Lifetime Communication

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Rate-lifetime tradeoff for reliable communicationin wireless sensor networks q

Junhua Zhu a, Ka-Lok Hung a, Brahim Bensaou a,*, Farid Nait-Abdesselam b

a Department of Computer Science and Engineering, The Hong Kong University of Science and Technology,

University Road, Clear Water Bay, Kowloon, Hong Kongb IRCICA/LIFL, CNRS UMR 8022, Batiment M3, University of Sciences and Technologies of Lille, 59655 Villeneuve dAscq Cedex, France

Available online 29 September 2007

Abstract

The network lifetime and application performance are two fundamental, yet conflicting, design objectives in wirelesssensor networks. There is an intrinsic tradeoff between network lifetime maximization and application performance max-imization, the latter being often correlated to the rate at which the application can send its data reliably in sensor networks.In this paper we study this tradeoff by investigating the interactions between the network lifetime maximization problemand the rate allocation problem with a reliable data delivery requirement. Severe bias on the allocated rates of some sensornodes may exist if only the total throughput of the sensor network is maximized, hence we enforce fairness on source ratesof sensor nodes by invoking the network utility maximization (NUM) framework. To guarantee reliable communication,

we adopt the hop-by-hop retransmission scheme. We formulate the network lifetime maximization and fair rate allocationboth as constrained maximization problems. We characterize the tradeoff between them, give the optimality condition, andderive a partially distributed algorithm to solve the problem. Furthermore, we propose an approximation of the tradeoffproblem using NUM framework, and derive a fully distributed algorithm to solve the problem. 2007 Elsevier B.V. All rights reserved.

Keywords: Wireless sensor networks; Network utility maximization; Rate allocation; Energy efficiency; Reliability

1. Introduction

Self-organized and -configurable wireless sensor

networks have invaluable potential in military andcivilian applications where distributed sensing, col-lection and dissemination of information are

required. Examples of such applications includebattlefield surveillance, environmental monitoring,home automation[1], and so on.

Typically, sensor nodes are battery operated, andhence have to run on a limited energy budget. Fur-ther, battery replacement is impossible in manywireless sensor network applications. Thus, energysupply is limited, and sensor networks have a finiteoperational lifetime. Although substantial improve-ments have been achieved in the chip design forenergy conservation, advances in battery design stilllag behind, making energy efficiency one of the fun-damental challenges in sensor networks.

1389-1286/$ - see front matter 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.comnet.2007.09.011

q This work is supported in part under grant Hong Kong RGCDAG05/06.EG42.* Corresponding author.

E-mail addresses: [email protected] (J. Zhu), [email protected] (K.-L. Hung), [email protected] (B. Bensaou),[email protected](F. Nait-Abdesselam).

Available online at www.sciencedirect.com

Computer Networks 52 (2008) 2543

www.elsevier.com/locate/comnet
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Tremendous research efforts have been spent onthe problem of energy conservation in wireless sen-sor networks. One important perspective is to max-imize the network lifetime [2,1113,30], wherenetwork lifetime usually refers to the time interval

between the initialization of the network and theexhaustion of the battery of the first sensor node.The lifetime of a sensor node is generally inverselyproportional to the average rate of information gen-erated and the information relayed by itself, andhence network lifetime is determined by the sourcerates of sensor nodes in the network.

The application performance is often evaluatedby the accuracy and latency of information obtainedin wireless sensor networks[3]. The precise seman-tics of accuracy and latency are application-depen-dent. For instance, in wireless video sensor

networks, the accuracy of information can be quan-tized by rate distortion analysis [4]. Roughly, thelarger the source rate, the smaller the distortionlevel. In general wireless sensor networks, both met-rics depend on the rates of application data reliablydelivered from the sensor nodes to the sink nodes.Thus, the application performance can be indirectlymeasured by the rate allocation in the network.Motivated by this reason, many works have beenconducted on the rate allocation problem withenergy constraint in wireless sensor networks

[5,14,16]. These works try to strike a balancebetween fully utilizing the network resources andmaintaining a certain fairness on source ratesamong sensor nodes.

Since the application performance correlates tothe rates of data obtained reliably in sensor net-works, we need to consider the data delivery reli-ability requirement in sensor networks. Manydefinitions of reliability have been proposed byresearchers, e.g., the end-to-end reliability, theevent-to-sink reliability proposed by Akan andAkyildiz for event-based sensor networks [6], andso on. Here we focus on the first one, that is, datafrom sensor nodes are delivered to the sink nodeswithout loss or error. Due to the error-prone natureof wireless channels, there are two approaches toimprove the reliability: reducing the probability ofdata loss or error and retransmitting data once lossor error occurs. Here we adopt the retransmissionscheme. Under this category, there exist twoapproaches: the end-to-end retransmission schemeand the hop-by-hop retransmission scheme. In thefirst method, the source node will initialize a

retransmission when loss or error is detected either

by the source or the destination, while in the secondmethod, the intermediate node will start a retrans-mission as soon as loss or error is detected. The sec-ond method is well known to be more effective thanthe first one in wireless networks since losses or

errors are recovered locally, and resources used todeliver data to the current node will not be wasted.Thus we use the hop-by-hop retransmission schemeto guarantee the reliable data delivery from sensornodes to sink nodes.

From the above discussion, we notice that thereare intrinsic tradeoffs between network lifetimemaximization and rate allocation in wireless sensornetworks. Although both problems have beenextensively studied in recent years separately, fewworks consider these two goals together, and studythe tradeoff between them with a reliability require-

ment. In this paper, we address this problem with amulti-path routing approach and observe that, onlymaximizing the throughput of the sensor network,leads sensor nodes that are far away from the sinknodes to suffer extremely small data rates, whichmakes the performance of these sensor nodes extre-mely bad. Thus, a certain fairness in rate allocationof sensor nodes is required, and should be achievedin a distributed way. This motivates the usage of anetwork utility maximization (NUM) framework[7]for our rate allocation problem, which has been

proved efficient in such a case.In this paper, we first consider network lifetime

of wireless sensor networks as global informationshared among all sensor nodes. We find that thesolution maximizing the network lifetime also min-imizes the maximum normalized power dissipationof sensor nodes with the latter one being relativelymore tractable. Thus, the problem of minimizingthe maximum normalized power dissipation of sen-sor nodes is solved instead. For the sake of simplic-ity, we still call this problem the network lifetimemaximization problem. Given a set of routes, thetradeoff between network lifetime maximizationand fair rate allocation is formulated as a multipleobjective programming problem with a set of con-vex constraints. By introducing a system parameterc, we can combine these two objectives together as asingle weighted objective. The tradeoff betweenthem can be characterized by the parameterc. Thenwe study the optimality condition, and derive a sub-gradient algorithm. Note that at this stage the infor-mation of network lifetime is shared among allsensor nodes, so global exchange of this information

cannot be avoided in this algorithm; for example, it

26 J. Zhu et al. / Computer Networks 52 (2008) 2543


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routing and rate allocation. They obtain a distrib-uted algorithm via dual decomposition of the origi-nal problem. In our paper, although we have thesame objective function, we focus on addressing thisproblem at the transport layer, and hence have dif-

ferent set of constraints. Also, the derived algorithmis different.

3. Problem formulation

We consider a wireless sensor network consistingof sensor nodes and sink nodes. Sensor nodes arebattery driven, non-rechargeable and irreplaceable.They periodically perform some sensing tasks, col-lect data about phenomena of interest, and reportthe measurements to the sink nodes. Data fusion

is not considered here, i.e., data will not be aggre-gated along the routes to the sink nodes. Thereare multiple sink nodes in the network, and eachsensor node is assigned to only one sink node. Theperformance of the sensor network is characterizedby two metrics, the network lifetime and the qualityof information received at the sink nodes. The qual-ity of information is roughly proportional to theallocated rate at the source node. The larger thesource rate, the more fine grained and accurate theinformation, and vice versa. We assume that sink

nodes accept information of any quality level. Thus,there is no source rate requirement for sensor nodes.

The wireless sensor network is modeled as adirected graph GN;L, where N S [T isthe set of all sensor and sink nodes in the network;S denotes the set of sensor nodes;T denotes the setof sink nodes andL is the set of all directed links inthe network. We define Lout(i) to be the set of alloutgoing links of node i, and Lin(i) to be the set ofall incoming links of node i.

In general, in wireless sensor networks, proactiverouting is preferred since nodes in the network arestatic and the traffic requirement is also fixed. Thus,each sensor node will first find a set of routes to deli-ver its collected data to the sink node, and thendetermine the data rate on each of its routes. LetR be the set of all routes in the network, andRs R be the set of routes for sensor node s.Let yr be the data rate on route r. Denoting thesource rate of sensor node s by xs, we have the fol-lowing flow conservation constraint:

xs Xr2Rs

yr 8s2 S: 1

Also the data rate on all routes should be non-neg-ative, i.e.,

yr P 0 8r2 R: 2

To ensure data from sensor nodes to be delivered tosink nodes reliably, we use the hop-by-hop retrans-mission method. At each node, a packet will beretransmitted at the MAC layer until it is success-fully received at the next hop node on the route.Assuming the network to be under-loaded, whichis common in most sensor networks applications,most packet losses are due to link errors becauseof interference. Under this assumption, it is reason-able to assume that the frame retransmission prob-ability is independent of the source rates vectory. Amodel that considers collision errors in a heavilyloaded networks and their dependence on the rate

vector is considered in[23]. To model this, for eachlink l, denoting the link error probability by sl, theaverage number of transmissions until successfuldelivery of one packet is 1

1sl.

Since in most types of sensor nodes, communica-tion modules dominate the energy consumption[24], we ignore energy consumed by other tasks suchas sensing and data processing. The same simpleenergy consumption model as in [25] is used forthe communication module of all nodes. Assumingall nodes have the power control functionality, the

power dissipation at node ifor transmission is deter-mined by

ptij esij fij; 3

where ptij is the power dissipated at nodeiwhen it istransmitting to nodej,fijis the rate of data transmit-ted at the physical layer, and esij, the transmissionenergy consumption cost of link (i,j), is given by

esij l g dmij : 4

Here, l denotes the energy cost of the transmit elec-tronics of node i, g is a coefficient term correspond-

ing to the energy cost of transmit amplifiers, dij isthe distance between nodeiand j, andm is the pathloss exponent, with values ranging between 2 and 4.Typically, l= 50nJ/b, g= 0.0013pJ/b/m4 (for m=4). The power dissipation at a receiver ican be ex-pressed as

pri er Xj6i

fji; 5

whereer is the energy consumption cost of the radioreceiver, and Pj6ifji is the rate of data received atthe physical layer of node i. Typically, er = 50nJ/b.



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Since we adopt a link layer retransmissionscheme, for a link l, the actual average data rateat the physical layer is the product of the data rateat the network layer and the average number oftransmissions at the MAC layer. Let Rl be the

set of routes on link l

, we have flP

r2Rlyr=1 sl. The total power dissipation at node i isthe sum of total power dissipation by radio trans-mitter and receiver. Thus, the total power dissipa-tion at node iis given by

pti X

l2Louti

esl

Pr2Rlyr

1 slX

l2Lini

er

Pr2Rlyr

1 sl: 6

Assuming that each sensor node ihas an initial en-ergy budgetEi, the lifetime of sensor node iis given

by

TiEi

pti: 7

3.1. Fair rate allocation problem

Since the accuracy of information depends on theallocated source rate, only maximizing the totalthroughput of the network is not good. It is not suf-ficient to guarantee the application performance asthis objective may be achieved at the expense ofsome source rates. For example, in a sensor networkthat tracks objects mobility in a large field, smallerrates means lesser location tracking accuracy andlarger rates equate to a finer grained accuracy. Bymaximizing the total throughput instead of consid-ering the fairness issue among source rates, onecan end up with a solution that shuts off manysources in the network and enables only thosesources whose transport energy-cost to the sink isthe smallest. Hence some fairness on the rates allo-cated to the different sensor nodes is desired. One

good methodology to achieve this goal is to adoptthe NUM framework, in which a concave, non-decreasing, and twice differentiable utility functionUs(xs) measures the satisfaction of sensor node swith the assigned rate xs, and the goal is to maxi-mize the sum of the individual utilities. One classof utility functions that has been extensively usedto achieve fair resource allocation [8] in economicsand distributed computing is

Uax logx; a 1;

11ax

1a

; a> 1: 8

When a = 1, the utility function leads to the so-called proportional fairness, whereas when a ! 1the utility function leads to maxmin fairness.

What class of fairness is required for rate alloca-tion in a sensor network is determined by the appli-

cation and is beyond the scope of this paper. Whilemaxmin fairness maximizes the smallest rate in thenetwork, allowing thus for the best possible accu-racy in monitoring the whole network, proportionalfairness may favor nodes that are nearer the sinksover those that are located further away. One canimagine many application scenarios, both militaryand civilian, that match these two requirements.

Typically, the average data rate generated byeach sensor node is small, so it is reasonable toassume that there is sufficient bandwidth for eachlink in the network. That is, resource constraints

below the network layer are not present. Under thisassumption, the fair rate allocation problem givennetwork lifetime requirements can be formulatedas a NUM problem

maximizeXs2S

wsUsxs

subject to Ti P Tnet 8i2 S;

constraints 1; 2; 6; 7;

9

wherewsis a weight associated with the utility func-tion, andTnetis the lifetime requirement of the sen-

sor network. With the help of parametersw, furtherdifferentiation among sensor nodes can be achieved.For example, sensor nodes deployed in the area thatattracts most interest can get fairly large allocatedrates by setting their weights relatively large. In thisway, we can achieve weighted fairness on sourcerates of sensor nodes.

3.2. Network lifetime maximization problem

All sensor nodes are assumed to be of equal impor-

tance, which is a reasonable assumption since thedeath of one sensor node may cause the network tobe partitioned, or some area requiring monitoringto be uncovered. Thus the lifetime of a sensor networkis defined as the time until the death of the first sensornode, i.e., Tnet mini2STi. Given source ratedemandsof all sensornodes, the problem of maximiz-ing the lifetime of a sensor network can be stated as

maximize mini2S

Ti

subject to xs x0

s 8s2 S;

constraints 1; 2; 6; 7;

10

J. Zhu et al. / Computer Networks 52 (2008) 2543 29


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where x0s is the source rate requirement of sensornode s. This formulation is very similar to that of[11], and the heuristic in[11]can be used. Generally,constraints(7) are not convex, and hence the opti-mization problem (10) is not convex. This usually

makes the problem difficult to solve. To overcomethis, we introduce a new variable zi= 1/Ti, whichis the inverse of the lifetime of sensor node i. Notethat from(7), it is also the normalized power dissi-pation of nodeiwith respect to initial energyEi. It iseasy to verify that maximizing the minimum lifetimeof all sensor nodes is equivalent to minimizing themaximum normalized power dissipation of all sen-sor nodes. Hence the original problem(10)is refor-mulated as

maximize maxi2S

zi

subject to pti Eizi 8i2 S

xs x0

s 8s2 S;

constraints 1; 2; 6:

11

Notice that problem (11) is a linear programmingproblem, which is much easier to solve than the ori-ginal problem in(10). For the sake of simplicity, wewill still call this problem the network lifetime max-

imization problem.

4. Tradeoff between network lifetime and fair rate

allocation

From problems(9) and (11), we observe that wehave two important but conflicting objectives whenoptimizing the sensor network performance, i.e.,achieving fair rate allocation among sensor nodes(problem (9)) and maximizing the network life-time (problem (11)). Both can be formulated asconstrained maximization problems. Hence, thetradeoff between them can be formulated as amulti-objective programming problem, and a simpleand efficient way to achieve desired tradeoff betweenthem is the weighting method[26]. That is, we intro-duce a new system parameterc 2 [0,1], and combinethese two objective functions into a single objectivefunction. Sinceccharacterizes the tradeoff, we call ittradeoff factor. With the constraint that data rateson all routes should be non-negative, the tradeoffbetween fair rate allocation and network lifetimemaximization is formulated as a constrained optimi-

zation problem

maximize cXs2S

wsUsxs 1 c maxi2S

zi

subject toX

l2Louti

esl

Pr2Rlyr

1 slX

l2Lini

erP

r2Rlyr

1 sl

Eizi 8i2 S;Xr2Rs

yr xs 8s2 S;

yr P 0 8r2 R:

12

The above weighted objective function is a concavefunction sinceU() is concave and the maximum of aset of linear functions is convex, and both x and zare linear functions ofy. The constraint sets are alsoconvex, thus it is a convex optimization problem,and there is only a unique optimal objective value,i.e., a locally optimal solution is also a globally opti-mal solution[27]. Denote the column vector of {yr},{xi}, and {zi} by y,x, andzrespectively. Note thatzandx depend ony, and can be represented by func-tions ofy. Therefore,x and z are dummy variablesin (12), and the only optimization variables are y.Define the weighted objective function as Qy cP

s2SwsUsxs 1 cmaxi2Szi. We have the fol-lowing optimality condition.

Proposition 1. Let znet maxi2Sfzig, J(z) ={ijzi=znet}, andRi be the set of routes on node i. then

(a) a solution {x*, y*, z*} is globally optimal if andonly if for alld 2 Rn, there holds

min cryXs2S

wsUsx

s

(

1 cryzi

!0dji2 Jz

)6 0; 13

(b) if a solution {x*, y*, z*} is optimal, then for alli2 J(z*), there holds

cryrXs2S

wsUsxs 1 cryrzi 6 0 8r2Ri:

14

Proof. The proof of the proposition is given inAppendix A.

Part (a) ofProposition 1 gives a necessary andsufficient condition for a solution to be optimal,however, it does not give us any insight on thestructure of the optimal solution. From part (b), wefind that at the optimal point, all the routes on the

sensor nodes with the minimum lifetime would not

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Problem(19)is very similar to that of[30], and hasthe following property:

Lemma 4. When b ! 1, the network lifetime deter-mined by the optimal solution y* of problem (19)approximates the maximum network lifetime of the

wireless sensor network.

Proof. The proof of the lemma is given inAppendixB. h

Thus, we call b the lifetime approximation expo-nent. The tradeoff between network lifetime maximi-zation and fair rate allocation of sensor nodes canbe unified in the network utility maximizationframework as

maximize

Xs2ScwsUsxs 1 cV

bs zs

subject toX

l2Louti

esl

Pr2Rlyr

1 slX

l2Lini

erP

r2Rlyr

1 sl

Eizi 8i2 S;Xr2Rs

yr xs; 8s2 S;

yr P 0 8r2 R:

20

Here, since both x and z are dummy variables, we

can define the weighted objective function asPy cP

s2SwsUsxs 1 cVbs zs. Note that

bothU() andV() are twice differentiable, and hencethe objective functionP(y) is differentiable. The gra-dient with respect to data rate yrexists and is givenby

ryrPy cwsU0

sxs 1 cX

i2r\S

zbi ryrzi; 21

where s is the source sensor node of route r. A fea-sible solution y* is locally optimal if and only if forall r2 R,

yrP 0;

ryrPy 6 0;

yrryrPy 0: 22

Since both U() and V() are strictly concave, andbothxand z are linear functions ofy, the objectivefunction P(y) is also concave. Hence a locally opti-mal solution is also a globally optimal solution.Generally, the optimal solutiony* is not necessarilyunique. However, the strict concavity of functions

U() and the strict convexity of functions V(

) imply

that the optimal solution x* and z* are unique. Wecan use the gradient projection method[28]to solve(20). Here we adopt a similar algorithm to the oneused in [31]. At each iteration t, the data rate yr isupdated by

Dyrt j yrt

cU0sxstryrPyt

jyrt ws 1 c

cU0sxst

Xi2r\S

zbi tryrzit

!:

23

From Eq.(23), we observe that the update rule alsosuffers from the same problem as previously viz.any route r cannot increase its rate once it reacheszero. Thus, we need to bound its data rate yraway

from zero. With AssumptionA1, which sets a smallminimum rate requirement d for each route, thisproblem can be solved. Hence, at each iteration t,the data rate yris updated by

yrt 1 maxd;yrt Dyrt: 24

The detailed gradient algorithm is given in Algo-rithm 2. We observe that this algorithm is a varia-tion of the scaled gradient projection algorithm.To demonstrate its convergence property, we firstintroduce an additional assumption:Assumption

A2: The data rate yr on each route r is upper-bounded by a constant cr.

This assumption is reasonable in wireless sensornetworks since the bandwidth at the MAC layer isfinite, and the data rate of routes cannot exceedthe bandwidth. With Assumptions A1 and A2, wefind that the source rates x are also bounded, andthe gradient $yP(y) is Lipschitz continuous, whichguarantees the convergence of the gradient projec-tion algorithm with constant step size. DefineR jRj, / maxs2SwsU

0sxs, and w maxr2Ryr=

U0

sxs. We have the following convergence result:Proposition 5. Let Y* be the set of optimal solutions

of problem (20). With Assumptions A1 and A2, ifconstant step size j satisfies 0< j < 2=R1=2/w,then for any initial feasible solution y0, Algorithm 2converges to an optimal solution y* 2 Y*.

Proof. The proof of the proposition is given inAppendix C. h

Algorithm 2. Fully distributed algorithm to solve

(20)

http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-http://-/?-


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At each iteration tStep 1: Each sensor node i computes its price1 czbi ryrzi, and its revenue for all of its routescwsU

0sxs.

Step 2: Compute the costs of all routes.for

each route r2R do

Routes a message from source to destination,aggregate the prices 1 czbi ryrzi charged bysensor nodes on it, and finally deliver this infor-mation back to the source node s.end for

Step 3: Update the rate of all routes.for each sensor node s2 S do

for each route r2 Rs do1. Compute Dyr(t) with formula(23).2. y(t+ 1) = max(d,y(t) + Dyr(t)).

end for

end for

InAlgorithm 2, a sensor node igives its route ran amount of revenue based on its marginal rateutility cwsU

0sxs, and charges any route on it a price

1 czbi ryrzi, which is a product of the marginalpower dissipation utility and normalized power dis-sipation for this route. A route rdelivers a messageto the sink node, and aggregates all the prices itreceives from the relaying nodes. Then the sink nodeforwards this message back to the source node i.After receiving this message, sensor node i willadd its price for route r to the aggregate price fromrelaying nodes, and then computes the new data rateyrbased on the revenue and aggregate price of router. When achieving the optimal solution, the revenueand price of each route with positive data rateshould be equal. By iteratively updating the datarates of the routes with a constant step size, oneoptimal solution can be guaranteed, and this canbe done asynchronously.

6. Performance evaluation

In this section, we use some numerical results toevaluate our rate allocation algorithms. We con-sider the wireless sensor network illustrated inFig. 1, where the topology is simple yet sufficientlylarge for us to investigate the performance of ouralgorithms. In this network, we have twenty sensornodes indexed 120, and one sink node. The loca-tions of sensor nodes were randomly generated over

a 300 m 300 m square area, and the location of the

sink node was randomly generated at the lower leftcorner of this square area.

We assume that each sensor node has a maxi-mum transmission range of 80 m, and the powerdissipation for transmitting to and receiving froma neighboring node is determined by (3) and (5) inSection 3, respectively, where l= 50nJ/b, g=0.0013pJ/b/m4,m = 4, and er = 50nJ/b[25]. The ini-tial energy budget of each sensor node is 5 kJ. Theutility function U() used here is logarithmic, andthe weight ws of each sensor node s is set to 1.

The initial data rates and minimum data raterequirement are set to 1 bps and 0.01 bps,respectively.

First we consider the scenario where the linkerror probabilities of all links are equal. We setthese probabilities to 0.01. To gain a better under-standing of the optimal performance of our rateallocation algorithms, we determine all the possibleroutes in the network for each sensor node.

6.1. Convergence results and tradeoff byAlgorithm 1

Fig. 2depicts the convergence property ofAlgo-rithm 1 described in Section 4 with a step sizej= 0.05.

Here the tradeoff factor c is set to 1.0 108. Weonly show the convergence results of sources rates;the convergence results of other variables are omit-ted due to space limitations. Here, we only showpart of the sensor nodes as we find that sensor nodes9, 15 and 18 have the same allocated rates as sensornode 4, and the remaining sensor nodes (not shownhere) have the same allocated rates as sensor node 1.

We can see that source rate allocation of sensor

0 50 100 150 200 2500

50

100

150

200

250

300

Xm

Ym

S16

S6

S12

S5 S4

S1

S2

S3

S7

S8

S9

S10

S11

S13

S14

S15

S17

S18

S19

S20

Fig. 1. Wireless sensor network topology.



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6.2. Convergence results and approximation ratio of

Algorithm 2

Figs. 6 and 7show the convergence results of thefully distributed Algorithm 2 derived in Section 5with step size j= 0.1 andj= 0.2, respectively.

Here we set the tradeoff factor c to 1.0 1080,and the lifetime approximation exponent b to 10.We see that if the step size j is selected accordingto the step size rule proposed in Proposition 5, thenthe larger the step size is, the faster the algorithmconverges. We also observe that the fully distributedAlgorithm 2 converges to an optimal solution ofproblem (20) in Section 5 in a few hundred itera-tions, and faster than the partially distributedAlgo-

rithm 1. Thus,Algorithm 2has a better convergenceproperty thanAlgorithm 1.

However,Algorithm 2may not achieve the opti-mal tradeoff since it is only an approximation algo-rithm. Thus it is interesting to investigate the impactof the lifetime approximation exponent b on theperformance ofAlgorithm 2. If the performance issatisfactory given a reasonably large b, thenAlgo-rithm 2 is preferred as it requires less communica-tion overhead, and is easier to implement than

Algorithm 1. We evaluate the performance by theapproximation ratio of the optimal solutionobtained byAlgorithm 2with respect to the optimalsolution of problem(12). Comparing the two trade-off curves in Fig. 5, which are obtained by Algo-rithms 1 and 2, respectively, we observe that thesame value ofc can result in different optimal trade-offs between network lifetime and fair rate alloca-tion of sensor nodes. Thus the naive way of fixingparameter c, and comparing the optimal solutionobtained byAlgorithm 2under different value ofbwith the optimal solution obtained byAlgorithm 1does not work. Hence we use another approach.First we fix the tradeoff factor c, and obtain theoptimal network lifetime and rate allocation ofproblem (12), then we set b for problem (20), andvary c such that the optimal aggregate rate utilityis the same as that of problem(12), or the differenceis extremely small. After that, we normalize theoptimal network lifetime of problem (20) withrespect to the optimal network lifetime of problem(12), and regard this ratio as the approximationratio. Here, we set c= 1.0 108 for problem(12),

and the results are shown inFig. 8.

0 5 10 15

x 107

60

80

100

120

140

160

180

200

Network Lifetime (s)

AggregateRateUtility

= 1.0109

= 1.0107

= 1.01090

= 1.01070

Algorithm 1

Algorithm 2 with = 10

Fig. 5. The impact of tradeoff factor c on the tradeoff betweennetwork lifetime and fair rate allocation of sensor nodes.

0 200 400 600 8000

0.2

0.4

0.6

0.8

1

1.2

1.4

Iteration Number

SourceRate(kbps

)

Sensor 1

Sensor 2

Sensor 3

Sensor 4Sensor 5

Sensor 12

Sensor 16

Fig. 6. The evolution of source rates of sensor nodes with step

size j = 0.1 in Algorithm 2.

0 200 400 600 8000

0.2

0.4

0.6

0.8

1

1.2

1.4

Iteration Number

SourceRate

(kbps)

Sensor 1

Sensor 2

Sensor 3

Sensor 4

Sensor 5

Sensor 12

Sensor 16

Fig. 7. The evolution of source rates of sensor nodes with stepsize j= 0.2 in Algorithm 2.



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From this figure, we can see that when b = 2, wealready guarantee a fairly good approximationratio, i.e., 90.7%, and when b P 14, the approxima-tion ratio exceeds 99%. Thus, the performance ofAlgorithm 2is very good. Another intuitive example

can be obtained fromFig. 5. Although the ranges ofc are different, the two tradeoff curves obtained byAlgorithms 1 and 2withb= 10 respectively are veryclose to each other.

We compare the lifetime and source rates of sen-sor nodes obtained byAlgorithms 1 and 2under dif-ferent value ofb with the same aggregate rate utilityinTable 1.

FromTable 1, we observe that when the tradeofffactor cis set to 1.0 108 inAlgorithm 1, the life-time of sensor nodes 2, 5, 12 and 16 is very close tothe network lifetime. This means that these sensor

nodes are bottleneck nodes in the sensor network(the slight difference between their individual life-time is due to the precision of the implementation).With Algorithm 2, we find that the larger the life-time approximation exponent b is, the closer thenetwork lifetime gets to the optimal network life-time obtained by Algorithm 1, and the larger thenumber of bottleneck nodes in the network is. Thisimplies when b grows large, Algorithm 2 can bal-ance the energy consumption rate of sensor nodesmore efficiently. Also, from the table, we see that

although the aggregate rate utilities are the same,

the sources rates allocated to sensor nodes are notthe same. We observe that the larger b is, the closerthe rate allocation obtained by Algorithm 2approaches to the rate allocation obtained byAlgo-rithm 1.

6.3. The impact of routing algorithm

In our formulation of the problem and in the pre-vious examples, we assumed that a sensor node

determines all its routes to the sink prior to starting

0 2 4 6 8 10 12 14 16 18 200.5

0.6

0.7

0.8

0.9

0.99

ApproximationRatio

Fig. 8. The impact of lifetime approximation exponentb on theperformance ofAlgorithm 2.

Table 1Comparison of the individual lifetime and rate of sensor nodes obtained by Algorithms 1 and 2with fixed aggregate rate utility

Sensor ID Algorithm 1(c= 1.0 108) Algorithm 2(b= 1) Algorithm 2(b= 10)

Lifetime 1.0 107 (s) Rate (kbps) Lifetime 1.0 107 (s) Rate (kbps) Lifetime 1.0 107 (s) Rate (kbps)

1 0.6340 0.7705 0.8026 0.7313 0.6433 0.76852 0.5021 0.9523 0.5592 1.4035 0.5059 1.01893 4.3908 1.3115 3.7722 1.6937 4.6587 1.38614 1.9810 1.6586 4.8375 1.1710 2.1207 1.54955 0.5034 3.1013 0.5308 5.8671 0.4994 3.19746 2.7067 0.7663 7.6066 0.5578 2.7449 0.75527 0.6387 1.6599 1.1499 2.5703 0.7280 1.69348 0.9393 0.7670 1.8214 0.6409 0.9652 0.75559 1.0925 1.6599 1.5025 1.2970 1.1696 1.5496

10 1.1645 0.7705 1.5345 1.0936 0.7893 0.817511 1.1893 0.7670 2.1460 0.6636 1.2083 0.755512 0.5033 2.0108 0.4032 2.7400 0.5033 2.125213 2.0641 0.7652 3.6006 0.5742 2.0932 0.755114 0.9595 0.7735 1.4780 0.6038 0.9779 0.755315 2.4748 1.6586 6.1513 1.1927 2.6492 1.549516 0.5047 2.5049 0.4516 3.1827 0.5113 2.581717 0.5463 0.7705 0.6808 0.9184 0.6561 0.801718 0.5386 1.6599 0.8301 1.6227 0.6545 1.594619 1.7395 0.7652 3.7552 0.5675 1.7641 0.7551

20 2.7520 0.7663 7.6826 0.5580 2.7908 0.7551



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data exchange. This in turn leads to the optimalmulti-path routing (OMPR) since we can prove thatour solutions are optimal. However, in real sensornetworks, it may be impossible to explore all possi-ble routes, and only a fairly small set of routes will

be used by the sensor nodes. Thus, the impact ofroute choices on the performance of our algorithmsis considered here. Obviously, convergence of ouralgorithms is still guaranteed since they do notdepend on the route choices. Nevertheless, differentroute choices other than those assumed in the modelmay lead to sub-optimal results. To investigate this,we compare the results of the optimal algorithmwith multi-path routing (including all possibleroutes) to the results obtained by two other routingalgorithms, namely, single path minimum energyrouting (MER) and 2-shortest path (link disjoint)

routing (2SPR).Table 2 compares the performance of the three

route choices with equal aggregate rate utility. Fromthe table, we observe that although the three routechoices obtain the same rate utility, the rate alloca-tion of sensor nodes with minimum energy routingis quite different from that of the optimal multi-pathrouting and is unfair. In contrast, the rate alloca-tions of sensor nodes with link-disjoint 2-shortestpath routing and optimal multi-path routing arevery close to each other. We observe that although

the network lifetime with minimum energy routing

is already close to the optimal network lifetime,i.e., around 80%, the lifetime of sensor nodes isnot well balanced. The network lifetime with thelink-disjoint 2-shortest path routing is even closerto the optimal network lifetime, and the power dis-

sipations of sensor nodes are close to the optimalpower dissipation with all the routes. From Table2, we find that the performance of minimum energyrouting is not satisfactory as it can result in a muchless fair rate allocation compared with the optimalrate allocation, and wastes more scarce energy ofsensor nodes. The performance of the link-disjoint2-shortest path routing is comparable with the opti-mal multi-path routing as it can achieve a rate allo-cation close to the optimal rate allocation, and atthe same time, a network lifetime close to the opti-mal network lifetime. This suggests that for real

implementation of our algorithms, it is sufficientto only enumerate and explore a small number ofdisjoint routes from each sensor to the sink toobtain results that are nearly optimal.

6.4. Non-uniform link error probability scenario

In the above, we assume the link error probabil-ities of all links are equal. This is not true in real net-works. Usually, the link error probability of aparticular link is related to the received power at

the receiver, the level of interference in the sur-

Table 2Impact of different route choices with equal aggregate rate utility

Sensor ID MER 2-SPR OMPR

Lifetime 1.0 107 (s) Rate (kbps) Lifetime 1.0 107 (s) Rate (kbps) Lifetime 1.0 107 (s) Rate (kbps)

1 0.7169 0.6787 0.6444 0.7551 0.6340 0.77052 0.6624 0.6787 0.4769 1.0841 0.5021 0.95233 9.8441 0.6787 5.8640 1.0648 4.3908 1.31154 3.2726 1.7933 5.4133 1.0841 1.9810 1.65865 0.4009 10.7050 0.4608 5.8547 0.5034 3.10136 11.5525 0.6787 10.3838 0.7551 2.7067 0.76637 1.6624 5.7349 1.0253 3.1381 0.6387 1.65998 0.8493 0.6787 0.7634 0.7551 0.9393 0.76709 1.0106 1.7933 1.6717 1.0841 1.0925 1.6599

10 12.1122 0.6787 3.2968 0.8091 1.1645 0.770511 13.1314 0.6787 11.8029 0.7551 1.1893 0.767012 0.3902 1.0405 0.4585 1.5564 0.5033 2.010813 2.7680 0.6787 2.4879 0.7551 2.0641 0.765214 1.5667 0.6787 1.4082 0.7551 0.9595 0.773515 5.0843 1.7933 8.4101 1.0841 2.4748 1.658616 0.3931 2.7843 0.4632 1.6269 0.5047 2.504917 0.5539 0.6787 0.4596 0.9570 0.5463 0.770518 0.6523 1.7933 0.7644 3.1381 0.5386 1.659919 8.2918 0.6787 7.4529 0.7551 1.7395 0.7652

20 12.2309 0.6787 10.9935 0.7551 2.7520 0.7663



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rounding environment, and so on. As these factorscannot be the same for all links, the error probabil-ities of links are different. Thus, we investigate herethe tradeoff between network lifetime and rate allo-cation of sensor nodes under the non-uniform link

error probabilities scenario.First we consider an extreme case, where theerror probabilities of all links except link (13, 1)are same, and set to 0.01. The error probability oflink (13,1) is set to 0.8.

Table 3shows the comparison of the lifetime andsource rates of sensor nodes under different errordistributions with tradeoff factor c= 1.0 108.We can observe that the rate allocation obtainedwith non-uniform link error probabilities differsgreatly from the one with no link errors. From thetable, we observe that traffic through the less reli-

able links is reduced by either redirection to otherlinks, or by decrease in the source rates. Hence thelifetime of sensor node 12 does not change toomuch, although the aggregate rate utility decreases.From this table, we also observe that the rate allo-cated to sensor node 12 in the extreme case isreduced significantly compared to the rate allocatedto it in the case where no link errors occur. Thisimplies that when we provide energy efficient andreliable communication in sensor networks by linklayer retransmission schemes, we should consider

the link error probability in the energy consumptionmodel. Otherwise, the obtained solution may bevery far from representing the optimal solution.

In real networks, the link error probabilities usu-ally relate indirectly to the number of neighboringnodes. Hence we study the case where the errorprobability of a link is determined by 0.01 + 0.1 n where n is the number of neighboring nodes ofthe receiver. The tradeoff curve is shown in Fig. 9.The tradeoff curve with the assumption of the error

probabilities of all links are 0.01 is also shown in thefigure. Comparing these two curves, we see thatalthough the network lifetime is still inversely pro-portional to the tradeoff factorc, they achieve differ-ent tradeoff between network lifetime and rateallocation of sensor nodes with the same value ofc. This is another reason why we need to considerthe impact of link error probabilities on the energyconsumption model of sensor nodes.

7. Conclusions

In this paper, we studied the tradeoff between thenetwork lifetime maximization and fair rate alloca-tion problem with a reliability requirement in wire-less sensor networks. We addressed the wholeproblem from the transport layer perspective, withmulti-path routing. To guarantee reliable end-to-end communication, because of its intrinsic energyefficiency, a hop-by-hop retransmission scheme isadopted instead of the traditional end-to-endretransmission scheme used in reliable transport.We formulated the network lifetime maximization

problem and the fair rate allocation problem as con-

Table 3Impact of error rate distribution on optimal result c = 1.0 108

Sensor ID No error With error

Lifetime1.0 107 (s)

Rate(kbps)

Lifetime1.0 107 (s)

Rate(kbps)

1 0.6379 0.7554 0.6890 0.70442 0.5054 0.9787 0.5033 0.80423 5.0520 1.2834 6.2568 0.87364 1.9788 1.6773 2.9587 1.11065 0.5068 3.1265 0.5000 2.08616 2.7201 0.7674 2.9206 0.71207 0.6518 1.6779 0.7006 1.11018 0.9436 0.7795 1.0200 0.70259 1.0913 1.6778 1.6320 1.1101

10 1.1784 0.7554 1.2676 0.704411 1.1946 0.7795 1.2917 0.702512 0.5071 1.9677 0.5242 0.934413 2.0743 0.7725 2.2272 0.707114 0.9646 0.7639 1.0396 0.701015 2.4719 1.6773 3.6961 1.110616 0.5047 2.5332 0.5028 1.647317 0.5519 0.7554 0.5944 0.704418 0.5210 1.6781 0.5621 1.110119 1.7481 0.7725 1.8770 0.7071

20 2.7655 0.7674 2.9695 0.7120

0 1 2 3 4 5

x 107

80

100

120

140

160

180

200

Network Lifetime (s)

AggregateRa

teUtility

= 1.0109

= 1.0107

Uniform Link Error Probability

Nonuniform Link Error Probability

Fig. 9. The impact of tradeoff factor c on the tradeoff betweennetwork lifetime and rate allocation of sensor nodes.



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Denote the network lifetime in problem (11) and(19) by Tnet and T

bnet, respectively. We observe

that Tnet= 1/k(A1 +A2)y*k1, and T

bnet 1=kA

1A2ybk1. Thus,T

bnet 6 Tnet. From the above inequal-

ities, we have

1

jSj1=b1

Tnet 6 Tbnet 6 Tnet:

Thus, limb!1Tbnet Tnet, and the lemma holds.

Appendix C. Proof ofProposition 5

Since the data rates y on the routes are bounded,the source rates x are also bounded. Therefore, /and w are finite as the gradient U

0

(xs) is boundedfor each sensor node s. We observe that function

P(y) is differentiable, hence the following inequali-ties regarding the gradient ryrPy hold:

ryrPy cwsU0

sxs 1 cX

i2r\S

zbi ryrzi

6 cwsU0

sxs 6 c/:

As function U() is strictly concave, and functionV() is strictly convex, function P(y) is strictly con-cave. By definition of concave functions, for anypoints y1 and y2, we have P(y1) P(y2) 6

$yP(y)T(y1 y2). Thus

kPy1 Py2k 6 kryPyT

y1 y2k

6 kryPy k ky1 y2k

6 R1=2c/ky1 y2k:

From the above inequalities, we see that functionP(y) is Lipschitz. Therefore, the limiting point ofthe sequences of points {y(t)} generated by the gra-dient algorithm with constant step size jis optimal[28, Proposition 1.2.3].

Next we derive the stepsize rule. Given theupdate direction d(t) at iteration t, we have

jdrtj yrt

cU0sxstryrPyt

6 wcjryrPytj:

Thus, we have

jryPytT

dtj

kdtk2

6 w=ckryPytk

2

w=c2kryPytk

2

c

w:

According to[28, Proposition 1.2.3], we find that if0< j < 2=R1=2/w, the gradient algorithm con-

verges to an optimal solution.

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Junhua Zhuis now a Ph.D. student in theDepartment of Computer Science andEngineering, the Hong Kong Universityof Science and Technology. He receivedhis B.Sc. degree in Computer Scienceand Technology from Fudan University,PR China, in 2004. His current researchinterests are in the areas of communica-tion quality and reliability in wirelesssensor networks, and cooperation issuesin wireless mesh and sensor networks.

John Ka-Lok Hung received his B.Eng.degree and M.Phil. degree in ComputerScience and Engineering in 2004 and2006, respectively, from the Hong KongUniversity of Science and Technology(HKUST). He is currently pursuing aPh.D. degreeat theHKUST. Hisresearchinterests include modeling and perfor-mance evaluation of multi-hop wirelessnetworks, cross-layer optimization, andprotocol design for wireless networks.

Brahim Bensaoureceived an EngineeringDegree in Computer Science (with dis-tinction) from the University of Scienceand Technology Houari Boumediene ofAlgiers Algeria in 1982, and a DEAdegree from University Paris XI inComputer Science in 1988. He earned hisDoctorate degree in Computer Sciencefrom the University Paris VI in 1993.From 1990 to 1994, he was a researchassistant at France Telecom Research

labs near Paris, where he was involved in the early designs andstudy of ATM technology. In mid 1995 he joined the Hong KongUniversity of Science and Technology as a Research Associatewhere he spent nearly 2 years working on various problems incongestion and traffic control. In 1997 he joined the Centre forWireless Communications, a national R& D center in Singapore(now known as the Institute of Infocomm Research, I2R A-Star)as a Member of Technical Staff, where he worked as a System

Architect on the design of Quality of Service (QoS) enabled MACprotocols and scheduling algorithms in a wireless ATM networkprototype. In 1998 he was promoted to a Senior Member ofTechnical Staff, and was instrumental in forming a small R&Dgroup in the area of wireless networking at the CWC. He led thegroup, for a year and a half, and then moved to Academia atHKUST in fall 2000 where he is now a faculty member in theDepartment of Computer Science and Engineering. His generalareas of research are in QoS-enabled wired/wireless networks,including ad hoc networks, sensor networks and wireless LAN,where he has published more than 80 research papers in promi-nent conferences and journals, received numerous researchgrants, graduated nearly 20 postgraduate students out of whichfour Ph.D.s and invented three US patents of which one is

licensed. He is an Associate Editor of The IEEE CommunicationsLetters and is a senior member of the IEEE, and a member of theACM.



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Farid Nat-Abdesselam obtained hisengineer degree in computer sciencesfrom University of Sciences and Tech-nologies of Algiers (USTHB) Algeria,in 1993 and a master degree in computersciences from University ReneDescartes(Paris 5) France, in 1994. After 2 yearsspent in the industry working as a soft-ware engineer, he joined the Universityof Versailles (UVSQ) France, and gothis Ph.D. degree in computer sciences in

2000. In 1998, he worked as an associate researcher at Universityof Western Ontario, London Ontario Canada. From 09/1999 to08/2000 he was an assistant professor at University of Sciences

and Technologies of Lille France. From 09/2000 to 08/2003 hewas an associate professor at INSA of Lyon and a researchmember of the INRIA Rhones Alpes. Since 09/2003 he is anassociate professor at University of Sciences and Technologies ofLille and research member of the INRIA Futurs. His researchinterests lie in the field of quality of service and security in IP-based networks, mobile ad hoc, sensor, and mesh networks, andpeer to peer networks. He has been on the technical programcommittee of different IEEE conferences, including Globecom,LCN, and ICC, and chaired some of their sessions. He has alsoserved as a technical program chair of the IEEE Workshop onWireless Local Networks 2007 (WLN07) and the InternationalWorkshop on Peer to Peer Networking 2007 (PPN07).


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