldquoLCARTlightweight congestion aware reliable transport protocol forWSN targeting hetero-geneous trafficrdquo in Proceedings of the17th International Conference on Neural Information Processing(ICONIP rsquo10) Sydney Australia November 2010

[19] B Yu J Li and Y Li ldquoDistributed data aggregation schedulingin wireless sensor networksrdquo in Proceedings of the 28th Confer-ence on Computer Communications (IEEE INFOCOM rsquo09) pp2159ndash2167 Rio de Janeiro Brazil April 2009

[20] M X Cheng and L Yin ldquoEnergy-efficient data gatheringalgorithm in sensor networks with partial aggregationrdquo Inter-national Journal of Sensor Networks vol 4 no 1-2 pp 48ndash542008

[21] A A Allen Probability Statistics and Queueing Theory withComputer Science Applications Gulf Professional Publishing1990

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DistributedSensor Networks

International Journal of

6 International Journal of Distributed Sensor Networks

Bringing these equations into (6) we determine the relation-ship between 119875

119894119896and 119875

1198940that

119875119894119896= (

12058210158401015840

119894+1

12058210158401015840

119894+1+ 120582agg

)

119896

1198751198940 (9)

As we knowinfin

sum

119896=1

119875119894119896= 1 (10)

Let

120572 =12058210158401015840

119894+1

12058210158401015840

119894+1+ 120582agg

(11)

Then we determineinfin

sum

119896=1

1205721198961198751198940= 1 (12)

Asinfin

sum

119896=1

120572119896=

1

1 minus 120572 (13)

henceinfin

sum

119896=1

1205721198961198751198940=

1

1 minus 1205721198751198940= 1 (14)

Accordingly we obtain

1198751198940= 1 minus 120572 (15)

Substituting 120572 we determine

1198751198940=

2120582119894+ 120583119894

12058210158401015840

119894+1+ 120582agg

(16)

119875119894119896=

(12058210158401015840

119894+1)119896

+ 120582agg

(12058210158401015840

119894+1+ 120582agg)

119896+1 (17)

Now we determine the amount of data that are received inQueuerx(119894) as

119876rx (119894) =infin

sum

119896=0

119896119875119894119896 (18)

Taking (17) into the above equation we obtain

119876rx (119894) =infin

sum

119896=0

119896

(12058210158401015840

119894+1)119896

+ 120582agg

(12058210158401015840

119894+1+ 120582agg)

119896+1 (19)

We express the above equation as

119876rx (119894) =infin

sum

119896=0

119896(12058210158401015840

119894+1

12058210158401015840

119894+1+ 120582agg

)

119896+1120582agg

12058210158401015840

119894+1

(20)

0

Y

X

Figure 9 Property distribution of119883 119884

Let 120573 = 12058210158401015840

119894+1(12058210158401015840

119894+1+ 120582agg) and 120574 = 120582agg120582

10158401015840

119894+1 then we

simplify (20) as

119876rx (119894) =infin

sum

119896=0

119896120573119896+1

120574 =

infin

sum

119896=0

119896120573119896minus1

1205732120574 = 1205732120574

infin

sum

119896=0

119896120573119896minus1

= 1205732120574120597

120597120573

infin

sum

119896=0

120573119896= 1205732120574120597

120597120573

1

1 minus 120573=

1205732120574

(1 minus 120573)2

(21)

As 120573 = 12058210158401015840119894+1(12058210158401015840

119894+1+ 120582agg) and 120582agg = 2120582119894 + 120583119894 so we get that

1 minus 120573 =2120582119894+ 120583119894

12058210158401015840

119894+1+ 2120582119894+ 120583119894

(22)

Therefore we determine that

1205732120574

(1 minus 120573)2

=

(12058210158401015840

119894+1 (12058210158401015840

119894+1+ 2120582119894+ 120583119894))2

(2120582119894+ 120583119894) 12058210158401015840

119894+1

((2120582119894+ 120583119894) (12058210158401015840

119894+1+ 2120582119894+ 120583119894))2

(23)

By calculating the above equations we determine that

119876rx (119894) =12058210158401015840

119894+1

2120582119894+ 120583119894

(24)

According to Littlersquos formula we determine the event waitingtime as follows

120591119890 (119894) =

119876rx (119894)

2120582119894+ 120583119894

=12058210158401015840

119894+1

(2120582119894+ 120583119894)2 (25)

343 Arrival Process to Queue119905119909 From the analytical process

of Queuerx(119894) we find that arrival data rate 1205821015840119894to Queuerx(119894)

is decided by the waterfalls pushing rate 120583119894and event data

generation rate 120582119894at node 119899

119894 The event generation rate 120582

119894

and waterfalls pushing rate 120583119894abide by discrete distribution

and they are independent of one another To determine theformulation of 1205821015840

119894 we calculate the property distribution of

120582119894and 120583

119894 At first we define that 120582

119894and 120583

119894are independent

distribution 119883 and 119884 as shown in Figure 9 We define thevalues of 119883 as 1 2 119864

119883and the values of 119884 as 1 2 119864

119884

Here we prove that the property 119884 is bigger than119883

International Journal of Distributed Sensor Networks 7

We determine the property as follows 119875119883and 119875

119884denote

the properties when119883 = 119894 and 119884 = 119894

119875 [119884 gt 119883]

= 119875119883 (119883 = 1) 119875119884 (119884 = 2) + 119875119884 (119884 = 3) + sdot sdot sdot + 119875119884 (119884

= 119864119884)

+ 119875119883 (119883

= 2) +119875119884 (119884 = 3) + sdot sdot sdot + 119875119884 (119884 = 119864

119884)

+ 119875119883(119883

= 119864119883) + sdot sdot sdot + 119875

119884(119884 = 119864

119883+ 1)

=

119864119883

sum

119894=1

119875119883 (119883 = 119894)

119864119884

sum

119895=119894+1

119875119884(119884 = 119895)

(26)

When applied to continuous distribution we obtain

119875 [119884 gt 119883] = int

infin

0

119875119883 (119905) [int

infin

0

119875119884 (119911) 119889119911] 119889119905 (27)

Here if 119875119883(119905) and 119875

119884(119905) are exponential distribution then

119875119883 (119905) = 120582119883119890

minus120582119883119905

119875119884 (119905) = 120582119884119890

minus120582119884119905

119875 [119884 gt 119883] = int

infin

0

119875119883 (119905) [int

infin

0

120582119884119890minus120582119884119911119889119911] 119889119905

= int

infin

0

119875119883 (119905) [minus119890

minus120582119884119911]infin

119905119889119905

= int

infin

0

119875119883 (119905) (119890

minus120582119884119905) 119889119905

= int

infin

0

120582119883119890minus120582119883119905119890minus120582119884119905119889119905

= int

infin

0

120582119883119890minus(120582119883+120582119884)119905119889119905

=120582119883

120582119883+ 120582119884

[minus119890minus(120582119883+120582119884)119905]

infin

0=

120582119883

120582119883+ 120582119884

(28)

Accordingly we determine the data arrival rate of Queuetx

1205821015840

119894= 120582119894+

120582119894

120582119894+ 120583119894

120583119894 (29)

344 Service Process Since the data generation rate is Pois-son distribution and the waterfalls pushing rate abides byexponential distribution the data arrival rate to Queuetx(119894)approximates to Poisson distribution and each node has oneserver The ACK packet transmission time is not consideredData aggregating time is very short and negligibleThereforethe service time is one-hop data transmission time 120591 In our

work data transmission rate is 119881119888and generated data size is

119878119894 Hence the service time for generated data is

120591 =119878119894

119881119888

(30)

Since 119881119888and 119878

119894are constant hence the service time

for data is fixed and constant According to the aboveanalysis we can determine that the queuing system on serverapproximates to 1198721198631 model As data transmission timeis determined and server utilization ratio is 120578 = 120582

1015840

119894times 120591tx(119894)

hence the average data transmission time 120591tx(119894) is

120591tx (119894) =120591

1 minus 1205821015840

119894120591 (31)

We figure up the probability density function of the servertime by means of Laplace transform in queuing theory andevaluate server waiting time 120591

119904(119894) that data wait for further

transmitting in server

120591119904 (119894) =

1205821015840

119894(120591tx (119894))

2

2 (1 minus 1205821015840

119894120591tx (119894))

(32)

345 Channel Waiting Time In general queuing systemcustomers can be served if there are no other customerswaiting in front of them However when we apply queuingtheory to model wireless communication it is necessary toconsider the impact of wireless channel caused by its proper-ties In wireless sensor networks because of the overhearingof omnidirectional antenna node has to wait a period oftime if its upper or lower neighbors are transmitting dataHere we call this time as channel waiting time 120591

119888(119894) and it

is determined as follows

120591119888 (119894) = 2 times 120578 times 120591tx (119894) = 2120582

1015840

119894times 1205912

tx (119894) (33)

346 Total Delay Total delay119863119899inWPA is the time interval

from when an event data is generated at a node until thedata are received by a sink node in 119873 hops network Theformulation is as follows

119863119899=

119873

sum

119894=1

(120591119890 (119894) + 120591119904 (119894) + 120591119888 (119894) + 120591tx (119894)) (34)

347 Energy Consumption The energy consumption 119864119894for

node 119899119894is the sum of transmission energy consumption

reception energy consumption and overhearing energy con-sumption Now we determine the three kinds of energyconsumption in the following paragraphs Note that 119875tx and119875rx are energy consumption of transmitting or receiving foreach data packet and 119864tx(119894) 119864rx(119894) and 119864oh(119894) are energyconsumption of data transmitting data receiving and over-hearing of node 119899

119894in 119863119899time As we are trying to obtain

the balanced energy consumption among nodes hence wecalculate energy consumption per 119863

119899time for each node

Therefore we get the data transmission energy consumptionaccording to Littlersquos formula as follows

119864tx (119894) = 1205821015840

119894119863119899119875tx (35)

8 International Journal of Distributed Sensor Networks

The reception energy consumption is

119864rx (119894) = 12058210158401015840

119894+1119863119899119875rx (36)

Here 12058210158401015840119894+1

is data arrival rate from upper neighboring node119899119894+1

as the service time is one data packet transmission timehence the data rate getting out from server is same with datatransmission rate which means

12058210158401015840

119894+1= 1205821015840

119894+1 (37)

The overhearing energy consumption is

119864oh (119894) = 1205821015840

119894minus1119863119899119875rx (38)

Here 1205821015840119894minus1

is data transmission rate of node 119899119894minus1

Hence wedetermine total energy consumption119864

119894for a node in time 119863

119899

as

119864119894= 119864tx (119894) + 119864rx (119894) + 119864oh (119894) (39)

4 Optimum Network Lifetime

41 Mathematical Solution According to the analysis in Sec-tion 32 the longest network lifetime is decided by the totaltransmission delay and waterfalls pushing rate vectors Herewe illustrate the relationship between transmission delayenergy consumption and network lifetime so that at lastwe determine the formulations of a set of waterfalls pushingrates Accordingly energy consumption can be written asfollows

119864119894= (1205821015840

119894119875tx + 120582

10158401015840

119894+1119875rx + 120582

1015840

119894minus1119875rx)

= 1205821015840

119894119875tx + (120582

10158401015840

119894+1+ 1205821015840

119894minus1) 119875rx

(40)

To balance energy among nodes we have

1198641= 1198642= sdot sdot sdot = 119864

119894= sdot sdot sdot = 119864

119899 (41)

As the service time is one-hop data transmitting time hence

12058210158401015840

119894= 1205821015840

119894 (42)

Here 119864119894is the energy consumption for node in time 119863

119899

Hence we assume the energy consumption for each unit oftime is 119864 We assume that 120582

119894= 120582 and the network in a finite

network Hence there are no arrival data to the last node inthe network so that it does not need a waterfalls pushing rateand moreover there is no overhearing for the first node thatcounted from the sink node Therefore for node 119899

119899 we have

120583119899= 0

1205821015840

119899= 120582

119864rx (119899) = 0

119864oh (1) = 0

(43)

Accordingly we obtain equations as follows

(120582 +1205821205831

120582 + 1205831

)119875tx + (120582 +1205821205832

120582 + 1205832

)119875rx = 119864

(120582 +1205821205832

120582 + 1205832

)119875tx + (2120582 +1205821205831

120582 + 1205831

+1205821205833

120582 + 1205833

)119875rx = 119864

(120582 +120582120583119894

120582 + 120583119894

)119875tx

+ (2120582 +120582120583119894minus1

120582 + 120583119894minus1

+120582120583119894+1

120582 + 120583119894+1

)119875rx = 119864

120582119875tx + (120582 +120582120583119899minus1

120582 + 120583119899minus1

)119875rx = 119864

(44)

By calculating the above equations we have

120583119899minus1

=120582119875rx (120582 + 120582120583119899minus2 (120582 + 120583119899minus2))

(120582120583119899minus2

(120582 + 120583119899minus2

)) 119875rx minus 120582119875tx(45)

120583119899minus2

=120582119875rx (120582120583119899minus3 (120582 + 120583119899minus3)) + 120582 (119875rx minus 119875tx) (120582120583119899minus1 (120582 + 120583119899minus1))

(119875rx minus 119875tx) (120582 minus 120582120583119899minus1 (120582 + 120583119899minus1)) minus 119875rx (120582120583119899minus3 (120582 + 120583119899minus3))

(46)

120583119894=

1205822120583119894+2

(119875rx minus 119875tx) (120582 + 120583119894+2) minus 1205822120583119894+1

(119875rx minus 119875tx) (120582 + 120583119894+1) minus 1205822120583119894+3119875rx (120582 + 120583119894+3)

120582120583119894+3119875rx (120582 + 120583119894+3) minus 120582120583119894+2 (119875rx minus 119875tx) (120582 + 120583119894+2) + 120582120583119894+1 (119875rx minus 119875tx) (120582 + 120583119894+1) minus 120582119875rx

1205831=

12058221205832(119875rx minus 119875tx) (120582 + 1205832) minus 120582

21205833119875rx (120582 + 1205833) minus 120582

2119875rx

1205821205832(119875tx minus 119875rx) (120582 + 1205832) minus 1205821205833119875rx (120582 + 1205833) + 2120582119875rx minus 120582119875tx

(47)

International Journal of Distributed Sensor Networks 9

Input 120582 120583119899minus1

120578119863appOutput [120583

1 120583119899minus1

]120583max = 2120582

120583119899minus1

= 120583max[1205831 120583119899minus1] = 119891 (120583

119899minus119894)

119863heu = 119892 (1205831 120583119899minus1)

While |119863app minus 119863heu| gt 120578 do120583119899minus2

= 120583119899minus1

minus Δ

[1205831 120583119899minus1] = 119891(120583

119899minus119894)

119863heu = 119892(1205831 120583119899minus1)

EndReturn [120583

1 120583119899minus1]

Algorithm 1 Heuristics algorithm

From the above equations we obtained the relationshipamong waterfalls pushing rate Equation (45) shows that if120583119899minus1

is given then 120583119899minus2

can be determined In (46) if 120583119899minus1

and120583119899minus2

are given then 120583119899minus3

can be determined In this way all 120583119894

can be determined if 120583119899minus1

is known To determine the valuesof 120583119894 we designed a heuristics algorithm as follows In light of

real applications ofWSNwhere the threshold of transmissiondelay is always given in the heuristics algorithm we assumethat the transmission delay is predefined by applicationHence we set the value of120583

119899minus1and then determine the others

However the value of 120583119899minus1

being too large or too little makesthe algorithm too complex to process Therefore it is criticalto determine the maximum value of 120583

119899minus1

42 Heuristics Algorithm In a given application we assumethat data generation rates at all nodes are known and are 120582Thus at node 119899

119899minus1 if the arrival data from node 119899

119899just does

not keep pace with the generated data then the node needswaterfalls pushing rate to transmit the arrival data Howeverthe processing period of waterfalls pushing rate is decided bynode 119899

119899minus1 hence there is again the case that the arrival data

just does not catch up the pace of the waterfalls pushing rate120583119899minus1

Therefore we determine simply that for the arrival datathe average catching up rate of waterfalls pushing rate at node119899119899minus1

is 1205822 Accordingly for transmitting all arrival data themaximum value of 120583

119899minus1at node 119899

119899minus1is

120583max = 2120582 (48)

Hence a heuristics algorithm is described as shown inAlgorithm 1

5 Evaluation

51 Validation of the Queuing Theory Model We implementWPA in the original network simulator written in C++language Analytic results in the previous section are shownas well as simulated result The parameters are shown inTable 1

In the simulation data occurs at each node randomly andindependently Buffer size of each node is infinite Althoughanalytic model assumes that transmission error is negligible

Table 1 Simulation parameters of WPA

Node distance 10 (m)Transmission range 11 (m)Transmission rate 250 (kbps)Data size 4096 (bit)Current consumption for transmission 174 (mA)Current consumption for reception 197 (mA)MAC CSMACARouting DSR

001

01

1

10

100

1000

01 1 10Data generation rate (datasec)

WPA-analysisWPA-simulation

Tota

l del

ay (120583

sec)

Figure 10 Total delay of WPA

transmission errors and retransmission may occur in thesimulation We set data generation rate 120582 = 5 and waterfallspushing rate as 120583

119894= 1 2 3 4 5 for 5-hop tandem network

From Figures 10 and 11 we find that our analytical modelmeets with simulation results so that we basically confirm thecorrectness of queuing theory model of WPA

52 Evaluation of OptimumNetwork Lifetime In this sectionwe evaluate the effectiveness of the proposedwaterfalls partialaggregation Here we consider network environment asnature monitoring network so that we obtain the maximumnetwork lifetime which is the same with network lifetime offull aggregation We set 119864 in (44) which is equal to 119864ful-largestwhere 119864ful-largest denotes the energy consumption of the nodethat consumes most energy in the network The optimalresults can be achieved by numerical packets however themathematical calculation results are too complex For moreunderstanding of our algorithm we obtain the pushing ratevectors applying Excel when the event generation rate is120582 = 1for 5-hop tandem network Nodes are counted from the sinknode which means that the nearest node with the sink isnode 119899

1and its waterfalls pushing rate is 120583

1 The optimum

waterfalls pushing rates are shown in Table 2 when WPAhas the same network lifetime with full aggregation Theanalytical model of nonaggregation and full aggregation isrealized by queuing theory however for the brevity of thepaper we omitted the derivation process

10 International Journal of Distributed Sensor Networks

0

1000

2000

3000

4000

5000

6000

7000

1 2 3 4 5

Ener

gy co

nsum

ptio

n (m

A)

Node ID

WPA-analysisWPA-simulation

Figure 11 Node energy consumption of WPA

0

05

1

15

2

25

3

35

4

1 2 3 4 5Node ID

NonaggregationPartial aggregationFull aggregation

Ener

gy co

nsum

ptio

n (m

A)

Figure 12 Lifetime of nodes

Table 2 Waterfalls pushing rate for optimum network lifetime

Waterfalls pushing rate Values1205831

0021205832

0041205833

0081205834

011205835

05

Figure 12 shows the lifetime of all nodes in the networkwhere we set the waterfalls pushing rate of nodes as the samewith Table 2 From the figure we find that all nodes almosthave the same lifetime in WPA In this case we can saythat the node energy is utilized optimally in WPA In fullaggregation the second and the third node consumed mostenergy and the other nodes have less energy consumptionwhich results in the suboptimum utilization of energy Thenonaggregation has very large energy consumption at thesecond node which results in the short lifetime of the entirenetwork We find from the figure that nonaggregation and

001

01

1

10

100

01 1 10 100

NonaggregationPartial aggregationFull aggregation

Data generation rate (datasec)

Tota

l del

ay (s

ec)

Figure 13 Transmission delay of network

full aggregation have the largest energy consumption attheir second node The reason is that the first node has noadjacent upper node which means that it does not consumeoverhearing energy compared with others

Figure 13 shows the corresponding total delay of thenetwork where we set the same waterfalls pushing rates withTable 1 From the figure we find that when it has the samenetwork lifetime with full aggregation WPA can shorten thetransmission delay by considerable amount When data gen-eration rate becomes larger (larger than 6 types of datasec)transmission delay of nonaggregation gets longer rapidly dueto data congestion at nodes However in WPA the rapidincrease of transmission delay arises later (at data generationrate of 30 types of datasec) than nonaggregation and fullaggregation on data generation line which indicates thatWPA can relieve data congestion at nodes This is becausewith waterfalls pushing rates proposed WPA can adjust thenumber of data transmissions Moreover we find from thefigure that with the given waterfalls pushing rates there isminimum transmission delay and we can determine it inWPA

53 Summary of Evaluation Firstly from the presentationof Section 51 we conclude that our analytical model ofWPA meets with simulation results the diversity of energyconsumption is that we did not consider retransmissionin analytical model and considered it in simulation Sec-ondly in Section 52 compared to nonaggregation and fullaggregation proposed WPA has the superiorities as followsnonaggregation consumes much energy and data congestionoccurs easily when data generation rate is larger In WPAafter data are sent to a neighboring node the data haveto be aggregated together or aggregated with local data forfurther transmitting hence WPA is more energy efficientthan nonaggregationWhen data generation rate is larger at anode data congestion occurs at nonaggregation which leadsto long transmission delay however proposed WPA canrelieve data congestion via reducing the number of data trans-missions by aggregating received data at nodes On the other

International Journal of Distributed Sensor Networks 11

hand full aggregation can save much energy consumptionand suffer long transmission delay Compared to full aggre-gation proposed WPA has similar energy consumption andis superior in balancing energy consumption among nodesMoreover with waterfalls pushing rates WPA can controldata waiting time more reasonably and efficiently than fullaggregation at nodes so that transmission delay of WPA issuperior to full aggregation

6 Conclusion

In this paper we proposed a waterfalls partial aggregationthat can achieve optimum network lifetime in WSN viaapplying data aggregation At first we modeled WPA inqueuing theory and then determined transmission delay ofthe network and energy consumption of a node Then wecontinued to model balanced energy consumption amongnodes The evaluation parts show the correctness of our ana-lytical model and demonstrate the superiorities of proposedWPA

Competing Interests

The authors declare no competing financial interests

Acknowledgments

Thiswork is supported in part by theNatural Science Founda-tion of Inner Mongolia Autonomous Region of China underGrant no 2014BS0603 and no 2014BS0601 and the Programof Higher-Level Talents of Inner Mongolia University underGrant no 30105-135134

References

[1] C-C Shen C Srisathapornphat and C Jaikaeo ldquoSensor infor-mation networking architecture and applicationsrdquo IEEE Per-sonal Communications vol 8 no 4 pp 52ndash59 2001

[2] S D Glaser ldquoSome real-world applications of wireless sen-sor nodesrdquo in Smart Structures and Materials 2004 Sensorsand Smart Structures Technologies for Civil Mechanical andAerospace Systems vol 5391 of Proceedings of SPIE pp 344ndash355San Diego Calif USA March 2004

[3] I F Akyildiz W Su Y Sankarasubramaniam and E CayircildquoWireless sensor networks a surveyrdquo Computer Networks vol38 no 4 pp 393ndash422 2002

[4] W Ye J Heidemann and D Estrin ldquoSensor-MAC (S-MAC)medium access control for wireless sensor networksrdquo in Pro-ceedings of the 21st International Annual Joint Conference of theIEEEComputer and Communications Societies (INFOCOM rsquo02)New York NY USA June 2002

[5] L J G Villalba A L S Orozco A T Cabrera andC J B AbbasldquoRouting protocols in wireless sensor networksrdquo Sensors vol 9no 11 pp 8399ndash8421 2009

[6] P Santi ldquoTopology control in wireless ad hoc and sensornetworksrdquo ACM Computing Surveys vol 37 no 2 pp 164ndash1942005

[7] R Rajagopalan and P K Varshney ldquoData-aggregation tech-niques in sensor networks a surveyrdquo IEEE CommunicationsSurveys and Tutorials vol 8 no 4 pp 48ndash63 2006

[8] W-S Jung K-W Lim Y-B Ko and S-J Park ldquoA hybridapproach for clustering-based data aggregation in wireless sen-sor networksrdquo in Proceedings of the 3rd International ConferenceonDigital Society (ICDS rsquo09) pp 112ndash117 IEEECancunMexicoFebruary 2009

[9] M Kumar S Verma and P S Pratap ldquoClustering approach todata aggregation in wireless sensor networksrdquo in Proceedingsof the 16th IEEE International Conference on Networks pp 1ndash6IEEE New Delhi India December 2008

[10] S Lindsey C Raghavendra and K M Sivalingam ldquoDatagathering algorithms in sensor networks using energy metricsrdquoIEEE Transactions on Parallel and Distributed Systems vol 13no 9 pp 924ndash935 2002

[11] JW Baek Y J Nam andD-W Seo ldquoATS-DA adaptive timeoutscheduling for data aggregation in wireless sensor networksrdquoin Proceedings of the International Conference on InformationNetworking pp 375ndash384 Estoril Portugal 2008

[12] H Chen H Mineno and T Mizuno ldquoAdaptive data aggrega-tion scheme in clustered wireless sensor networksrdquo ComputerCommunications vol 31 no 15 pp 3579ndash3585 2008

[13] I Solis and K Obraczka ldquoIn-network aggregation trade-offsfor data collection in wireless sensor networksrdquo InternationalJournal of Sensor Networks vol 1 no 3-4 pp 200ndash212 2006

[14] F Mirian and M Sabaei ldquoA delay and accuracy sensitive dataaggregation structure in wireless sensor networksrdquo in Proceed-ings of the International Conference on InformationManagementand Engineering (ICIME rsquo09) pp 231ndash235 IEEE Kuala LumpurMalaysia April 2009

[15] P Sun X Zhang Z Dong et al ldquoA novel energy efficientwireless sensor MAC protocolrdquo in Proceedings of the 4th Inter-national Conference on Networked Computing and AdvancedInformation Management pp 68ndash72 Gyeongju South KoreaSeptember 2008

[16] N Li J C Hou and L Sha ldquoDesign and analysis of an mst-based topology control algorithmrdquo in Proceedings of the IEEESocieties 22nd Annual Joint Conference of the IEEE Computerand Communications (INFOCOM rsquo03) vol 3 pp 1702ndash1712 SanFrancisco Calif USA April 2003

[17] B Yin H Shi and Y Shang ldquoA two-level topology controlstrategy for energy efficiency in wireless sensor networksrdquoInternational Journal of Wireless and Mobile Computing vol 4no 1 pp 41ndash49 2010

[18] A Sharif V M Potdar and A J D Rathnayaka ldquoLCARTlightweight congestion aware reliable transport protocol forWSN targeting hetero-geneous trafficrdquo in Proceedings of the17th International Conference on Neural Information Processing(ICONIP rsquo10) Sydney Australia November 2010

[19] B Yu J Li and Y Li ldquoDistributed data aggregation schedulingin wireless sensor networksrdquo in Proceedings of the 28th Confer-ence on Computer Communications (IEEE INFOCOM rsquo09) pp2159ndash2167 Rio de Janeiro Brazil April 2009

[20] M X Cheng and L Yin ldquoEnergy-efficient data gatheringalgorithm in sensor networks with partial aggregationrdquo Inter-national Journal of Sensor Networks vol 4 no 1-2 pp 48ndash542008

[21] A A Allen Probability Statistics and Queueing Theory withComputer Science Applications Gulf Professional Publishing1990

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DistributedSensor Networks

International Journal of

International Journal of Distributed Sensor Networks 7

We determine the property as follows 119875119883and 119875

119884denote

the properties when119883 = 119894 and 119884 = 119894

119875 [119884 gt 119883]

= 119875119883 (119883 = 1) 119875119884 (119884 = 2) + 119875119884 (119884 = 3) + sdot sdot sdot + 119875119884 (119884

= 119864119884)

+ 119875119883 (119883

= 2) +119875119884 (119884 = 3) + sdot sdot sdot + 119875119884 (119884 = 119864

119884)

+ 119875119883(119883

= 119864119883) + sdot sdot sdot + 119875

119884(119884 = 119864

119883+ 1)

=

119864119883

sum

119894=1

119875119883 (119883 = 119894)

119864119884

sum

119895=119894+1

119875119884(119884 = 119895)

(26)

When applied to continuous distribution we obtain

119875 [119884 gt 119883] = int

infin

0

119875119883 (119905) [int

infin

0

119875119884 (119911) 119889119911] 119889119905 (27)

Here if 119875119883(119905) and 119875

119884(119905) are exponential distribution then

119875119883 (119905) = 120582119883119890

minus120582119883119905

119875119884 (119905) = 120582119884119890

minus120582119884119905

119875 [119884 gt 119883] = int

infin

0

119875119883 (119905) [int

infin

0

120582119884119890minus120582119884119911119889119911] 119889119905

= int

infin

0

119875119883 (119905) [minus119890

minus120582119884119911]infin

119905119889119905

= int

infin

0

119875119883 (119905) (119890

minus120582119884119905) 119889119905

= int

infin

0

120582119883119890minus120582119883119905119890minus120582119884119905119889119905

= int

infin

0

120582119883119890minus(120582119883+120582119884)119905119889119905

=120582119883

120582119883+ 120582119884

[minus119890minus(120582119883+120582119884)119905]

infin

0=

120582119883

120582119883+ 120582119884

(28)

Accordingly we determine the data arrival rate of Queuetx

1205821015840

119894= 120582119894+

120582119894

120582119894+ 120583119894

120583119894 (29)

344 Service Process Since the data generation rate is Pois-son distribution and the waterfalls pushing rate abides byexponential distribution the data arrival rate to Queuetx(119894)approximates to Poisson distribution and each node has oneserver The ACK packet transmission time is not consideredData aggregating time is very short and negligibleThereforethe service time is one-hop data transmission time 120591 In our

work data transmission rate is 119881119888and generated data size is

119878119894 Hence the service time for generated data is

120591 =119878119894

119881119888

(30)

Since 119881119888and 119878

119894are constant hence the service time

for data is fixed and constant According to the aboveanalysis we can determine that the queuing system on serverapproximates to 1198721198631 model As data transmission timeis determined and server utilization ratio is 120578 = 120582

1015840

119894times 120591tx(119894)

hence the average data transmission time 120591tx(119894) is

120591tx (119894) =120591

1 minus 1205821015840

119894120591 (31)

We figure up the probability density function of the servertime by means of Laplace transform in queuing theory andevaluate server waiting time 120591

119904(119894) that data wait for further

transmitting in server

120591119904 (119894) =

1205821015840

119894(120591tx (119894))

2

2 (1 minus 1205821015840

119894120591tx (119894))

(32)

345 Channel Waiting Time In general queuing systemcustomers can be served if there are no other customerswaiting in front of them However when we apply queuingtheory to model wireless communication it is necessary toconsider the impact of wireless channel caused by its proper-ties In wireless sensor networks because of the overhearingof omnidirectional antenna node has to wait a period oftime if its upper or lower neighbors are transmitting dataHere we call this time as channel waiting time 120591

119888(119894) and it

is determined as follows

120591119888 (119894) = 2 times 120578 times 120591tx (119894) = 2120582

1015840

119894times 1205912

tx (119894) (33)

346 Total Delay Total delay119863119899inWPA is the time interval

from when an event data is generated at a node until thedata are received by a sink node in 119873 hops network Theformulation is as follows

119863119899=

119873

sum

119894=1

(120591119890 (119894) + 120591119904 (119894) + 120591119888 (119894) + 120591tx (119894)) (34)

347 Energy Consumption The energy consumption 119864119894for

node 119899119894is the sum of transmission energy consumption

reception energy consumption and overhearing energy con-sumption Now we determine the three kinds of energyconsumption in the following paragraphs Note that 119875tx and119875rx are energy consumption of transmitting or receiving foreach data packet and 119864tx(119894) 119864rx(119894) and 119864oh(119894) are energyconsumption of data transmitting data receiving and over-hearing of node 119899

119894in 119863119899time As we are trying to obtain

the balanced energy consumption among nodes hence wecalculate energy consumption per 119863

119899time for each node

Therefore we get the data transmission energy consumptionaccording to Littlersquos formula as follows

119864tx (119894) = 1205821015840

119894119863119899119875tx (35)

8 International Journal of Distributed Sensor Networks

The reception energy consumption is

119864rx (119894) = 12058210158401015840

119894+1119863119899119875rx (36)

Here 12058210158401015840119894+1

is data arrival rate from upper neighboring node119899119894+1

as the service time is one data packet transmission timehence the data rate getting out from server is same with datatransmission rate which means

12058210158401015840

119894+1= 1205821015840

119894+1 (37)

The overhearing energy consumption is

119864oh (119894) = 1205821015840

119894minus1119863119899119875rx (38)

Here 1205821015840119894minus1

is data transmission rate of node 119899119894minus1

Hence wedetermine total energy consumption119864

119894for a node in time 119863

119899

as

119864119894= 119864tx (119894) + 119864rx (119894) + 119864oh (119894) (39)

4 Optimum Network Lifetime

41 Mathematical Solution According to the analysis in Sec-tion 32 the longest network lifetime is decided by the totaltransmission delay and waterfalls pushing rate vectors Herewe illustrate the relationship between transmission delayenergy consumption and network lifetime so that at lastwe determine the formulations of a set of waterfalls pushingrates Accordingly energy consumption can be written asfollows

119864119894= (1205821015840

119894119875tx + 120582

10158401015840

119894+1119875rx + 120582

1015840

119894minus1119875rx)

= 1205821015840

119894119875tx + (120582

10158401015840

119894+1+ 1205821015840

119894minus1) 119875rx

(40)

To balance energy among nodes we have

1198641= 1198642= sdot sdot sdot = 119864

119894= sdot sdot sdot = 119864

119899 (41)

As the service time is one-hop data transmitting time hence

12058210158401015840

119894= 1205821015840

119894 (42)

Here 119864119894is the energy consumption for node in time 119863

119899

Hence we assume the energy consumption for each unit oftime is 119864 We assume that 120582

119894= 120582 and the network in a finite

network Hence there are no arrival data to the last node inthe network so that it does not need a waterfalls pushing rateand moreover there is no overhearing for the first node thatcounted from the sink node Therefore for node 119899

119899 we have

120583119899= 0

1205821015840

119899= 120582

119864rx (119899) = 0

119864oh (1) = 0

(43)

Accordingly we obtain equations as follows

(120582 +1205821205831

120582 + 1205831

)119875tx + (120582 +1205821205832

120582 + 1205832

)119875rx = 119864

(120582 +1205821205832

120582 + 1205832

)119875tx + (2120582 +1205821205831

120582 + 1205831

+1205821205833

120582 + 1205833

)119875rx = 119864

(120582 +120582120583119894

120582 + 120583119894

)119875tx

+ (2120582 +120582120583119894minus1

120582 + 120583119894minus1

+120582120583119894+1

120582 + 120583119894+1

)119875rx = 119864

120582119875tx + (120582 +120582120583119899minus1

120582 + 120583119899minus1

)119875rx = 119864

(44)

By calculating the above equations we have

120583119899minus1

=120582119875rx (120582 + 120582120583119899minus2 (120582 + 120583119899minus2))

(120582120583119899minus2

(120582 + 120583119899minus2

)) 119875rx minus 120582119875tx(45)

120583119899minus2

=120582119875rx (120582120583119899minus3 (120582 + 120583119899minus3)) + 120582 (119875rx minus 119875tx) (120582120583119899minus1 (120582 + 120583119899minus1))

(119875rx minus 119875tx) (120582 minus 120582120583119899minus1 (120582 + 120583119899minus1)) minus 119875rx (120582120583119899minus3 (120582 + 120583119899minus3))

(46)

120583119894=

1205822120583119894+2

(119875rx minus 119875tx) (120582 + 120583119894+2) minus 1205822120583119894+1

(119875rx minus 119875tx) (120582 + 120583119894+1) minus 1205822120583119894+3119875rx (120582 + 120583119894+3)

120582120583119894+3119875rx (120582 + 120583119894+3) minus 120582120583119894+2 (119875rx minus 119875tx) (120582 + 120583119894+2) + 120582120583119894+1 (119875rx minus 119875tx) (120582 + 120583119894+1) minus 120582119875rx

1205831=

12058221205832(119875rx minus 119875tx) (120582 + 1205832) minus 120582

21205833119875rx (120582 + 1205833) minus 120582

2119875rx

1205821205832(119875tx minus 119875rx) (120582 + 1205832) minus 1205821205833119875rx (120582 + 1205833) + 2120582119875rx minus 120582119875tx

(47)

International Journal of Distributed Sensor Networks 9

Input 120582 120583119899minus1

120578119863appOutput [120583

1 120583119899minus1

]120583max = 2120582

120583119899minus1

= 120583max[1205831 120583119899minus1] = 119891 (120583

119899minus119894)

119863heu = 119892 (1205831 120583119899minus1)

While |119863app minus 119863heu| gt 120578 do120583119899minus2

= 120583119899minus1

minus Δ

[1205831 120583119899minus1] = 119891(120583

119899minus119894)

119863heu = 119892(1205831 120583119899minus1)

EndReturn [120583

1 120583119899minus1]

Algorithm 1 Heuristics algorithm

From the above equations we obtained the relationshipamong waterfalls pushing rate Equation (45) shows that if120583119899minus1

is given then 120583119899minus2

can be determined In (46) if 120583119899minus1

and120583119899minus2

are given then 120583119899minus3

can be determined In this way all 120583119894

can be determined if 120583119899minus1

is known To determine the valuesof 120583119894 we designed a heuristics algorithm as follows In light of

real applications ofWSNwhere the threshold of transmissiondelay is always given in the heuristics algorithm we assumethat the transmission delay is predefined by applicationHence we set the value of120583

119899minus1and then determine the others

However the value of 120583119899minus1

being too large or too little makesthe algorithm too complex to process Therefore it is criticalto determine the maximum value of 120583

119899minus1

42 Heuristics Algorithm In a given application we assumethat data generation rates at all nodes are known and are 120582Thus at node 119899

119899minus1 if the arrival data from node 119899

119899just does

not keep pace with the generated data then the node needswaterfalls pushing rate to transmit the arrival data Howeverthe processing period of waterfalls pushing rate is decided bynode 119899

119899minus1 hence there is again the case that the arrival data

just does not catch up the pace of the waterfalls pushing rate120583119899minus1

Therefore we determine simply that for the arrival datathe average catching up rate of waterfalls pushing rate at node119899119899minus1

is 1205822 Accordingly for transmitting all arrival data themaximum value of 120583

119899minus1at node 119899

119899minus1is

120583max = 2120582 (48)

Hence a heuristics algorithm is described as shown inAlgorithm 1

5 Evaluation

51 Validation of the Queuing Theory Model We implementWPA in the original network simulator written in C++language Analytic results in the previous section are shownas well as simulated result The parameters are shown inTable 1

In the simulation data occurs at each node randomly andindependently Buffer size of each node is infinite Althoughanalytic model assumes that transmission error is negligible

Table 1 Simulation parameters of WPA

Node distance 10 (m)Transmission range 11 (m)Transmission rate 250 (kbps)Data size 4096 (bit)Current consumption for transmission 174 (mA)Current consumption for reception 197 (mA)MAC CSMACARouting DSR

001

01

1

10

100

1000

01 1 10Data generation rate (datasec)

WPA-analysisWPA-simulation

Tota

l del

ay (120583

sec)

Figure 10 Total delay of WPA

transmission errors and retransmission may occur in thesimulation We set data generation rate 120582 = 5 and waterfallspushing rate as 120583

119894= 1 2 3 4 5 for 5-hop tandem network

From Figures 10 and 11 we find that our analytical modelmeets with simulation results so that we basically confirm thecorrectness of queuing theory model of WPA

52 Evaluation of OptimumNetwork Lifetime In this sectionwe evaluate the effectiveness of the proposedwaterfalls partialaggregation Here we consider network environment asnature monitoring network so that we obtain the maximumnetwork lifetime which is the same with network lifetime offull aggregation We set 119864 in (44) which is equal to 119864ful-largestwhere 119864ful-largest denotes the energy consumption of the nodethat consumes most energy in the network The optimalresults can be achieved by numerical packets however themathematical calculation results are too complex For moreunderstanding of our algorithm we obtain the pushing ratevectors applying Excel when the event generation rate is120582 = 1for 5-hop tandem network Nodes are counted from the sinknode which means that the nearest node with the sink isnode 119899

1and its waterfalls pushing rate is 120583

1 The optimum

waterfalls pushing rates are shown in Table 2 when WPAhas the same network lifetime with full aggregation Theanalytical model of nonaggregation and full aggregation isrealized by queuing theory however for the brevity of thepaper we omitted the derivation process

10 International Journal of Distributed Sensor Networks

0

1000

2000

3000

4000

5000

6000

7000

1 2 3 4 5

Ener

gy co

nsum

ptio

n (m

A)

Node ID

WPA-analysisWPA-simulation

Figure 11 Node energy consumption of WPA

0

05

1

15

2

25

3

35

4

1 2 3 4 5Node ID

NonaggregationPartial aggregationFull aggregation

Ener

gy co

nsum

ptio

n (m

A)

Figure 12 Lifetime of nodes

Table 2 Waterfalls pushing rate for optimum network lifetime

Waterfalls pushing rate Values1205831

0021205832

0041205833

0081205834

011205835

05

Figure 12 shows the lifetime of all nodes in the networkwhere we set the waterfalls pushing rate of nodes as the samewith Table 2 From the figure we find that all nodes almosthave the same lifetime in WPA In this case we can saythat the node energy is utilized optimally in WPA In fullaggregation the second and the third node consumed mostenergy and the other nodes have less energy consumptionwhich results in the suboptimum utilization of energy Thenonaggregation has very large energy consumption at thesecond node which results in the short lifetime of the entirenetwork We find from the figure that nonaggregation and

001

01

1

10

100

01 1 10 100

NonaggregationPartial aggregationFull aggregation

Data generation rate (datasec)

Tota

l del

ay (s

ec)

Figure 13 Transmission delay of network

full aggregation have the largest energy consumption attheir second node The reason is that the first node has noadjacent upper node which means that it does not consumeoverhearing energy compared with others

Figure 13 shows the corresponding total delay of thenetwork where we set the same waterfalls pushing rates withTable 1 From the figure we find that when it has the samenetwork lifetime with full aggregation WPA can shorten thetransmission delay by considerable amount When data gen-eration rate becomes larger (larger than 6 types of datasec)transmission delay of nonaggregation gets longer rapidly dueto data congestion at nodes However in WPA the rapidincrease of transmission delay arises later (at data generationrate of 30 types of datasec) than nonaggregation and fullaggregation on data generation line which indicates thatWPA can relieve data congestion at nodes This is becausewith waterfalls pushing rates proposed WPA can adjust thenumber of data transmissions Moreover we find from thefigure that with the given waterfalls pushing rates there isminimum transmission delay and we can determine it inWPA

53 Summary of Evaluation Firstly from the presentationof Section 51 we conclude that our analytical model ofWPA meets with simulation results the diversity of energyconsumption is that we did not consider retransmissionin analytical model and considered it in simulation Sec-ondly in Section 52 compared to nonaggregation and fullaggregation proposed WPA has the superiorities as followsnonaggregation consumes much energy and data congestionoccurs easily when data generation rate is larger In WPAafter data are sent to a neighboring node the data haveto be aggregated together or aggregated with local data forfurther transmitting hence WPA is more energy efficientthan nonaggregationWhen data generation rate is larger at anode data congestion occurs at nonaggregation which leadsto long transmission delay however proposed WPA canrelieve data congestion via reducing the number of data trans-missions by aggregating received data at nodes On the other

International Journal of Distributed Sensor Networks 11

hand full aggregation can save much energy consumptionand suffer long transmission delay Compared to full aggre-gation proposed WPA has similar energy consumption andis superior in balancing energy consumption among nodesMoreover with waterfalls pushing rates WPA can controldata waiting time more reasonably and efficiently than fullaggregation at nodes so that transmission delay of WPA issuperior to full aggregation

6 Conclusion

In this paper we proposed a waterfalls partial aggregationthat can achieve optimum network lifetime in WSN viaapplying data aggregation At first we modeled WPA inqueuing theory and then determined transmission delay ofthe network and energy consumption of a node Then wecontinued to model balanced energy consumption amongnodes The evaluation parts show the correctness of our ana-lytical model and demonstrate the superiorities of proposedWPA

Competing Interests

The authors declare no competing financial interests

Acknowledgments

Thiswork is supported in part by theNatural Science Founda-tion of Inner Mongolia Autonomous Region of China underGrant no 2014BS0603 and no 2014BS0601 and the Programof Higher-Level Talents of Inner Mongolia University underGrant no 30105-135134

References

[1] C-C Shen C Srisathapornphat and C Jaikaeo ldquoSensor infor-mation networking architecture and applicationsrdquo IEEE Per-sonal Communications vol 8 no 4 pp 52ndash59 2001

[2] S D Glaser ldquoSome real-world applications of wireless sen-sor nodesrdquo in Smart Structures and Materials 2004 Sensorsand Smart Structures Technologies for Civil Mechanical andAerospace Systems vol 5391 of Proceedings of SPIE pp 344ndash355San Diego Calif USA March 2004

[3] I F Akyildiz W Su Y Sankarasubramaniam and E CayircildquoWireless sensor networks a surveyrdquo Computer Networks vol38 no 4 pp 393ndash422 2002

[4] W Ye J Heidemann and D Estrin ldquoSensor-MAC (S-MAC)medium access control for wireless sensor networksrdquo in Pro-ceedings of the 21st International Annual Joint Conference of theIEEEComputer and Communications Societies (INFOCOM rsquo02)New York NY USA June 2002

[5] L J G Villalba A L S Orozco A T Cabrera andC J B AbbasldquoRouting protocols in wireless sensor networksrdquo Sensors vol 9no 11 pp 8399ndash8421 2009

[6] P Santi ldquoTopology control in wireless ad hoc and sensornetworksrdquo ACM Computing Surveys vol 37 no 2 pp 164ndash1942005

[7] R Rajagopalan and P K Varshney ldquoData-aggregation tech-niques in sensor networks a surveyrdquo IEEE CommunicationsSurveys and Tutorials vol 8 no 4 pp 48ndash63 2006

[8] W-S Jung K-W Lim Y-B Ko and S-J Park ldquoA hybridapproach for clustering-based data aggregation in wireless sen-sor networksrdquo in Proceedings of the 3rd International ConferenceonDigital Society (ICDS rsquo09) pp 112ndash117 IEEECancunMexicoFebruary 2009

[9] M Kumar S Verma and P S Pratap ldquoClustering approach todata aggregation in wireless sensor networksrdquo in Proceedingsof the 16th IEEE International Conference on Networks pp 1ndash6IEEE New Delhi India December 2008

[10] S Lindsey C Raghavendra and K M Sivalingam ldquoDatagathering algorithms in sensor networks using energy metricsrdquoIEEE Transactions on Parallel and Distributed Systems vol 13no 9 pp 924ndash935 2002

[11] JW Baek Y J Nam andD-W Seo ldquoATS-DA adaptive timeoutscheduling for data aggregation in wireless sensor networksrdquoin Proceedings of the International Conference on InformationNetworking pp 375ndash384 Estoril Portugal 2008

[12] H Chen H Mineno and T Mizuno ldquoAdaptive data aggrega-tion scheme in clustered wireless sensor networksrdquo ComputerCommunications vol 31 no 15 pp 3579ndash3585 2008

[13] I Solis and K Obraczka ldquoIn-network aggregation trade-offsfor data collection in wireless sensor networksrdquo InternationalJournal of Sensor Networks vol 1 no 3-4 pp 200ndash212 2006

[14] F Mirian and M Sabaei ldquoA delay and accuracy sensitive dataaggregation structure in wireless sensor networksrdquo in Proceed-ings of the International Conference on InformationManagementand Engineering (ICIME rsquo09) pp 231ndash235 IEEE Kuala LumpurMalaysia April 2009

[15] P Sun X Zhang Z Dong et al ldquoA novel energy efficientwireless sensor MAC protocolrdquo in Proceedings of the 4th Inter-national Conference on Networked Computing and AdvancedInformation Management pp 68ndash72 Gyeongju South KoreaSeptember 2008

[16] N Li J C Hou and L Sha ldquoDesign and analysis of an mst-based topology control algorithmrdquo in Proceedings of the IEEESocieties 22nd Annual Joint Conference of the IEEE Computerand Communications (INFOCOM rsquo03) vol 3 pp 1702ndash1712 SanFrancisco Calif USA April 2003

[17] B Yin H Shi and Y Shang ldquoA two-level topology controlstrategy for energy efficiency in wireless sensor networksrdquoInternational Journal of Wireless and Mobile Computing vol 4no 1 pp 41ndash49 2010

[18] A Sharif V M Potdar and A J D Rathnayaka ldquoLCARTlightweight congestion aware reliable transport protocol forWSN targeting hetero-geneous trafficrdquo in Proceedings of the17th International Conference on Neural Information Processing(ICONIP rsquo10) Sydney Australia November 2010

[19] B Yu J Li and Y Li ldquoDistributed data aggregation schedulingin wireless sensor networksrdquo in Proceedings of the 28th Confer-ence on Computer Communications (IEEE INFOCOM rsquo09) pp2159ndash2167 Rio de Janeiro Brazil April 2009

[20] M X Cheng and L Yin ldquoEnergy-efficient data gatheringalgorithm in sensor networks with partial aggregationrdquo Inter-national Journal of Sensor Networks vol 4 no 1-2 pp 48ndash542008

[21] A A Allen Probability Statistics and Queueing Theory withComputer Science Applications Gulf Professional Publishing1990

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DistributedSensor Networks

International Journal of

8 International Journal of Distributed Sensor Networks

The reception energy consumption is

119864rx (119894) = 12058210158401015840

119894+1119863119899119875rx (36)

Here 12058210158401015840119894+1

is data arrival rate from upper neighboring node119899119894+1

as the service time is one data packet transmission timehence the data rate getting out from server is same with datatransmission rate which means

12058210158401015840

119894+1= 1205821015840

119894+1 (37)

The overhearing energy consumption is

119864oh (119894) = 1205821015840

119894minus1119863119899119875rx (38)

Here 1205821015840119894minus1

is data transmission rate of node 119899119894minus1

Hence wedetermine total energy consumption119864

119894for a node in time 119863

119899

as

119864119894= 119864tx (119894) + 119864rx (119894) + 119864oh (119894) (39)

4 Optimum Network Lifetime

41 Mathematical Solution According to the analysis in Sec-tion 32 the longest network lifetime is decided by the totaltransmission delay and waterfalls pushing rate vectors Herewe illustrate the relationship between transmission delayenergy consumption and network lifetime so that at lastwe determine the formulations of a set of waterfalls pushingrates Accordingly energy consumption can be written asfollows

119864119894= (1205821015840

119894119875tx + 120582

10158401015840

119894+1119875rx + 120582

1015840

119894minus1119875rx)

= 1205821015840

119894119875tx + (120582

10158401015840

119894+1+ 1205821015840

119894minus1) 119875rx

(40)

To balance energy among nodes we have

1198641= 1198642= sdot sdot sdot = 119864

119894= sdot sdot sdot = 119864

119899 (41)

As the service time is one-hop data transmitting time hence

12058210158401015840

119894= 1205821015840

119894 (42)

Here 119864119894is the energy consumption for node in time 119863

119899

Hence we assume the energy consumption for each unit oftime is 119864 We assume that 120582

119894= 120582 and the network in a finite

network Hence there are no arrival data to the last node inthe network so that it does not need a waterfalls pushing rateand moreover there is no overhearing for the first node thatcounted from the sink node Therefore for node 119899

119899 we have

120583119899= 0

1205821015840

119899= 120582

119864rx (119899) = 0

119864oh (1) = 0

(43)

Accordingly we obtain equations as follows

(120582 +1205821205831

120582 + 1205831

)119875tx + (120582 +1205821205832

120582 + 1205832

)119875rx = 119864

(120582 +1205821205832

120582 + 1205832

)119875tx + (2120582 +1205821205831

120582 + 1205831

+1205821205833

120582 + 1205833

)119875rx = 119864

(120582 +120582120583119894

120582 + 120583119894

)119875tx

+ (2120582 +120582120583119894minus1

120582 + 120583119894minus1

+120582120583119894+1

120582 + 120583119894+1

)119875rx = 119864

120582119875tx + (120582 +120582120583119899minus1

120582 + 120583119899minus1

)119875rx = 119864

(44)

By calculating the above equations we have

120583119899minus1

=120582119875rx (120582 + 120582120583119899minus2 (120582 + 120583119899minus2))

(120582120583119899minus2

(120582 + 120583119899minus2

)) 119875rx minus 120582119875tx(45)

120583119899minus2

=120582119875rx (120582120583119899minus3 (120582 + 120583119899minus3)) + 120582 (119875rx minus 119875tx) (120582120583119899minus1 (120582 + 120583119899minus1))

(119875rx minus 119875tx) (120582 minus 120582120583119899minus1 (120582 + 120583119899minus1)) minus 119875rx (120582120583119899minus3 (120582 + 120583119899minus3))

(46)

120583119894=

1205822120583119894+2

(119875rx minus 119875tx) (120582 + 120583119894+2) minus 1205822120583119894+1

(119875rx minus 119875tx) (120582 + 120583119894+1) minus 1205822120583119894+3119875rx (120582 + 120583119894+3)

120582120583119894+3119875rx (120582 + 120583119894+3) minus 120582120583119894+2 (119875rx minus 119875tx) (120582 + 120583119894+2) + 120582120583119894+1 (119875rx minus 119875tx) (120582 + 120583119894+1) minus 120582119875rx

1205831=

12058221205832(119875rx minus 119875tx) (120582 + 1205832) minus 120582

21205833119875rx (120582 + 1205833) minus 120582

2119875rx

1205821205832(119875tx minus 119875rx) (120582 + 1205832) minus 1205821205833119875rx (120582 + 1205833) + 2120582119875rx minus 120582119875tx

(47)

International Journal of Distributed Sensor Networks 9

Input 120582 120583119899minus1

120578119863appOutput [120583

1 120583119899minus1

]120583max = 2120582

120583119899minus1

= 120583max[1205831 120583119899minus1] = 119891 (120583

119899minus119894)

119863heu = 119892 (1205831 120583119899minus1)

While |119863app minus 119863heu| gt 120578 do120583119899minus2

= 120583119899minus1

minus Δ

[1205831 120583119899minus1] = 119891(120583

119899minus119894)

119863heu = 119892(1205831 120583119899minus1)

EndReturn [120583

1 120583119899minus1]

Algorithm 1 Heuristics algorithm

From the above equations we obtained the relationshipamong waterfalls pushing rate Equation (45) shows that if120583119899minus1

is given then 120583119899minus2

can be determined In (46) if 120583119899minus1

and120583119899minus2

are given then 120583119899minus3

can be determined In this way all 120583119894

can be determined if 120583119899minus1

is known To determine the valuesof 120583119894 we designed a heuristics algorithm as follows In light of

real applications ofWSNwhere the threshold of transmissiondelay is always given in the heuristics algorithm we assumethat the transmission delay is predefined by applicationHence we set the value of120583

119899minus1and then determine the others

However the value of 120583119899minus1

being too large or too little makesthe algorithm too complex to process Therefore it is criticalto determine the maximum value of 120583

119899minus1

42 Heuristics Algorithm In a given application we assumethat data generation rates at all nodes are known and are 120582Thus at node 119899

119899minus1 if the arrival data from node 119899

119899just does

not keep pace with the generated data then the node needswaterfalls pushing rate to transmit the arrival data Howeverthe processing period of waterfalls pushing rate is decided bynode 119899

119899minus1 hence there is again the case that the arrival data

just does not catch up the pace of the waterfalls pushing rate120583119899minus1

Therefore we determine simply that for the arrival datathe average catching up rate of waterfalls pushing rate at node119899119899minus1

is 1205822 Accordingly for transmitting all arrival data themaximum value of 120583

119899minus1at node 119899

119899minus1is

120583max = 2120582 (48)

Hence a heuristics algorithm is described as shown inAlgorithm 1

5 Evaluation

51 Validation of the Queuing Theory Model We implementWPA in the original network simulator written in C++language Analytic results in the previous section are shownas well as simulated result The parameters are shown inTable 1

In the simulation data occurs at each node randomly andindependently Buffer size of each node is infinite Althoughanalytic model assumes that transmission error is negligible

Table 1 Simulation parameters of WPA

Node distance 10 (m)Transmission range 11 (m)Transmission rate 250 (kbps)Data size 4096 (bit)Current consumption for transmission 174 (mA)Current consumption for reception 197 (mA)MAC CSMACARouting DSR

001

01

1

10

100

1000

01 1 10Data generation rate (datasec)

WPA-analysisWPA-simulation

Tota

l del

ay (120583

sec)

Figure 10 Total delay of WPA

transmission errors and retransmission may occur in thesimulation We set data generation rate 120582 = 5 and waterfallspushing rate as 120583

119894= 1 2 3 4 5 for 5-hop tandem network

From Figures 10 and 11 we find that our analytical modelmeets with simulation results so that we basically confirm thecorrectness of queuing theory model of WPA

52 Evaluation of OptimumNetwork Lifetime In this sectionwe evaluate the effectiveness of the proposedwaterfalls partialaggregation Here we consider network environment asnature monitoring network so that we obtain the maximumnetwork lifetime which is the same with network lifetime offull aggregation We set 119864 in (44) which is equal to 119864ful-largestwhere 119864ful-largest denotes the energy consumption of the nodethat consumes most energy in the network The optimalresults can be achieved by numerical packets however themathematical calculation results are too complex For moreunderstanding of our algorithm we obtain the pushing ratevectors applying Excel when the event generation rate is120582 = 1for 5-hop tandem network Nodes are counted from the sinknode which means that the nearest node with the sink isnode 119899

1and its waterfalls pushing rate is 120583

1 The optimum

waterfalls pushing rates are shown in Table 2 when WPAhas the same network lifetime with full aggregation Theanalytical model of nonaggregation and full aggregation isrealized by queuing theory however for the brevity of thepaper we omitted the derivation process

10 International Journal of Distributed Sensor Networks

0

1000

2000

3000

4000

5000

6000

7000

1 2 3 4 5

Ener

gy co

nsum

ptio

n (m

A)

Node ID

WPA-analysisWPA-simulation

Figure 11 Node energy consumption of WPA

0

05

1

15

2

25

3

35

4

1 2 3 4 5Node ID

NonaggregationPartial aggregationFull aggregation

Ener

gy co

nsum

ptio

n (m

A)

Figure 12 Lifetime of nodes

Table 2 Waterfalls pushing rate for optimum network lifetime

Waterfalls pushing rate Values1205831

0021205832

0041205833

0081205834

011205835

05

Figure 12 shows the lifetime of all nodes in the networkwhere we set the waterfalls pushing rate of nodes as the samewith Table 2 From the figure we find that all nodes almosthave the same lifetime in WPA In this case we can saythat the node energy is utilized optimally in WPA In fullaggregation the second and the third node consumed mostenergy and the other nodes have less energy consumptionwhich results in the suboptimum utilization of energy Thenonaggregation has very large energy consumption at thesecond node which results in the short lifetime of the entirenetwork We find from the figure that nonaggregation and

001

01

1

10

100

01 1 10 100

NonaggregationPartial aggregationFull aggregation

Data generation rate (datasec)

Tota

l del

ay (s

ec)

Figure 13 Transmission delay of network

full aggregation have the largest energy consumption attheir second node The reason is that the first node has noadjacent upper node which means that it does not consumeoverhearing energy compared with others

Figure 13 shows the corresponding total delay of thenetwork where we set the same waterfalls pushing rates withTable 1 From the figure we find that when it has the samenetwork lifetime with full aggregation WPA can shorten thetransmission delay by considerable amount When data gen-eration rate becomes larger (larger than 6 types of datasec)transmission delay of nonaggregation gets longer rapidly dueto data congestion at nodes However in WPA the rapidincrease of transmission delay arises later (at data generationrate of 30 types of datasec) than nonaggregation and fullaggregation on data generation line which indicates thatWPA can relieve data congestion at nodes This is becausewith waterfalls pushing rates proposed WPA can adjust thenumber of data transmissions Moreover we find from thefigure that with the given waterfalls pushing rates there isminimum transmission delay and we can determine it inWPA

53 Summary of Evaluation Firstly from the presentationof Section 51 we conclude that our analytical model ofWPA meets with simulation results the diversity of energyconsumption is that we did not consider retransmissionin analytical model and considered it in simulation Sec-ondly in Section 52 compared to nonaggregation and fullaggregation proposed WPA has the superiorities as followsnonaggregation consumes much energy and data congestionoccurs easily when data generation rate is larger In WPAafter data are sent to a neighboring node the data haveto be aggregated together or aggregated with local data forfurther transmitting hence WPA is more energy efficientthan nonaggregationWhen data generation rate is larger at anode data congestion occurs at nonaggregation which leadsto long transmission delay however proposed WPA canrelieve data congestion via reducing the number of data trans-missions by aggregating received data at nodes On the other

International Journal of Distributed Sensor Networks 11

hand full aggregation can save much energy consumptionand suffer long transmission delay Compared to full aggre-gation proposed WPA has similar energy consumption andis superior in balancing energy consumption among nodesMoreover with waterfalls pushing rates WPA can controldata waiting time more reasonably and efficiently than fullaggregation at nodes so that transmission delay of WPA issuperior to full aggregation

6 Conclusion

In this paper we proposed a waterfalls partial aggregationthat can achieve optimum network lifetime in WSN viaapplying data aggregation At first we modeled WPA inqueuing theory and then determined transmission delay ofthe network and energy consumption of a node Then wecontinued to model balanced energy consumption amongnodes The evaluation parts show the correctness of our ana-lytical model and demonstrate the superiorities of proposedWPA

Competing Interests

The authors declare no competing financial interests

Acknowledgments

Thiswork is supported in part by theNatural Science Founda-tion of Inner Mongolia Autonomous Region of China underGrant no 2014BS0603 and no 2014BS0601 and the Programof Higher-Level Talents of Inner Mongolia University underGrant no 30105-135134

References

[1] C-C Shen C Srisathapornphat and C Jaikaeo ldquoSensor infor-mation networking architecture and applicationsrdquo IEEE Per-sonal Communications vol 8 no 4 pp 52ndash59 2001

[2] S D Glaser ldquoSome real-world applications of wireless sen-sor nodesrdquo in Smart Structures and Materials 2004 Sensorsand Smart Structures Technologies for Civil Mechanical andAerospace Systems vol 5391 of Proceedings of SPIE pp 344ndash355San Diego Calif USA March 2004

[3] I F Akyildiz W Su Y Sankarasubramaniam and E CayircildquoWireless sensor networks a surveyrdquo Computer Networks vol38 no 4 pp 393ndash422 2002

[4] W Ye J Heidemann and D Estrin ldquoSensor-MAC (S-MAC)medium access control for wireless sensor networksrdquo in Pro-ceedings of the 21st International Annual Joint Conference of theIEEEComputer and Communications Societies (INFOCOM rsquo02)New York NY USA June 2002

[5] L J G Villalba A L S Orozco A T Cabrera andC J B AbbasldquoRouting protocols in wireless sensor networksrdquo Sensors vol 9no 11 pp 8399ndash8421 2009

[6] P Santi ldquoTopology control in wireless ad hoc and sensornetworksrdquo ACM Computing Surveys vol 37 no 2 pp 164ndash1942005

[7] R Rajagopalan and P K Varshney ldquoData-aggregation tech-niques in sensor networks a surveyrdquo IEEE CommunicationsSurveys and Tutorials vol 8 no 4 pp 48ndash63 2006

[8] W-S Jung K-W Lim Y-B Ko and S-J Park ldquoA hybridapproach for clustering-based data aggregation in wireless sen-sor networksrdquo in Proceedings of the 3rd International ConferenceonDigital Society (ICDS rsquo09) pp 112ndash117 IEEECancunMexicoFebruary 2009

[9] M Kumar S Verma and P S Pratap ldquoClustering approach todata aggregation in wireless sensor networksrdquo in Proceedingsof the 16th IEEE International Conference on Networks pp 1ndash6IEEE New Delhi India December 2008

[10] S Lindsey C Raghavendra and K M Sivalingam ldquoDatagathering algorithms in sensor networks using energy metricsrdquoIEEE Transactions on Parallel and Distributed Systems vol 13no 9 pp 924ndash935 2002

[11] JW Baek Y J Nam andD-W Seo ldquoATS-DA adaptive timeoutscheduling for data aggregation in wireless sensor networksrdquoin Proceedings of the International Conference on InformationNetworking pp 375ndash384 Estoril Portugal 2008

[12] H Chen H Mineno and T Mizuno ldquoAdaptive data aggrega-tion scheme in clustered wireless sensor networksrdquo ComputerCommunications vol 31 no 15 pp 3579ndash3585 2008

[13] I Solis and K Obraczka ldquoIn-network aggregation trade-offsfor data collection in wireless sensor networksrdquo InternationalJournal of Sensor Networks vol 1 no 3-4 pp 200ndash212 2006

[14] F Mirian and M Sabaei ldquoA delay and accuracy sensitive dataaggregation structure in wireless sensor networksrdquo in Proceed-ings of the International Conference on InformationManagementand Engineering (ICIME rsquo09) pp 231ndash235 IEEE Kuala LumpurMalaysia April 2009

[15] P Sun X Zhang Z Dong et al ldquoA novel energy efficientwireless sensor MAC protocolrdquo in Proceedings of the 4th Inter-national Conference on Networked Computing and AdvancedInformation Management pp 68ndash72 Gyeongju South KoreaSeptember 2008

[16] N Li J C Hou and L Sha ldquoDesign and analysis of an mst-based topology control algorithmrdquo in Proceedings of the IEEESocieties 22nd Annual Joint Conference of the IEEE Computerand Communications (INFOCOM rsquo03) vol 3 pp 1702ndash1712 SanFrancisco Calif USA April 2003

[17] B Yin H Shi and Y Shang ldquoA two-level topology controlstrategy for energy efficiency in wireless sensor networksrdquoInternational Journal of Wireless and Mobile Computing vol 4no 1 pp 41ndash49 2010

[18] A Sharif V M Potdar and A J D Rathnayaka ldquoLCARTlightweight congestion aware reliable transport protocol forWSN targeting hetero-geneous trafficrdquo in Proceedings of the17th International Conference on Neural Information Processing(ICONIP rsquo10) Sydney Australia November 2010

[19] B Yu J Li and Y Li ldquoDistributed data aggregation schedulingin wireless sensor networksrdquo in Proceedings of the 28th Confer-ence on Computer Communications (IEEE INFOCOM rsquo09) pp2159ndash2167 Rio de Janeiro Brazil April 2009

[20] M X Cheng and L Yin ldquoEnergy-efficient data gatheringalgorithm in sensor networks with partial aggregationrdquo Inter-national Journal of Sensor Networks vol 4 no 1-2 pp 48ndash542008

[21] A A Allen Probability Statistics and Queueing Theory withComputer Science Applications Gulf Professional Publishing1990

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of Distributed Sensor Networks 9

Input 120582 120583119899minus1

120578119863appOutput [120583

1 120583119899minus1

]120583max = 2120582

120583119899minus1

= 120583max[1205831 120583119899minus1] = 119891 (120583

119899minus119894)

119863heu = 119892 (1205831 120583119899minus1)

While |119863app minus 119863heu| gt 120578 do120583119899minus2

= 120583119899minus1

minus Δ

[1205831 120583119899minus1] = 119891(120583

119899minus119894)

119863heu = 119892(1205831 120583119899minus1)

EndReturn [120583

1 120583119899minus1]

Algorithm 1 Heuristics algorithm

From the above equations we obtained the relationshipamong waterfalls pushing rate Equation (45) shows that if120583119899minus1

is given then 120583119899minus2

can be determined In (46) if 120583119899minus1

and120583119899minus2

are given then 120583119899minus3

can be determined In this way all 120583119894

can be determined if 120583119899minus1

is known To determine the valuesof 120583119894 we designed a heuristics algorithm as follows In light of

real applications ofWSNwhere the threshold of transmissiondelay is always given in the heuristics algorithm we assumethat the transmission delay is predefined by applicationHence we set the value of120583

119899minus1and then determine the others

However the value of 120583119899minus1

being too large or too little makesthe algorithm too complex to process Therefore it is criticalto determine the maximum value of 120583

119899minus1

42 Heuristics Algorithm In a given application we assumethat data generation rates at all nodes are known and are 120582Thus at node 119899

119899minus1 if the arrival data from node 119899

119899just does

not keep pace with the generated data then the node needswaterfalls pushing rate to transmit the arrival data Howeverthe processing period of waterfalls pushing rate is decided bynode 119899

119899minus1 hence there is again the case that the arrival data

just does not catch up the pace of the waterfalls pushing rate120583119899minus1

Therefore we determine simply that for the arrival datathe average catching up rate of waterfalls pushing rate at node119899119899minus1

is 1205822 Accordingly for transmitting all arrival data themaximum value of 120583

119899minus1at node 119899

119899minus1is

120583max = 2120582 (48)

Hence a heuristics algorithm is described as shown inAlgorithm 1

5 Evaluation

51 Validation of the Queuing Theory Model We implementWPA in the original network simulator written in C++language Analytic results in the previous section are shownas well as simulated result The parameters are shown inTable 1

In the simulation data occurs at each node randomly andindependently Buffer size of each node is infinite Althoughanalytic model assumes that transmission error is negligible

Table 1 Simulation parameters of WPA

Node distance 10 (m)Transmission range 11 (m)Transmission rate 250 (kbps)Data size 4096 (bit)Current consumption for transmission 174 (mA)Current consumption for reception 197 (mA)MAC CSMACARouting DSR

001

01

1

10

100

1000

01 1 10Data generation rate (datasec)

WPA-analysisWPA-simulation

Tota

l del

ay (120583

sec)

Figure 10 Total delay of WPA

transmission errors and retransmission may occur in thesimulation We set data generation rate 120582 = 5 and waterfallspushing rate as 120583

119894= 1 2 3 4 5 for 5-hop tandem network

From Figures 10 and 11 we find that our analytical modelmeets with simulation results so that we basically confirm thecorrectness of queuing theory model of WPA

52 Evaluation of OptimumNetwork Lifetime In this sectionwe evaluate the effectiveness of the proposedwaterfalls partialaggregation Here we consider network environment asnature monitoring network so that we obtain the maximumnetwork lifetime which is the same with network lifetime offull aggregation We set 119864 in (44) which is equal to 119864ful-largestwhere 119864ful-largest denotes the energy consumption of the nodethat consumes most energy in the network The optimalresults can be achieved by numerical packets however themathematical calculation results are too complex For moreunderstanding of our algorithm we obtain the pushing ratevectors applying Excel when the event generation rate is120582 = 1for 5-hop tandem network Nodes are counted from the sinknode which means that the nearest node with the sink isnode 119899

1and its waterfalls pushing rate is 120583

1 The optimum

waterfalls pushing rates are shown in Table 2 when WPAhas the same network lifetime with full aggregation Theanalytical model of nonaggregation and full aggregation isrealized by queuing theory however for the brevity of thepaper we omitted the derivation process

10 International Journal of Distributed Sensor Networks

0

1000

2000

3000

4000

5000

6000

7000

1 2 3 4 5

Ener

gy co

nsum

ptio

n (m

A)

Node ID

WPA-analysisWPA-simulation

Figure 11 Node energy consumption of WPA

0

05

1

15

2

25

3

35

4

1 2 3 4 5Node ID

NonaggregationPartial aggregationFull aggregation

Ener

gy co

nsum

ptio

n (m

A)

Figure 12 Lifetime of nodes

Table 2 Waterfalls pushing rate for optimum network lifetime

Waterfalls pushing rate Values1205831

0021205832

0041205833

0081205834

011205835

05

Figure 12 shows the lifetime of all nodes in the networkwhere we set the waterfalls pushing rate of nodes as the samewith Table 2 From the figure we find that all nodes almosthave the same lifetime in WPA In this case we can saythat the node energy is utilized optimally in WPA In fullaggregation the second and the third node consumed mostenergy and the other nodes have less energy consumptionwhich results in the suboptimum utilization of energy Thenonaggregation has very large energy consumption at thesecond node which results in the short lifetime of the entirenetwork We find from the figure that nonaggregation and

001

01

1

10

100

01 1 10 100

NonaggregationPartial aggregationFull aggregation

Data generation rate (datasec)

Tota

l del

ay (s

ec)

Figure 13 Transmission delay of network

full aggregation have the largest energy consumption attheir second node The reason is that the first node has noadjacent upper node which means that it does not consumeoverhearing energy compared with others

Figure 13 shows the corresponding total delay of thenetwork where we set the same waterfalls pushing rates withTable 1 From the figure we find that when it has the samenetwork lifetime with full aggregation WPA can shorten thetransmission delay by considerable amount When data gen-eration rate becomes larger (larger than 6 types of datasec)transmission delay of nonaggregation gets longer rapidly dueto data congestion at nodes However in WPA the rapidincrease of transmission delay arises later (at data generationrate of 30 types of datasec) than nonaggregation and fullaggregation on data generation line which indicates thatWPA can relieve data congestion at nodes This is becausewith waterfalls pushing rates proposed WPA can adjust thenumber of data transmissions Moreover we find from thefigure that with the given waterfalls pushing rates there isminimum transmission delay and we can determine it inWPA

53 Summary of Evaluation Firstly from the presentationof Section 51 we conclude that our analytical model ofWPA meets with simulation results the diversity of energyconsumption is that we did not consider retransmissionin analytical model and considered it in simulation Sec-ondly in Section 52 compared to nonaggregation and fullaggregation proposed WPA has the superiorities as followsnonaggregation consumes much energy and data congestionoccurs easily when data generation rate is larger In WPAafter data are sent to a neighboring node the data haveto be aggregated together or aggregated with local data forfurther transmitting hence WPA is more energy efficientthan nonaggregationWhen data generation rate is larger at anode data congestion occurs at nonaggregation which leadsto long transmission delay however proposed WPA canrelieve data congestion via reducing the number of data trans-missions by aggregating received data at nodes On the other

International Journal of Distributed Sensor Networks 11

hand full aggregation can save much energy consumptionand suffer long transmission delay Compared to full aggre-gation proposed WPA has similar energy consumption andis superior in balancing energy consumption among nodesMoreover with waterfalls pushing rates WPA can controldata waiting time more reasonably and efficiently than fullaggregation at nodes so that transmission delay of WPA issuperior to full aggregation

6 Conclusion

In this paper we proposed a waterfalls partial aggregationthat can achieve optimum network lifetime in WSN viaapplying data aggregation At first we modeled WPA inqueuing theory and then determined transmission delay ofthe network and energy consumption of a node Then wecontinued to model balanced energy consumption amongnodes The evaluation parts show the correctness of our ana-lytical model and demonstrate the superiorities of proposedWPA

Competing Interests

The authors declare no competing financial interests

Acknowledgments

Thiswork is supported in part by theNatural Science Founda-tion of Inner Mongolia Autonomous Region of China underGrant no 2014BS0603 and no 2014BS0601 and the Programof Higher-Level Talents of Inner Mongolia University underGrant no 30105-135134

References

[1] C-C Shen C Srisathapornphat and C Jaikaeo ldquoSensor infor-mation networking architecture and applicationsrdquo IEEE Per-sonal Communications vol 8 no 4 pp 52ndash59 2001

[2] S D Glaser ldquoSome real-world applications of wireless sen-sor nodesrdquo in Smart Structures and Materials 2004 Sensorsand Smart Structures Technologies for Civil Mechanical andAerospace Systems vol 5391 of Proceedings of SPIE pp 344ndash355San Diego Calif USA March 2004

[3] I F Akyildiz W Su Y Sankarasubramaniam and E CayircildquoWireless sensor networks a surveyrdquo Computer Networks vol38 no 4 pp 393ndash422 2002

[4] W Ye J Heidemann and D Estrin ldquoSensor-MAC (S-MAC)medium access control for wireless sensor networksrdquo in Pro-ceedings of the 21st International Annual Joint Conference of theIEEEComputer and Communications Societies (INFOCOM rsquo02)New York NY USA June 2002

[5] L J G Villalba A L S Orozco A T Cabrera andC J B AbbasldquoRouting protocols in wireless sensor networksrdquo Sensors vol 9no 11 pp 8399ndash8421 2009

[6] P Santi ldquoTopology control in wireless ad hoc and sensornetworksrdquo ACM Computing Surveys vol 37 no 2 pp 164ndash1942005

[7] R Rajagopalan and P K Varshney ldquoData-aggregation tech-niques in sensor networks a surveyrdquo IEEE CommunicationsSurveys and Tutorials vol 8 no 4 pp 48ndash63 2006

[8] W-S Jung K-W Lim Y-B Ko and S-J Park ldquoA hybridapproach for clustering-based data aggregation in wireless sen-sor networksrdquo in Proceedings of the 3rd International ConferenceonDigital Society (ICDS rsquo09) pp 112ndash117 IEEECancunMexicoFebruary 2009

[9] M Kumar S Verma and P S Pratap ldquoClustering approach todata aggregation in wireless sensor networksrdquo in Proceedingsof the 16th IEEE International Conference on Networks pp 1ndash6IEEE New Delhi India December 2008

[10] S Lindsey C Raghavendra and K M Sivalingam ldquoDatagathering algorithms in sensor networks using energy metricsrdquoIEEE Transactions on Parallel and Distributed Systems vol 13no 9 pp 924ndash935 2002

[11] JW Baek Y J Nam andD-W Seo ldquoATS-DA adaptive timeoutscheduling for data aggregation in wireless sensor networksrdquoin Proceedings of the International Conference on InformationNetworking pp 375ndash384 Estoril Portugal 2008

[12] H Chen H Mineno and T Mizuno ldquoAdaptive data aggrega-tion scheme in clustered wireless sensor networksrdquo ComputerCommunications vol 31 no 15 pp 3579ndash3585 2008

[13] I Solis and K Obraczka ldquoIn-network aggregation trade-offsfor data collection in wireless sensor networksrdquo InternationalJournal of Sensor Networks vol 1 no 3-4 pp 200ndash212 2006

[14] F Mirian and M Sabaei ldquoA delay and accuracy sensitive dataaggregation structure in wireless sensor networksrdquo in Proceed-ings of the International Conference on InformationManagementand Engineering (ICIME rsquo09) pp 231ndash235 IEEE Kuala LumpurMalaysia April 2009

[15] P Sun X Zhang Z Dong et al ldquoA novel energy efficientwireless sensor MAC protocolrdquo in Proceedings of the 4th Inter-national Conference on Networked Computing and AdvancedInformation Management pp 68ndash72 Gyeongju South KoreaSeptember 2008

[16] N Li J C Hou and L Sha ldquoDesign and analysis of an mst-based topology control algorithmrdquo in Proceedings of the IEEESocieties 22nd Annual Joint Conference of the IEEE Computerand Communications (INFOCOM rsquo03) vol 3 pp 1702ndash1712 SanFrancisco Calif USA April 2003

[17] B Yin H Shi and Y Shang ldquoA two-level topology controlstrategy for energy efficiency in wireless sensor networksrdquoInternational Journal of Wireless and Mobile Computing vol 4no 1 pp 41ndash49 2010

[18] A Sharif V M Potdar and A J D Rathnayaka ldquoLCARTlightweight congestion aware reliable transport protocol forWSN targeting hetero-geneous trafficrdquo in Proceedings of the17th International Conference on Neural Information Processing(ICONIP rsquo10) Sydney Australia November 2010

[19] B Yu J Li and Y Li ldquoDistributed data aggregation schedulingin wireless sensor networksrdquo in Proceedings of the 28th Confer-ence on Computer Communications (IEEE INFOCOM rsquo09) pp2159ndash2167 Rio de Janeiro Brazil April 2009

[20] M X Cheng and L Yin ldquoEnergy-efficient data gatheringalgorithm in sensor networks with partial aggregationrdquo Inter-national Journal of Sensor Networks vol 4 no 1-2 pp 48ndash542008

[21] A A Allen Probability Statistics and Queueing Theory withComputer Science Applications Gulf Professional Publishing1990

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 International Journal of Distributed Sensor Networks

0

1000

2000

3000

4000

5000

6000

7000

1 2 3 4 5

Ener

gy co

nsum

ptio

n (m

A)

Node ID

WPA-analysisWPA-simulation

Figure 11 Node energy consumption of WPA

0

05

1

15

2

25

3

35

4

1 2 3 4 5Node ID

NonaggregationPartial aggregationFull aggregation

Ener

gy co

nsum

ptio

n (m

A)

Figure 12 Lifetime of nodes

Table 2 Waterfalls pushing rate for optimum network lifetime

Waterfalls pushing rate Values1205831

0021205832

0041205833

0081205834

011205835

05

Figure 12 shows the lifetime of all nodes in the networkwhere we set the waterfalls pushing rate of nodes as the samewith Table 2 From the figure we find that all nodes almosthave the same lifetime in WPA In this case we can saythat the node energy is utilized optimally in WPA In fullaggregation the second and the third node consumed mostenergy and the other nodes have less energy consumptionwhich results in the suboptimum utilization of energy Thenonaggregation has very large energy consumption at thesecond node which results in the short lifetime of the entirenetwork We find from the figure that nonaggregation and

001

01

1

10

100

01 1 10 100

NonaggregationPartial aggregationFull aggregation

Data generation rate (datasec)

Tota

l del

ay (s

ec)

Figure 13 Transmission delay of network

full aggregation have the largest energy consumption attheir second node The reason is that the first node has noadjacent upper node which means that it does not consumeoverhearing energy compared with others

Figure 13 shows the corresponding total delay of thenetwork where we set the same waterfalls pushing rates withTable 1 From the figure we find that when it has the samenetwork lifetime with full aggregation WPA can shorten thetransmission delay by considerable amount When data gen-eration rate becomes larger (larger than 6 types of datasec)transmission delay of nonaggregation gets longer rapidly dueto data congestion at nodes However in WPA the rapidincrease of transmission delay arises later (at data generationrate of 30 types of datasec) than nonaggregation and fullaggregation on data generation line which indicates thatWPA can relieve data congestion at nodes This is becausewith waterfalls pushing rates proposed WPA can adjust thenumber of data transmissions Moreover we find from thefigure that with the given waterfalls pushing rates there isminimum transmission delay and we can determine it inWPA

53 Summary of Evaluation Firstly from the presentationof Section 51 we conclude that our analytical model ofWPA meets with simulation results the diversity of energyconsumption is that we did not consider retransmissionin analytical model and considered it in simulation Sec-ondly in Section 52 compared to nonaggregation and fullaggregation proposed WPA has the superiorities as followsnonaggregation consumes much energy and data congestionoccurs easily when data generation rate is larger In WPAafter data are sent to a neighboring node the data haveto be aggregated together or aggregated with local data forfurther transmitting hence WPA is more energy efficientthan nonaggregationWhen data generation rate is larger at anode data congestion occurs at nonaggregation which leadsto long transmission delay however proposed WPA canrelieve data congestion via reducing the number of data trans-missions by aggregating received data at nodes On the other

International Journal of Distributed Sensor Networks 11

hand full aggregation can save much energy consumptionand suffer long transmission delay Compared to full aggre-gation proposed WPA has similar energy consumption andis superior in balancing energy consumption among nodesMoreover with waterfalls pushing rates WPA can controldata waiting time more reasonably and efficiently than fullaggregation at nodes so that transmission delay of WPA issuperior to full aggregation

6 Conclusion

In this paper we proposed a waterfalls partial aggregationthat can achieve optimum network lifetime in WSN viaapplying data aggregation At first we modeled WPA inqueuing theory and then determined transmission delay ofthe network and energy consumption of a node Then wecontinued to model balanced energy consumption amongnodes The evaluation parts show the correctness of our ana-lytical model and demonstrate the superiorities of proposedWPA

Competing Interests

The authors declare no competing financial interests

Acknowledgments

Thiswork is supported in part by theNatural Science Founda-tion of Inner Mongolia Autonomous Region of China underGrant no 2014BS0603 and no 2014BS0601 and the Programof Higher-Level Talents of Inner Mongolia University underGrant no 30105-135134

References

[1] C-C Shen C Srisathapornphat and C Jaikaeo ldquoSensor infor-mation networking architecture and applicationsrdquo IEEE Per-sonal Communications vol 8 no 4 pp 52ndash59 2001

[2] S D Glaser ldquoSome real-world applications of wireless sen-sor nodesrdquo in Smart Structures and Materials 2004 Sensorsand Smart Structures Technologies for Civil Mechanical andAerospace Systems vol 5391 of Proceedings of SPIE pp 344ndash355San Diego Calif USA March 2004

[3] I F Akyildiz W Su Y Sankarasubramaniam and E CayircildquoWireless sensor networks a surveyrdquo Computer Networks vol38 no 4 pp 393ndash422 2002

[4] W Ye J Heidemann and D Estrin ldquoSensor-MAC (S-MAC)medium access control for wireless sensor networksrdquo in Pro-ceedings of the 21st International Annual Joint Conference of theIEEEComputer and Communications Societies (INFOCOM rsquo02)New York NY USA June 2002

[5] L J G Villalba A L S Orozco A T Cabrera andC J B AbbasldquoRouting protocols in wireless sensor networksrdquo Sensors vol 9no 11 pp 8399ndash8421 2009

[6] P Santi ldquoTopology control in wireless ad hoc and sensornetworksrdquo ACM Computing Surveys vol 37 no 2 pp 164ndash1942005

[7] R Rajagopalan and P K Varshney ldquoData-aggregation tech-niques in sensor networks a surveyrdquo IEEE CommunicationsSurveys and Tutorials vol 8 no 4 pp 48ndash63 2006

[8] W-S Jung K-W Lim Y-B Ko and S-J Park ldquoA hybridapproach for clustering-based data aggregation in wireless sen-sor networksrdquo in Proceedings of the 3rd International ConferenceonDigital Society (ICDS rsquo09) pp 112ndash117 IEEECancunMexicoFebruary 2009

[9] M Kumar S Verma and P S Pratap ldquoClustering approach todata aggregation in wireless sensor networksrdquo in Proceedingsof the 16th IEEE International Conference on Networks pp 1ndash6IEEE New Delhi India December 2008

[10] S Lindsey C Raghavendra and K M Sivalingam ldquoDatagathering algorithms in sensor networks using energy metricsrdquoIEEE Transactions on Parallel and Distributed Systems vol 13no 9 pp 924ndash935 2002

[11] JW Baek Y J Nam andD-W Seo ldquoATS-DA adaptive timeoutscheduling for data aggregation in wireless sensor networksrdquoin Proceedings of the International Conference on InformationNetworking pp 375ndash384 Estoril Portugal 2008

[12] H Chen H Mineno and T Mizuno ldquoAdaptive data aggrega-tion scheme in clustered wireless sensor networksrdquo ComputerCommunications vol 31 no 15 pp 3579ndash3585 2008

[13] I Solis and K Obraczka ldquoIn-network aggregation trade-offsfor data collection in wireless sensor networksrdquo InternationalJournal of Sensor Networks vol 1 no 3-4 pp 200ndash212 2006

[14] F Mirian and M Sabaei ldquoA delay and accuracy sensitive dataaggregation structure in wireless sensor networksrdquo in Proceed-ings of the International Conference on InformationManagementand Engineering (ICIME rsquo09) pp 231ndash235 IEEE Kuala LumpurMalaysia April 2009

[15] P Sun X Zhang Z Dong et al ldquoA novel energy efficientwireless sensor MAC protocolrdquo in Proceedings of the 4th Inter-national Conference on Networked Computing and AdvancedInformation Management pp 68ndash72 Gyeongju South KoreaSeptember 2008

[16] N Li J C Hou and L Sha ldquoDesign and analysis of an mst-based topology control algorithmrdquo in Proceedings of the IEEESocieties 22nd Annual Joint Conference of the IEEE Computerand Communications (INFOCOM rsquo03) vol 3 pp 1702ndash1712 SanFrancisco Calif USA April 2003

[17] B Yin H Shi and Y Shang ldquoA two-level topology controlstrategy for energy efficiency in wireless sensor networksrdquoInternational Journal of Wireless and Mobile Computing vol 4no 1 pp 41ndash49 2010

[18] A Sharif V M Potdar and A J D Rathnayaka ldquoLCARTlightweight congestion aware reliable transport protocol forWSN targeting hetero-geneous trafficrdquo in Proceedings of the17th International Conference on Neural Information Processing(ICONIP rsquo10) Sydney Australia November 2010

[19] B Yu J Li and Y Li ldquoDistributed data aggregation schedulingin wireless sensor networksrdquo in Proceedings of the 28th Confer-ence on Computer Communications (IEEE INFOCOM rsquo09) pp2159ndash2167 Rio de Janeiro Brazil April 2009

[20] M X Cheng and L Yin ldquoEnergy-efficient data gatheringalgorithm in sensor networks with partial aggregationrdquo Inter-national Journal of Sensor Networks vol 4 no 1-2 pp 48ndash542008

[21] A A Allen Probability Statistics and Queueing Theory withComputer Science Applications Gulf Professional Publishing1990

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of Distributed Sensor Networks 11

hand full aggregation can save much energy consumptionand suffer long transmission delay Compared to full aggre-gation proposed WPA has similar energy consumption andis superior in balancing energy consumption among nodesMoreover with waterfalls pushing rates WPA can controldata waiting time more reasonably and efficiently than fullaggregation at nodes so that transmission delay of WPA issuperior to full aggregation

6 Conclusion

In this paper we proposed a waterfalls partial aggregationthat can achieve optimum network lifetime in WSN viaapplying data aggregation At first we modeled WPA inqueuing theory and then determined transmission delay ofthe network and energy consumption of a node Then wecontinued to model balanced energy consumption amongnodes The evaluation parts show the correctness of our ana-lytical model and demonstrate the superiorities of proposedWPA

Competing Interests

The authors declare no competing financial interests

Acknowledgments

Thiswork is supported in part by theNatural Science Founda-tion of Inner Mongolia Autonomous Region of China underGrant no 2014BS0603 and no 2014BS0601 and the Programof Higher-Level Talents of Inner Mongolia University underGrant no 30105-135134

References

[1] C-C Shen C Srisathapornphat and C Jaikaeo ldquoSensor infor-mation networking architecture and applicationsrdquo IEEE Per-sonal Communications vol 8 no 4 pp 52ndash59 2001

[2] S D Glaser ldquoSome real-world applications of wireless sen-sor nodesrdquo in Smart Structures and Materials 2004 Sensorsand Smart Structures Technologies for Civil Mechanical andAerospace Systems vol 5391 of Proceedings of SPIE pp 344ndash355San Diego Calif USA March 2004

[3] I F Akyildiz W Su Y Sankarasubramaniam and E CayircildquoWireless sensor networks a surveyrdquo Computer Networks vol38 no 4 pp 393ndash422 2002

[4] W Ye J Heidemann and D Estrin ldquoSensor-MAC (S-MAC)medium access control for wireless sensor networksrdquo in Pro-ceedings of the 21st International Annual Joint Conference of theIEEEComputer and Communications Societies (INFOCOM rsquo02)New York NY USA June 2002

[5] L J G Villalba A L S Orozco A T Cabrera andC J B AbbasldquoRouting protocols in wireless sensor networksrdquo Sensors vol 9no 11 pp 8399ndash8421 2009

[6] P Santi ldquoTopology control in wireless ad hoc and sensornetworksrdquo ACM Computing Surveys vol 37 no 2 pp 164ndash1942005

[7] R Rajagopalan and P K Varshney ldquoData-aggregation tech-niques in sensor networks a surveyrdquo IEEE CommunicationsSurveys and Tutorials vol 8 no 4 pp 48ndash63 2006

[8] W-S Jung K-W Lim Y-B Ko and S-J Park ldquoA hybridapproach for clustering-based data aggregation in wireless sen-sor networksrdquo in Proceedings of the 3rd International ConferenceonDigital Society (ICDS rsquo09) pp 112ndash117 IEEECancunMexicoFebruary 2009

[9] M Kumar S Verma and P S Pratap ldquoClustering approach todata aggregation in wireless sensor networksrdquo in Proceedingsof the 16th IEEE International Conference on Networks pp 1ndash6IEEE New Delhi India December 2008

[10] S Lindsey C Raghavendra and K M Sivalingam ldquoDatagathering algorithms in sensor networks using energy metricsrdquoIEEE Transactions on Parallel and Distributed Systems vol 13no 9 pp 924ndash935 2002

[11] JW Baek Y J Nam andD-W Seo ldquoATS-DA adaptive timeoutscheduling for data aggregation in wireless sensor networksrdquoin Proceedings of the International Conference on InformationNetworking pp 375ndash384 Estoril Portugal 2008

[12] H Chen H Mineno and T Mizuno ldquoAdaptive data aggrega-tion scheme in clustered wireless sensor networksrdquo ComputerCommunications vol 31 no 15 pp 3579ndash3585 2008

[13] I Solis and K Obraczka ldquoIn-network aggregation trade-offsfor data collection in wireless sensor networksrdquo InternationalJournal of Sensor Networks vol 1 no 3-4 pp 200ndash212 2006

[14] F Mirian and M Sabaei ldquoA delay and accuracy sensitive dataaggregation structure in wireless sensor networksrdquo in Proceed-ings of the International Conference on InformationManagementand Engineering (ICIME rsquo09) pp 231ndash235 IEEE Kuala LumpurMalaysia April 2009

[15] P Sun X Zhang Z Dong et al ldquoA novel energy efficientwireless sensor MAC protocolrdquo in Proceedings of the 4th Inter-national Conference on Networked Computing and AdvancedInformation Management pp 68ndash72 Gyeongju South KoreaSeptember 2008

[16] N Li J C Hou and L Sha ldquoDesign and analysis of an mst-based topology control algorithmrdquo in Proceedings of the IEEESocieties 22nd Annual Joint Conference of the IEEE Computerand Communications (INFOCOM rsquo03) vol 3 pp 1702ndash1712 SanFrancisco Calif USA April 2003

[17] B Yin H Shi and Y Shang ldquoA two-level topology controlstrategy for energy efficiency in wireless sensor networksrdquoInternational Journal of Wireless and Mobile Computing vol 4no 1 pp 41ndash49 2010

[18] A Sharif V M Potdar and A J D Rathnayaka ldquoLCARTlightweight congestion aware reliable transport protocol forWSN targeting hetero-geneous trafficrdquo in Proceedings of the17th International Conference on Neural Information Processing(ICONIP rsquo10) Sydney Australia November 2010

[19] B Yu J Li and Y Li ldquoDistributed data aggregation schedulingin wireless sensor networksrdquo in Proceedings of the 28th Confer-ence on Computer Communications (IEEE INFOCOM rsquo09) pp2159ndash2167 Rio de Janeiro Brazil April 2009

[20] M X Cheng and L Yin ldquoEnergy-efficient data gatheringalgorithm in sensor networks with partial aggregationrdquo Inter-national Journal of Sensor Networks vol 4 no 1-2 pp 48ndash542008

[21] A A Allen Probability Statistics and Queueing Theory withComputer Science Applications Gulf Professional Publishing1990

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of