Design of a Probability Density Function Targeting Energy- Efficient Node Deployment in Wireless...

Design of a Probability Density Function Targeting Energy-

Efficient Node Deployment in Wireless Sensor Networks

Journal: IEEE Transactions on Network and Service Management

Manuscript ID: TNSM-2013-00583

Manuscript Type: Original Article

Date Submitted by the Author: 10-Oct-2013

Complete List of Authors: Halder, Subir; Dr. B. C. Roy Engineering College, Computer Science and Engineering Das Bit, Sipra; Bengal Engineering and Science University, Computer Science and Technology

Keywords:

Ad-Hoc and sensor networks < Management applications and case studies, Energy management < Management Functions, Network planning and

service deployment < Service Provisioning & Reliability and Quality Assurance

IEEE Transactions on Network and Service Management

1

Abstract—In wireless sensor networks the issue of preserving

energy requires utmost attention. One primary way of conserving

energy is judicious deployment of sensor nodes within the

network area so that the energy flow remains balanced

throughout the network and prevents the problem of occurrence

of energy holes. Firstly, we have analyzed network lifetime, found

node density as the parameter which has significant influence on

network lifetime and derived the desired parameter values for

balanced energy consumption. Then to meet the requirement of

energy balancing, we have proposed a probability density

function (PDF), derived the PDF’s intrinsic characteristics and

shown its suitability to model the network architecture considered

for the work. A node deployment algorithm is also developed

based on this PDF. Performance of the deployment scheme is

evaluated in terms of coverage-connectivity, energy balance and

network lifetime. In qualitative analysis, we have shown the

extent to which our proposed PDF has been able to provide

desired node density derived from the analysis on network

lifetime. Finally, the scheme is compared with three existing

deployment schemes based on various distributions. Simulation

results confirm our scheme’s supremacy over all the existing

schemes in terms of all the three performance metrics.

Index Terms—Energy balance, Node deployment, Network

lifetime, Probability density function, Wireless sensor networks.

I. INTRODUCTION

HE advent of efficient short range radio communication

and advances in miniaturization of computing devices

have made possible for large-volume commercial production

of wireless sensor nodes as well as large-scale real-world

deployment of the same to form a wireless sensor network

(WSN). Such a network typically suffers from a number of

unavoidable problems such as resource constrained nodes,

random node deployment some times in an unattended open

field where it is very difficult to replace/ recharge battery etc.

So the network as a whole must minimizes the energy usage in

order to enable untethered and unattended operation for an

extended period of time. Therefore, a critical consideration in

designing such WSNs is conserving energy to maximize the

post deployment network lifetime [1]. The rate of energy

S. Halder is with the Department of Computer Science and Engineering,

Dr. B. C. Roy Engineering College, Durgapur, India 713206 (fax: +91 343-

250-4059; e-mail: [email protected]).

S. DasBit is with Department of Computer Science and Technology,

Bengal Engineering and Science University, Shibpur, India 711103 (e-mail:

[email protected]).

depletion in the network primarily depends on the deployment

nature of the nodes. The nature of deployment, on the other

hand, mainly depends on the application environment [2].

In WSNs, nodes can be deployed either randomly or in pre-

determined manner. In random deployment, nodes are

deployed randomly, generally in an inaccessible terrain. For

example, in the application domain of disaster recovery or in

forest fire detection, nodes are dropped by helicopter in

random manner [3]. On the contrary, in pre-determined

deployment, the locations of the nodes are specified. This type

of deployment is used in applications when sensors are

expensive or when their operations are significantly affected

by their positions. The applications include placing imaging

and video sensors, populating an area with highly precise

seismic sensors, underwater WSN applications, monitoring

manufacturing plants etc.

One important way of conserving energy is by uniform

energy consumption or load distribution throughout the

network. Non-uniform dissipation of energy in any part of the

network may stop functioning of that part of the network

leading to a phenomenon known as the energy hole problem.

Sometimes even after the network lifetime is over, due to

energy hole problem a substantial amount of energy still

remains in the nodes leading to significant wastage of energy

[4].

The energy hole problem arises when more data are

transmitted by certain nodes of the network than the other

nodes resulting in extra energy dissipation of those nodes [5].

Therefore, if any part of the network is affected by the energy

hole problem, the whole network gets affected badly as uneven

consumption of energy in the network leads to premature

shortening of the network lifetime. To avoid this, care should

be taken during node deployment such that energy dissipation

of all the nodes takes place uniformly ensuring load balancing

throughout the network.

A. Motivation

Due to the nature of operation of WSN, nodes near the sink

bear the major share of data forwarding compared to the nodes

in rest of the network. So it is a common problem that nodes

near the sink get drained off more quickly than the other

nodes, thereby creating energy holes near the sink resulting in

loss of connectivity while most of the nodes are alive. Further,

it is well known that sensor nodes have limited battery life and

it is sometimes infeasible to replenish energy via battery

Design of a Probability Density Function

Targeting Energy-Efficient Node Deployment in

Wireless Sensor Networks

Subir Halder and Sipra DasBit, Member IEEE

T

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replacements and therefore prolonging network lifetime is of

utmost importance. In such a network with energy constraints

in order to prolong network lifetime nodes should be deployed

with varying densities depending upon its position from the

sink. One important issue that arises in such energy-

constrained networks is to avoid energy holes in order to

improve a network lifetime. To be more specific, we are

interested in finding the answer for the following question-“Is

it possible that all sensor nodes die simultaneously irrespective

of their positions from the sink so that there will be no energy

hole in a WSN thereby prolonging network lifetime?” This

motivates us to explore a solution by providing varying node

density in different parts of the network area based upon its

proximity from the sink.

B. Contributions and Organization

One of the solutions to avoid energy hole problem in WSN

while maintaining coverage and connectivity is to deploy

nodes with varying densities based on their positions from the

sink. On the other hand, another promising approach may be to

design a dedicated distribution function which perfectly

conforms to such a need of varying densities. We have

proposed a probability distribution function (PDF) based on

which we have developed a node deployment strategy which

not only keeps energy hole problem away from the network

but also ensures enhancement of network lifetime while

maintaining coverage and connectivity of the network. To the

best of our knowledge we are the first to propose such a

tailored-made PDF to be used in node deployment to avoid

energy hole problem. The preliminary version of this work was

reported in [6]. Our proposed PDF based node deployment

strategy is pre-determined in nature. Here, by pre-determined

strategy we mean that the number of nodes to be deployed in

an area is determined a priori. We extend our earlier work [6]

in several aspects. The main contributions of this paper are as

follows:

— Unlike [6] here, we analyze the method of controlling network lifetime and found node density as a major parameter which has significant role to control network lifetime. The main question to be addressed is: How many nodes per unit area (i.e., the node density) should be deployed in different parts of the sensor field in order to achieve energy balancing throughout the network and enhancement of network lifetime? The desired node densities derived out of the said analysis guarantee that all the nodes exhaust their energy at the same time, and hence, energy balancing is achieved.

— Based on the analysis, we propose a PDF and derive the

PDF’s intrinsic characteristics e.g. expectation, covariance

etc.

— We develop a node deployment algorithm based on the

PDF. It provides the node density to avoid energy hole

problem thereby achieves enhanced network lifetime

while maintaining coverage and connectivity of the

network.

— Performance of the scheme is evaluated both through qualitative and quantitative analyses. In qualitative analysis unlike [6], we analyze whether the PDF based scheme has been able to achieve the desired target set prior to the designing of PDF, towards energy hole

elimination.

— Similar to qualitative analysis, we show through

simulation the extent to which our proposed PDF has been

able to provide desired network lifetime. In addition to the

contribution in [6], here we show the impacts of routing

and medium access control (MAC) protocols on the

performance of the scheme.

The rest of the paper is organized as follows. In section II,

results from the literature review are presented. The network

model considered for the present work is presented in section

III. Analysis on network lifetime is done in section IV. Section

V presents the proposed node deployment scheme along with

the proposed PDF based on which the scheme is developed. In

section VI, the performance of the scheme is evaluated based

on both qualitative and quantitative analyses. Finally the paper

is concluded with some mention about the future scope of the

work in section VII.

II. LITERATURE REVIEW

Many works have been reported so far that deal with the

issue of balancing the load throughout the network with a goal

to reduce the energy hole problem for prolonging network

lifetime. All these works have been conducted through

different approaches for achieving this goal. Each type of the

above schemes has their own strengths and limitations. We

have categorized the existing works based on their

commonality in approaches which are presented below.

A. Transmission Paths Based Strategies

In each of the works described below, nodes choose routing

path differently.

1) Data traffic and distance based strategies

In these types of strategies, network lifetime is prolonged by

making the nodes choose routing path judiciously considering

data traffic along the path and distance. The distance

considered is either transmission distance between the node

and the sink or the distance between the node and

neighbouring nodes towards sink.

Azad and Kamruzzaman [7] have proposed energy balanced

transmission range regulation policies for maximizing network

lifetime in WSNs with corona based architecture. Firstly, they

have analyzed and found two parameters- ring thickness and

hop size - responsible for energy balancing. They have

proposed a transmission range regulation scheme of each node

and determined the optimal ring thickness and hop size for

maximizing network lifetime. Simulation results show

substantial improvements in terms of network lifetime and

energy usage distribution over existing policies. However,

before implementation of the proposed transmission policies it

requires significant computation to determine the optimal ring

thickness and hop size. Song et al. [8] have presented two

algorithms viz. centralized and distributed for the same corona

based network architecture. Network lifetime has been

optimized using a proposed decision factor computed by

selecting right transmission range of nodes in each corona.

They have claimed that the algorithms not only reduce the

complexity for searching right transmission range of node but

also obtain results approximated to the optimal solution.

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2) Probability based strategies

These types of strategies employ probability based routing

path selection with a target to balance energy consumption and

enhance network lifetime.

Boukerche et al. [9] have studied the problem of energy-

balanced data propagation in corona based WSNs for uniform

and non-uniform deployments. The authors have proposed a

density based probabilistic data propagation protocol towards

balancing the energy consumption. In each step a node in a

corona that holds data on-line, calculates the probability of

data delivery either by hop-by-hop or directly to the sink based

on the density information of the neighbouring coronas.

Finally the authors have shown that the proposed protocol

works well for both uniform and non-uniform network

deployments. Powell et al. [10] have proposed a probability

based data propagation algorithm where the nodes compute

off-line the probability of number of times data sent directly to

the sink and data sent to a next hop neighbour. They have

observed that by controlling the ratio of these two, energy

consumption in each layer/slice is balanced but individual

node in each slice is not well balanced. It is due to the unequal

number of neighbouring nodes in the adjacent slices. Jarry et

al. [11] have analyzed the data gathering and network lifetime

maximization problem and based on the analysis, the authors

have designed probabilistic on-line distributed routing

strategies for different network structures. They have

formulated the conditions to compute an energy-balanced data

propagation pattern for wireless network including WSN.

B. Deployment Based Strategies

The following works on enhancement of network lifetime

are based on judicious node deployment.

Olariu and Stojmenovic [12] have explored network design

guideline for maximizing lifetime while avoiding energy hole

considering uniform node deployment strategy. They have

shown that uneven energy depletion due to an energy hole is

unavoidable for free-space model, but it can be prevented in

two-ray model. Wu et al. [13] have proposed a non-uniform

node distribution where the number of nodes to be distributed

in a layer is determined based on the minimum number of

nodes required in the upper adjacent layer. They have

concluded that only sub-balanced energy depletion in the

network is possible. Liu et al. [14] have proposed a non-

uniform, power-aware distribution scheme where nodes are

deployed with the help of a proposed distribution function.

The authors have shown that their scheme increases network

lifetime and improves network service quality by maintaining a

continuous connectivity-coverage pattern. Chang et al. [15]

have proposed two node deployment strategies viz. distance-

based and density-based strategies, with an objective for

balanced energy consumption among the nodes. They have

shown that the proposed strategies can efficiently balance

energy consumption of each node and prolong the network

lifetime. Zou et al. [16] have formulated a problem on

uncertainty-aware sensor node placement and proposed two

algorithms using which one can determine the minimum

number of sensor nodes required and their locations such that

coverage is ensured. They have shown that the algorithms

provide optimized coverage compared to the random

deployment under the constraint of imprecise terrain

properties. Agarwal et al. [17] have also proposed a node

placement algorithm based on landmarks that are the set of

finite points in a 2-D space. They have claimed that the

algorithm requires minimum number of nodes for covering a

region. Wang et al. [18] have given an analytical model for the

coverage and the network lifetime issues using a 2-D Gaussian

distribution. They have proposed two deployment algorithms

that achieve larger coverage and longer network lifetime using

limited number of sensor nodes. They have concluded that the

proposed algorithms effectively increase network lifetime with

polynomial time complexity.

C. Mobile Sink Based Strategies

In these types of strategies, load among all the nodes are

distributed in a balanced manner by changing the position of

the sink so that network lifetime is enhanced.

Luo and Hubaux [19] have proposed a deployment

algorithm by considering mobile sink where the nodes lying

nearer the sink keep on changing resulting in even distribution

of load among the nodes. They have concluded that sink

mobility helps in optimizing the network lifetime. The authors

have further extended their work in [20] by introducing a

routing protocol which supports sink mobility as well as

minimizes packet loss during sink movement. Ammari and Das

[21] have also provided three different solutions for

eliminating energy hole problem. In one of the three solutions,

the authors have proposed a localized energy aware Voronoi

diagram based data forwarding protocol considering

homogeneous nodes and mobile sink.

In the above, a number of existing works on node

deployment are reviewed under different categories. Our

scheme generally belongs to the category of ‘Deployment

Based Strategies’. The existing works reviewed under this

category either have not used any mathematical model to

implement the node deployment scheme or even if any such

model is used, it has not addressed energy balance and

network lifetime together. Our scheme, on the contrary,

presents a node deployment strategy based on a proposed PDF

and addresses energy balance and network lifetime together.

III. NETWORK MODEL

In this section, we describe the network architecture along

with the assumptions. Definitions of coverage and connectivity

in association with sensing and communication model

respectively are described next. The energy model is presented

at the end of this section.

A. Architecture

The authors in [22] have proposed a realistic network model

for WSNs where the network area is covered by a set of

concentric circles centered at the sink. They have also

provided an in-depth discussion of its real life implementation.

Further, in [19] it is proved that in case of the said network

model, for enhancement of network lifetime, the best position

for a sink is the center of the circle. Primarily these two works

[19], [22] along with other relevant works [12], [13], [21],

[23] motivate us to consider present network architecture.


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However, in this architecture, energy consumption in annuli

can be kept balanced but individual nodes in the annuli may

remain imbalanced [10] and this has been mitigated by

considering q-switch [13] routing policy elaborated in section

VI.B.1.

Fig. 1. Layered network area.

We consider a square shaped network area a×a which is

covered by a set of uniform-width coronas or annuli (Fig. 1)

[22], [23]. Each such annulus is designated with width r as

layer. The sink is considered to be located at the center of the

network area and responsible for collecting data from the

sensors nodes. Nodes are placed in different layers

surrounding the sink. A layer is identified as iL where

i=1,2,...,N. Here i=1 indicates the layer nearest to the sink and

i=N indicates the layer farthest from the sink where 2

=

aN

r.

We assume that all the sensor nodes are homogeneous with

respect to their initial energy, sensing and communication

ranges while an unlimited amount of energy is set for the sink.

The nodes are static and need to be distributed in each layer

within the network with certain node density. The node density

[5], [13] is the ratio of number of nodes in an area (layer) and

the area (of the layer). It is also assumed that there is no local

coordination among the nodes and therefore, the nearby nodes

report same events to the sink. Periodic data gathering

applications are considered where sensory data generation rate

is proportional to the area (1 square unit) irrespective of the

shape of the network area. Given a unit area, if the data

generation rate is ρ bits per second, it is meant that this unit

area generates ρ bits/second of data to be transmitted towards

the sink. So, the data that needs to be reported from a given

area a×a is ρ×a×a bits. The data is collected by the nodes and

sent to the sink through multi-hop communication after a unit

time-interval. A single sink is responsible for gathering the

sensors’ data and controlling network operations. Further, we

have assumed that both the time-interval between two

successive generations of data per area and two successive

collections of data at the node as 1 sec. During theoretical

analysis, for the sake of simplicity, an ideal MAC layer with

no collision and retransmission is assumed and that does not

loss any generality. The reason for not considering real MAC

in theoretical analysis is the complexity involved in real MAC

such as need of control packet (TDMA scheduling),

retransmission in case of collision (CSMA/CA) etc. Moreover,

some of the overheads such as retransmission is a real time

parameter which needs probabilistic formulation adding

further complexity in theoretical analysis. However, in

simulations to make the assumption realistic, additionally we

consider a real MAC protocol for investigating the impact of

the protocol on network lifetime.

B. Sensing and Communication Model

1) Sensing model

We define a unit area to be covered, if every point in that

area is within the sensing range of at least one active node.

Alternatively, if each point in an area is covered by at least k

nodes, it is known as k-coverage. The nodes perform

observation [25] at an angle of 360o . The maximal circular

area centered around a node v that can be covered by it is

defined as its sensing area S(v). The radius of S(v) is called the

v’s sensing range [15] sR . We assume the relationship

between r and sR must satisfy the condition 2≤ sr R [6], [18]

for covering the network area (Fig. 1) under consideration. In

case of 1-coverage, node density (λ) 1 ( )= S v . Also, the

coverage area C(X) of a set of nodes X is the union of the

sensing areas covered by each node in X i.e.

( ) ( )∀ ∈= U v XC X S v [25].

2) Communication model

We define a network as connected if any active node can

communicate with any other active node, possibly using

intermediate nodes as relays, so that the information collected

by the nodes can be relayed back to sink [24]. We assume that

two nodes can directly exchange messages if their Euclidean

distance is not larger than the communication range cR .

Further, we assume that the relationship between r and cR

must satisfy the condition ≤ cr R [6], [18] for ensuring

connectivity in the network area (Fig. 1).

Lemma 1: For a given network area a×a, in order to

maintain connectivity of the network, the number of layers (N)

stands in relation with cR as 2

=

c

aN

R and to maintain

coverage the number of layers (N) stands in relation with sR

as 4

=

s

aN

R.

Proof: If the radius of each layer in the layered architecture

is r, then the distance between the center of the network area

and the periphery of a layer-i is i×r (Fig. 1). Now for the

farthest layer (i.e., layer-N), the distance between the center

and the periphery of layer-N is N×r and 2

= aN r (section III.A)

or, 2

=c

aR

N (putting = cr R ). Therefore, for a given network

area a×a, in order to maintain connectivity of the network the

relationship between cR and N should stand as 2

=

c

aN

R.

Similarly, we have 2 4

= =s

a ar R

N (replacing 2= sr R ).

Therefore, for a given network area a×a, in order to maintain

coverage of the network the relationship between sR and N

should stand as 4

=

s

aN

R.

C. Energy Model

We have considered the first order radio model [18] as the

energy model where energy consumption of a node is

dominated by its wireless transmissions and receptions; so the

other energy consumption factors such as for sensing and

processing are neglected. According to this radio model,

energy consumed by a node for transmission and reception is

as follows:

Energy consumption for transmitting n-bits data over a

Sink r

2r

Nr

L2

LN

L1

a

a


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distance d is ( ) 2, = +t elec ampe n d e n e n d .

Since we have assumed the transmission range of a node as

cR , the above can be rewritten by replacing d with cR :

( ) 2, = + =t c elec amp c te n R e n e n R e n (1)

where 2= +t elec amp ce e e R and te is energy required to

transmit one bit of data.

Energy consumption for receiving n-bits data is

( ) = =r elec re n e n e n (2)

where re is energy required to receive one bit of data and

=r elece e .

IV. ANALYSIS ON NETWORK LIFETIME

This section presents an analysis on network lifetime with

an objective to find out the parameter(s) which have

significant influence on network lifetime so that the lifetime

can be extended by controlling the parameter values. In

presence of several existing state-of-the-art definitions of

network lifetime [1] and lifetime of a node, the present work

considers the following definitions throughout the paper.

Definition 1. The node lifetime (measured in unit time) is

defined as the time when the initial energy of a node is

completely drained out so that it is neither able to transmit its

own data nor able to forward any data. It is measured as the

ratio of the initial energy of a node and the energy

consumption rate of the node. Considering initial energy of

each sensor node as 0ε and the energy consumption rate of a

node in layer-i as iECR , the lifetime of a node in layer-i is 0

ii

LTECR

=ε

.

Definition 2. The network lifetime is defined as the time

interval from the beginning of the network operation until the

proportion of dead nodes exceeds a certain threshold, which

may result in loss of coverage of a certain region, and/or

network partitioning [5]. If the total number of nodes in layer-i

is iT , the lifetime of a layer-i in the network is 0 0×=

×i

i i i

T

T ECR ECR

ε ε.

It is same as the lifetime of a node in layer-i as defined

previously. If any of the layer’s lifetime terminates which

causes loss of coverage within the layer, it results in

termination of network lifetime. Therefore, the network

lifetime can be determined by the shortest lifetime of a layer

and it is expressed as { }0min∀i

i ECR

ε.

We assume that a node requires minimum energy

consumption for transmitting/forwarding data to the sink.

Therefore, the node should transmit data to the sink via the

shortest path i.e. a node in layer-i (section III.A) requires i

hops to transmit data to the sink. This routing policy is

basically a simplified version of the q-switch, shortest path

routing policy proposed in [13] which is used in our scheme

and briefed in section VI.B.1. In this analysis the

simplification is made by considering a node in layer-i finds

only one node in layer-(i-1) to forward data towards sink.

Further, we assume that area of a layer-i (Fig. 1) is Ai=π(2i-

1)r2 where r is the width of a layer and iλ is the node density

of layer-i. Therefore, the total number of nodes in layer-i is

given as = ×i i iT Aλ for i=1,2,…,N. The nodes of all the layers

except the farthest layer from the sink spend their energy by

transmitting their own sensory data, receiving data from the

nodes of adjacent layers farther away from the sink and

forwarding the received data. Nodes in the farthest layer spend

energy only for transmitting their own data.

Therefore, the data transmission rate of a node (for

transmitting its own sensory data) in layer-i is

=××

i

i i i

A

A

ρ ρλ λ

. (3)

Further, a node in layer-i receives data from the nodes of

layer-(i+1) i.e. the nodes in layer-(i+1) transmit data towards

the nodes of layer-i to forward the same towards the sink. So,

the average data transmission rate of layer-(i+1) towards a

node in layer-i is 2

1 1 1 1

2

(2 1) (2 1)

(2 1)(2 1)

+ + + ++ += =

−−i i i i

i i ii

A i r i

A ii r

λ ρ λ π ρ λ ρλ λλ π

.

The rate of data relayed by a node of layer-i is the sum of the

above quantity for all the farther layers i.e.,

1 (2 1)

(2 1)

= + −∑−

Nhh i

i

h

i

λ ρλ

. (4)

Therefore, the total data transmission rate per node in layer-i

(mi) can be obtained from (3) and (4) as follows:

1 (2 1)1,2,..., ( 1)

(2 1)

= + −∑+ = −

−=

=

Nhh i

i ii

N

hfor i N

im

for i N

λ ρρλ λ

ρλ

. (5)

The first component of the above expression of mi i.e. iρ λ is

for transmitting the node’s sensory data and the second

component is for forwarding the outward adjacent layers’ data.

Now the energy consumption rate of a node in layer-i for

transmission is:

For transmitting the node’s own data:

× ti

eρλ

(6)

where te is energy required to transmit one bit of data.

For transmitting the relay data received from the farther

adjacent layers:

1 (2 1)

(2 1)

= + −∑×

−

Nhh i

ti

he

i

λ ρλ

. (7)

So the energy consumption rate of each node in layer-i due to

transmission ( )TxiECR is computed from (6) and (7) as

follows:

1 (2 1)1,2, ,( 1)

(2 1)

= + −∑+ = − − =

=

KN

hh it

i iTxi

tN

he for i N

iECR

e for i N

λ ρρλ λ

ρλ

.(8)

Similarly we calculate the energy consumption rate of each

node in layer-i for receiving ( )RxiECR data from the farther

layers as follows:

1 (2 1)

(2 1)

= + −∑= −

NhRx h i

i ri

hECR e

i

λ ρλ

, for i=1,2,…,(N-1) (9)

where re is the energy required to receive one bit of data.

Hence the total energy consumption rate of each node in a

layer-i is = +Tx Rxi i iECR ECR ECR .


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So the total energy consumption rate of the nodes of layer-i is

1,2, , ( 1)Tx Rxi i

iTxN

ECR ECR for i NECR

ECR for i N

+ = −=

=

K. (10)

We know, energy depletion across the network is balanced

[12] when all the nodes of the network exhaust their energy at

the same time. To be more specific, if balanced energy

depletion is attained in the network then all nodes located in

any layer have the same lifetime. Alternatively, all the nodes

exhaust their energy at the same time. Therefore, for energy

balancing, the following condition must be satisfied-

1i i NECR ECR ECR+= = =L .

Now rewriting (10) with the help of (8) and (9), we have

( ) ( )1 2

1 1

(2 1) (2 1)

(2 1) (2 1)

= + = +

+ +

− −∑ ∑+ + = + + − +

N Nh hh i h i

t t r t t ri i i i

h he e e e e e

i i

λ ρ λ ρρ ρλ λ λ λ

.

After simplification and basic transformations, we obtain

( )( )

11

2

(2 1) (2 1)2 1

2 1 (2 1) (2 1)

= ++

= +

− + + −+ = − + + + −

∑∑

Nt t r hh i

i i Nt t r hh i

e i e e hi

i e i e e h

λλ λ

λ. (11)

The above expression implies that the ratio of node density

between two consecutive layers depends on layer number i and

the total number of layers N. Further, the node density in a

layer is uniform but this node density varies in different layers.

The nature of variation is such, that the node density is

maximum at the layer nearest to the sink and it decreases in the

layers farther away form the sink i.e., 1 2> > >L Nλ λ λ . Now,

(11) is a non-linear equation and computation of iλ is fairly

complex. However, it suggests that one can compute iλ , for

i=1,2,…,(N-1), if Nλ is known. Considering 1-coverage

(section III.B.1), 1 ( )λ =N S v where S(v)=π2sR . Moreover,

the balanced energy consumption can be obtained in different

layers of the network if the nodes are distributed in accordance

with the desired node density given in (11).

When the nodes are distributed according to (11) to get

balanced energy consumption, we can ensure that all the nodes

deployed in the sensor field completely deplete their energy at

the same time. Now, from the Definition 1/2, the lifetime of a

node/the network lifetime (LT) can be expressed as: 0

i

LTi ECR=

ε

for i=1,2,…,N. Replacing the denominator with the help of

(10), (8) and (9) we have LTi as follows:

( )0

1

0

(2 1)1,2, ,( 1)

(2 1) (2 1)

(2 1)

= +

− = − − + + − =

= −

∑K

i

Nt t r hh i

i

N

t

ifor i N

i e e e hLT

for i Ni e

λ ε

ρ λ ρ

λ ερ

.(12)

As we have assumed earlier that the nodes in a layer report

data to the sink in minimum hops, therefore the derived

network lifetime (see (12)) provides the upper bound of the

network lifetime. Also we can say that the upper bound of the

network lifetime is achievable by controlling node density iλ

in each layer, as given in (11).

V. PROBABILITY DENSITY FUNCTION BASED NODE

DEPLOYMENT (PDFND)

From the analysis of network lifetime provided in the

previous section, we have found that the ratio of node density

between two consecutive layers depends on layer number i and

the total number of layers N. Further, for balanced energy

consumption the required node density is maximum in the

layer nearest to the sink and it decreases in the layers farther

away from the sink. Considering these observations and taking

guidelines from the analysis, in this section, we have designed

a PDF targeting its implementation in lifetime-enhancing node

distribution in WSNs. Also, we have presented a node

deployment algorithm based on the proposed PDF [6].

A. Proposed Probability Density Function [6]

The mathematical domain under consideration is divided

into a number of concentric circles having radii increasing

arithmetically from ′r to ( )′×N r with a difference of ′r . In

the mathematical domain, if (x, y) be a point and it lies

between circles (i-1) and i, then the probability density at that

point is

( ) ( )( ) ( )2 4

2 1−′ =

k if x, y; N ,i,r

N i, ( ) ( ) ( )2 2 22 21 ′ ′∀ − < + ≤i r x y ir (13)

where i=1,2,…,N and k is a constant as follows:

( )

( ) ( )( )

( )

( ) ( )( )

2 2

222 222

44 4 41

2 12 13 51

2 3

N

i

N Nk

iNrr

iN =

= = −′ −

′′ + + + +′

∑L ππ

.

Fig. 2 is the 3-D graph of the proposed PDF. The

characteristics of the PDF show decrease of the functional

value with increase in the value of i implying lower probability

and vice versa.

Theorem 1: The value of constant k is:

( )

( ) ( )( )

2

22 22

4 4 4

2 13 51

2 3

= −

′ + + + +

L

Nk

Nr

Nπ

.

Proof: Let ip denotes the probability of x and y in the given

area of the domain for given value of i. From the proposed

PDF, the probability ip is given as

( )( ) ( )

( )2 4

2 1,i

k ip f x y dx dy

N i

−= ∫∫ .

In the above relation ( ),∫∫ f x y dx dy is nothing but the area of

the domain. If the area of the domain is circular, then

( ) ( ) ( )2, 2 1f x y dx dy i r′= −∫∫ π , where r ′ is the radius of the

Fig. 2. 3-D graph (surface plot) of the PDF


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7

circular domain. The probability of x and y is given as

( ) ( ) ( )( ) ( )

2

2 4

2 1 2 1i

k i i rp

N i

′− × −=

π.

By fundamental rule of probability, ( ), 1=∑ ∑ f x y

1

1N

ii

p=

=∑ or, ( ) ( )

( ) ( )

2 2

2 41

2 11

N

i

k i r

N i=

′−=∑

π

( )

( ) ( )( )

( )

( ) ( )( )

2 2

2 22 22 2

4 4 4 41

2 1 2 13 51

2 3

N

i

N Nk

i Nr r

i N=

= = − −

′ ′ ∑ + + + +

Lπ π

.

Theorem 2: If the random variables X and Y follow a

proposed PDF with parameters N and i, then the cumulative

distribution function (CDF) of X and Y is given as

[ ] ( )( )

( ) ( )

( )

222 2

2 4 41

2 1,

i

j

ik r jF X x Y y

jN i=

− ′ − ≤ ≤ = +∑

ηπ.

We choose (x, y) such that ( )22 2 20 x y r′≤ + ≤ η , where

1≤ ≤ +i iη .

Proof: Let us consider the probability of two discrete random

variables X and Y for a particular value within a given range of

i as

( )( )

( )2 2

2 41

2 1i

j

k r j

jN =

′ −∑

π. (14)

The probability of the variables X and Y between a given area

of domain ir′ and r′η , where r ir′ ′>η is given as

( ) ( )( ) ( )2 2

2 4

kr ir

N i

′ ′− π η π . (15)

So, the CDF of X and Y is obtained using (14) and (15)

[ ] ( )( )

( ) ( )

( )

222 2

2 4 41

2 1,

i

j

ik r jF X x Y y

jN i=

− ′ − ≤ ≤ = +∑

ηπ.

Theorem 3: If the two random variables X and Y follow the

proposed PDF with parameters N and i, then the expectation of

X and Y is given as

[ ] ( )( ) ( ) ( ) ( )

4

2 3 2 41

2 2 3 1

2

N

i

k rE XY

iN i i i=

′ = + − −∑

.

Proof: The expectation of the two variables X and Y with

parameters N and i can be given as

[ ] [ ] [ ] [ ] [ ]1 21=

= + + + = ∑LN

N ii

E XY E XY E XY E XY E XY (16)

where [ ]iE XY is the expectation of X and Y for a given value

of i.

Now, [ ]( ) ( )2 4

= ∫∫ik

E XY xy dy dxN i

[ ]( ) ( ) ( )( )

( )( )

( ) ( )

( )

( )

2 21

2 40 2 21

2 2

2 41 0

4

4

ir xi r

i

i r x

ir xir

i r

kE XY x y dy dx

N i

kx y dy dx

N i

′ −′−

′− −

′ −′

′−

=

+

∫ ∫

∫ ∫

[ ] ( )( ) ( ) ( ) ( ) ( )

4

2 4 3 2 4

2 2 3 1

2i

k rE XY

iN i i i i

′ = + − −

. (17)

So, replacing [ ]iE XY in (16) with the value of (17) we get

[ ] [ ]1== ∑NiiE XY E XY

[ ] ( )( ) ( ) ( ) ( )

4

2 3 2 41

2 2 3 1

2

N

i

k rE XY

iN i i i=

′ = + − −∑

. (18)

Theorem 4: If the two random variables X and Y follow a

proposed PDF with parameters N and i, then the covariance of

X and Y is given as

( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( )

23

2 3 2 4 2 3 41

2 2 3 1 4 3 3 1,

32

N

i

k rCov X Y r

iN i i i i i i=

′ ′= + − − − + −

∑ .

Proof: From the definition of covariance [26] we know that

( ) [ ] [ ] [ ], = −Cov X Y E XY E X E Y .

We can find the covariance of the two discrete and random variables X and Y for a particular value i in the domain. The covariance of the two discrete and random variables X and Y for the entire domain is obtained by summing different values of the parameter i, where i=1,2,…,N

( ) ( ) [ ] [ ] [ ]1 1

, ,= =

= = −∑ ∑ N N

i i i ii i

Cov X Y Cov X Y E XY E X E Y .

We get the expectation of X and Y for a particular value of i,

i.e. [ ]iE XY from (17)-

[ ] ( )( ) ( ) ( ) ( ) ( )

4

2 4 3 2 4

2 2 3 1

2i

k rE XY

iN i i i i

′ = + − −

.

The expectation of X for a particular value of i, i.e. [ ]iE X can

be calculated as

[ ]( ) ( )2 4

= ∫∫ik

E X x dy dxN i

[ ]( ) ( )

( )( )

( )( )

( )

( )

2 21

0 2 21

2 42 2

1 0

4

ir xi r

i r x

i

ir xir

i r

x dy dx

kE X

N i

x dy dx

′ −′−

′− −

′ −′

′−

= +

∫ ∫

∫ ∫

[ ] ( )( ) ( ) ( ) ( )

3

2 2 3 4

4 3 3 1

3i

k rE X

N i i i

′ = + −

.

From the above equations, the expectation of X in the entire

domain is


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8

[ ] ( )( ) ( ) ( ) ( )

3

2 2 3 41

4 3 3 1

3

N

i

k rE X

N i i i=

′ = + −∑

. (19)

As our network model is symmetric, so the expectation of the

random variables X and Y in the entire domain is same. We can

say that [ ] [ ]=E X E Y . The covariance of random variables X

and Y is

( ) [ ] [ ] [ ]1 1

,= =

= −∑ ∑ N N

i i ii i

Cov X Y E XY E X E Y

( ) [ ] [ ] [ ] [ ]2

2

1 1

, [ ] .= =

= − = −∑ ∑

N N

i ii i

Cov X Y E XY E X E XY E X (20)

In (20), replacing by (18) and (19) we get covariance of

random variables X and Y as

( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( )

23

2 3 2 4 2 3 41

2 2 3 1 4 3 3 1,

32

N

i

k rCov X Y r

iN i i i i i i=

′ ′= + − − − + −

∑ .

Although in preliminary version [6] of this work, the PDF was

proposed, the proofs of the Theorem 3 & 4 are provided in this

present work only.

B. Proposed PDF-based Node Deployment

The PDF proposed in the previous section is discrete in

nature. Our objective is to deploy sensor nodes in the layered

network area (Fig. 1) with the proposed PDF. The PDF is

mapped with the node deployment in a layered network area as

follows: the parameter i represents the layer number for both

the proposed PDF and layered network area (see Fig. 1) where

i=1,2,…,N; the parameter r ′ of the proposed PDF corresponds

to the width r of the annuli/layer. Therefore, the relationship

between r ′ and cR is cr R≤′ whereas between r ′ and sR is

2 sr R≤′ . The density function is designed as per the analysis

in section IV. It is a non-uniform one i.e. the value of PDF is

higher for the nodes deployed around the sink whereas the

value is lower as one moves away from the sink. The PDF of

deploying a sensor node at point f(x, y) located in layer-i is

given as follows:

( )2 4

2 1k i

N i

− where i=1,2,…,N and k is the proportionality

constant.

Further, the probability for the nodes deployed within a

layer is ( ) 2 42 1− ik i A N i , where iA is the area of layer-i and k

is as follows:

( )

2

22 41 2 1N

i

Nk

r i i=

= −∑ π

(21)

where r is the width of the layer-i.

The area of layer-i is given as

( ) ( )22 2 2 21 2 1iA i r i r i r = − − = − π π π .

By replacing the value of iA , the probability ( )ip of

deploying nodes at a layer-i is given as

( )2 2

2 4

2 1i

k i rp

N i

−=

π. (22)

The number of nodes in a layer-i ( )iT ′ is equal to the

probability ( )ip of deployment of nodes at layer-i multiplied

by the total number of nodes ( )totalT i.e. i i totalT p T′= × .

Therefore, the node density of a layer-i according to our

proposed PDF is given as

1i i N

i total j jji i

p pT A

A A=′ ′= × = × ×∑λ λ

( ) ( ) ( )2

12 4

2 12 1N

i jj

k i rj

N i=

′−′ ′= −∑

πλ λ , for i=1,2,…,N. (23)

The above expression implies that the node density determined

by the proposed PDF is controlled by the parameters i and N

and that conforms to the guideline provided in section IV.

Further, the node density in a layer is uniform and probability

of deploying nodes is equal, but this node density as well as

probability varies in different layers. The nature of variation is

that the node density is maximum in the layer nearest to the

sink and it decreases in the layers farther away form the sink.

Therefore, we have 1 2 N′ ′ ′> > >Lλ λ λ .

An application area for our proposed PDF-based node

deployment may be a battlefield which requires more detailed

information around the monitoring station (sink) where the

sink is located around the centre position of the battlefield.

C. Algorithm for Node Deployment

Input: a, r, Ttotal; Output: i′λ // area parameter, width of layer,

and total number of nodes to be deployed

1: compute 2 c

aN

R

= ×

// N: number of layers; Lemma 1

2: compute k // using (21)

3: for i=N; i ≤ 1; i - -

4: compute ip // using (22)

5: compute i′λ // using (23)

6: end for

D. Illustrative Example

Let us consider a 200×200 sq unit area where 100 nodes

with 25cR = unit are deployed employing the proposed

probability density function. The number of layers N is:

2004

2 2 25c

aN

R

= = = × × (using Lemma 1).

Now replacing the values of N and cR in equation (21), the

value of k can be computed as

( )

2

2 2 22

4 4 4

40.004

3 5 725 1

2 3 4

k = =

+ + +

π

[where r=Rc].

Now the probability of deploying nodes in each of the four

layers is obtained by replacing k by 0.004, r by 25 and N by 4

in equation (22). For example, ip is obtained as,

( )2 40.49 2 1ip i i= × − i.e. 1 0.49p = . Similarly 2 0.27p = ,

3 0.15p = and 4 0.09p = . Using (23), node density in each of

the 4 layers is as follows: in layer-1,

1 0.00024 100 0.024′ = × =λ , in layer-2, 2 0.0045′ =λ , in layer-

3, 3 0.0015′ =λ , in layer-4, 4 0.00065′ =λ .

We observe that the node density in each layer obtained

from the algorithm conforms to the non-uniform nature of the

PDF. Therefore, it fulfils our objective of deploying more

number of nodes towards the sink and decreasing the number

of nodes as the distance from the sink increases.

Finally, we claim that the proposed deployment is feasible.


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9

As reported in a state-of-the-art work [27] on design and

deployment of WSN, air-dropped deployment in a controllable

manner is feasible even in an inaccessible terrain. We propose

to compute the node density in each part (layer/annuli) of the

network off-line prior to the actual deployment. At last, the

nodes are to be dropped (e.g. from helicopter) using a point

(sink) as the center following the pre-computed node densities

of the proposed PDF.

Unlike the preliminary version [6], in the proposed node

deployment (V.B, V.C and V.D) we have considered node

density instead of number of nodes in each layer as the

parameter of concern for making the scheme energy balanced

thereby getting enhanced network lifetime. This is as per the

guideline of the analysis done in section IV.

VI. PERFORMANCE ANALYSIS

Performance of the present node deployment strategy is

measured based on two parameters such as energy balance and

network lifetime. Both qualitative and quantitative analyses are

presented here.

A. Qualitative Analysis

In this section we have analyzed the performance of the

proposed PDF qualitatively to show in which extent our node

deployment scheme (section V) is close to fulfill the desired

objective (section IV).

1) Energy balance

The network is said to be energy balanced when the nodes

located in any layer have the same lifetime. Based on this

condition we have derived the desired node density ( )iλ of a

layer using (11) (see section IV) as follows:

( )( )

11

2

(2 1) (2 1)2 1

2 1 (2 1) (2 1)

Nt t r hh i

i i Nt t r hh i

e i e e hi

i e i e e h

= ++

= +

− + + −∑+ = − + + + −∑

λλ λ

λ.

On the other hand, in our node deployment strategy, nodes are

deployed in different layers with varying node density ( )′iλ

which is achieved node density and is given as (section V.B,

see (23))

( ) ( )2

12 4

2 12 1N

i jj

k i rj

N i=

−′ ′= −∑

πλ λ .

It is observed from both the above equations that node

density among the layers is non-uniform in nature but uniform

within the layers. Moreover, as 1 2 NA A A< < <L , the nature

of variation of node density is that it is maximum in the layer

nearest to the sink and it decreases in the layers farther away

form the sink.

To see the effectiveness of the proposed scheme, Fig. 3

plots both the desired and achieved node densities for two

different network sizes. After deriving (see section IV) the

node density in the farthest layer for both the cases, desired

and achieved node densities of the remaing layers are

calculated iteratively using (11) and (23) respectively. It is

clear from the plot, in all the cases desired and achieved node

densities are almost same and that indicates the proposed

scheme has been able to achieve energy balancing.

2) Network lifetime

The desired lifetime, according to (12), is as follows:

( )

0

1

0

(2 1)1,2, ,( 1)

(2 1) (2 1)

(2 1)

= +

− = − − + + − =

= −

∑K

i

Nt t r hh i

i

N

t

ifor i N

i e e e hLT

for i Ni e

λ ε

ρ λ ρ

λ ερ

.

Now following the method for calculating iLT (section IV, see

(12)) we derive iLT ′ as follows:

( )0

1

0

(2 1)1,2, ,( 1)

(2 1) (2 1)

i

Nt t r hh i

i

N

t

ifor i N

i e e e hLT

for i Ne

= +

′− = − ′− + + −∑′= ′ =

Kε λ

ρ λ ρ

ε λρ

.

If we compare iLT (desired) and iLT ′ (achieved), as iλ and i′λ

are found almost same, iLT and iLT ′ are also same.

3) Coverage and connectivity

In addition to the energy balance and network lifetime, we

have also measured coverage and connectivity to show the

extent of maintaining coverage and connectivity by the

proposed node deployment strategy. This section formulates

necessary constraints to be satisfied for maintaining coverage

and connectivity. It also contains a couple of Lemmas along

with the proofs with an objective to show the extent of

maintaining coverage and connectivity by the PDF. To

measure the coverage, the concept of coverage density ( )iC

[28] has been used. If the sensing area S(v) (refer section

III.B.1) of each node is mutually exclusive, the coverage

density iC of layer-i is defined as

( )ii

i

T S vC

A

′×= .

If 1iC = i.e.1-coverage (section III.B.1), we say that iA is

covered by minimum number of nodes and coverage area of

each node is mutually exclusive. If 1iC > i.e. k-coverage, we

say that iA is covered by more than the minimum number of

-1

-0.5

0

0.5

1

1.5

2

1 2 3 4 5Layer number

Node

den

sity

(node/

sq. m

)

Achieved node density

Desired node density

×10-3

(a) 5-layer network.

-1

1

3

5

7

9

11

1 2 3 4 5 6 7 8 9 10

Layer number

Node

den

sity

(node/

sq. m

)

Achieved node density

Desired node density

×10-3

(b) 10-layer network.

Fig. 3. Node densities in each layer for various network sizes.


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10

nodes and therefore, coverage area of a node is overlapped

with the coverage area of the other nodes in the area. The

sensing accuracy would increase proportionally with the

overlapping of coverage area, thus making the scheme more

robust against sensing failure.

Lemma 2: For a given network area, the proposed PDF

gives the coverage density of a layer-i as ( )4 2 1i iC T i′= − .

Proof: Let us consider iT ′ numbers of nodes are deployed

in layer-i. The S(v) of each sensor is calculated as 2sRπ where

sR is the sensing radius of each sensor. So,

( ) ( ) ( ) 2ii v T i i sC T S v T S v T R∀ ∈′ ′ ′= = × = ×U π .

From the definition of Ci we have

( )

2

2

( )

2 1

i i si

i

T S v T RC

A i r

′ ′× ×= =

−

π

π.

Replacing r by 2 sR , we have ( ) ( )

2

2 4 2 12 1 4

i s ii

s

T R TC

ii R

′ ′×= =

−−

π

π.

From the above expression it is observed that, the coverage density of layer-i depends on the number of nodes deployed in layer-i and it is inversely proportional to layer number i, which suits the requirement for energy balancing (section IV). Also, one can achieve the desired coverage density by controlling the number of deployed nodes ( )iT ′ in various layers within the network.

Lemma 3: For a network, if coverage is ensured,

connectivity of the network is also ensured.

Proof: We have discussed that when the network area is

covered by the minimum number of nodes, then 1iC = . Now

when the network is covered by minimum number of nodes,

the maximum distance between the two nodes is 2 sR . In our

communication model we have assumed that two nodes can

communicate with each other if the Euclidean distance

between them is less or equal to cR and we have considered

2c sR R≤ . As the maximum distance between the nodes is

2 sR and 2c sR R≤ so we can say that the connectivity is

guaranteed if coverage is ensured.

B. Quantitative Analysis

The effectiveness of the proposed node deployment scheme,

reported in section V.B is evaluated through simulation.

Moreover all the theoretical claims made through qualitative

analysis presented in section VI.A are justified by simulation

results. Simulation results of our scheme PDFND are

compared with three existing node deployment schemes

namely non-uniform node distribution strategy (NNDS) [13],

node deployment with Gaussian distribution (NDGD) [18] and

node deployment with Uniform distribution (NDUD) [29].

1) Simulation environment

The simulation is performed using MATLAB (version 7.1).

We have done qualitative analysis considering simplified q-

switch routing [13]. However, in simulation we have used the

same routing protocol with no simplification for all the three

schemes. This routing protocol is briefed along with the

scheme [13] in the next paragraph.

In NNDS the authors have proposed a non-uniform node

distribution strategy for the uniform-width corona model. Here

the nodes are deployed in such a way that the node densities in

a corona increases in geometric proportion with common ratio

q (>1) from corona (N-1) to corona 1. We assume that the

number of nodes deployed at the farthest corona and the

common ratio are known a priori. Once the number of nodes

deployed at the farthest corona and the common ratio are

known, nodes for the remaining coronas are computed and

these computed numbers are exponentially increasing function

of the common ratio q. The NNDS uses q-switch routing

where the source node always selects one reachable relay node

with maximum remaining energy in its subsequent inner layer

to forward data. If there is more than one relay node with the

same maximum remaining energy, one of them is chosen

randomly. Once the source node selects the relay node, it

forwards the data of its own as well as those received from the

nodes of adjacent layers farther away from the sink. The

selected relay node repeats this process until the data arrives at

a node in layer-1, after which the data is sent to the sink.

Hence, the routing itself takes care of individual node’s load

balancing and that eliminates the problem [10] of annuli

architecture as stated in section III.A.

In NDGD, the authors have considered the nodes are

deployed using two dimensional Gaussian distribution and

node density function at point ( )i if x , y as-

( )2 2

2 22 21

2

i i

x y

( x x ) ( y y )

x y

f x, y e

− − − + = σ σ

πσ σ,

where xσ and yσ are the standard deviations for x and y

dimensions. Further, the authors have considered two

deployment types: x y=σ σ and x y≠σ σ . However, during

simulation we have considered x y= =σ σ σ , which conforms

to a disk model and that is similar to our network model. So,

the probability density function of deploying a sensor node for

point (x, y) is:

( )2 2

222

1,

2

x y

f x y e+

−= σ

πσ.

It is evident from the above equation that any two points in the

disk having same distance from the center-point, have the

same deployment probability.

In NDUD, nodes are uniformly and independently

distributed in the layered network area, the probability af that

a point is covered by sensor nodes is- 21 sR

af e−= − λπ

where sR is the sensing range of the nodes and λ is the node

density.

We simulate our work both under ideal scenario and

realistic scenario. Here, by the ideal scenario we mean the

scenario considered during theoretical analysis (section IV and

section VI.A) i.e., simplified q-switch routing protocol, ideal

MAC layer and the energy consumption only for transmission

and reception. On the other hand, in the realistic scenario we

consider q-switch routing protocol and real MAC protocol

which includes idle/sleep schedule of the nodes. Moreover,

unlike ideal scenario, in realistic scenario energy consumption

is considered for idle, sleeping and sensing in addition to

transmission and reception. The real MAC protocol has been

implemented by funneling-MAC [30]. The funneling-MAC is

a hybrid MAC protocol where TDMA (schedule-based) is

used in nodes located within a few hops from the sink whereas


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11

CSMA/CA (contention-based) is used in nodes located far

away from the sink. The sink broadcasts a beacon for nodes

located within a smaller number of hops by controlling the

transmission power of the beacon. The nodes which receive

the beacon are considered as f-nodes and perform TDMA

while the nodes that do not receive the beacon perform

CSMA/CA. During simulation we have considered the nodes

located within layer-2 use TDMA schedule whereas nodes

beyond layer-2 use CSMA/CA. We have considered energy

consumption rates for sensing, remaining idle and remaining

sleeping are 20%, 5%, 2.5% of the energy consumption rate of

reception respectively. Further, in simulation, all the

funneling-MAC implementing parameters (e.g. slot size,

superframe size, moving average factor) values are considered

same as in [30].

During implementation of the NNDS, NDGD, NDUD, and

PDFND, we have deployed 500 and 2000 nodes for network

with 5 and 10 layers respectively. For all the schemes, in order

to have an integer number of sensor nodes for each layer the

upper ceil function is employed. For NNDS, the common ratio

q is considered as 2. Extensive simulation has been performed

with a confidence level of 95%. The average results of 2000

independent runs have been taken while plotting the simulation

graphs.

2) Simulation metrics

Similar to qualitative analysis (section VI.A) energy

balance, network lifetime and coverage-connectivity have been

considered as performance metrics in simulation. We define

two more parameters namely energy consumption rate per

node and average residual energy per node for evaluating the

extent of energy balance in the network. Further, we evaluate

coverage-connectivity using the parameter coverage density

(VI.A.3).

Energy consumption rate per node (ER): It is defined as

energy consumption of a node per unit time.

Average residual energy per node (Avg RE per node): It is

defined as the residual energy in a node of a particular layer

after the network lifetime ends. It is evaluated as follows:

Sum of residual energy of nodes in a layerAvg RE per node=

Number of nodes in the layer.

We have conducted two sets of experiments. One set of

experiments is to compare simulation results with analytical

results considering network lifetime as the parameter. In the

other set of experiments our scheme is compared with the three

other competing schemes considering both ideal and realistic

scenarios. In this set of experiments again energy balance,

network lifetime and coverage-connectivity are considered as

performance metrics. All the parameters and their

corresponding values used for simulation are listed in Table I.

3) Comparison of results (Analytical vs Simulated)

In this section, the analytical (section IV) performance of

the scheme in terms of network lifetime is compared with the

simulated (section V) performance and the results are plotted

in Fig. 4. Here both the set of results consider ideal scenario.

We observe from the plot that the nature of graph for the

analytical result is perfectly straight irrespective of network

sizes whereas the simulation result is fairly straight and that

indicates the algorithm has been able to provide almost perfect

energy-balanced network lifetime as desired by the theoretical

analysis. We also observe that the network lifetime decreases

with the increase of network sizes. This is because the data

traffic increases while the network size increases, especially

for the layers nearer the sink. Finally, the most important

observation is, for both the 5-layer (Fig. 4(a)) and 10-layer

(Fig. 4(b)) network sizes, analytical results and the average of

simulation results are almost same. The slight differences

between the analytical and simulated results are due to the

minute variation of desired and achieved node densities (refer

Fig. 3).

4) Comparison of results (Competing schemes)

This section compares our scheme’s performance with the

three competing schemes considering both ideal and realistic

scenarios.

a) Energy balancing [6]

In this section energy balancing of the scheme is evaluated

in terms of the following two parameters.

(1) ER (Energy consumption rate per node)

Figure 5 shows the ER for different network sizes. We

observe that in PDFND, for both ideal and realistic scenarios,

the ER for a particular network size is constant for all the

layers and this rate varies with network sizes. Precisely ER

increases with increase in network size. For example, in case

of ideal scenario, ER is 1.01 mJ/sec for 5-layer network

whereas for 10-layer network it is 1.19 mJ/sec. Similarly, in

realistic scenario, ER is 1.21 mJ/sec for 5-layer network

whereas for 10-layer network it is 1.45 mJ/sec. On the

contrary, in NNDS, NDGD and NDUD it is observed that the

ER varies in different layers for a given network size. Further,

in NNDS, NDGD and NDUD, irrespective of network size,

nodes in the layer-1 have the maximum ER and nodes in the

farthest layer have the lowest ER. Therefore, nodes deployed

in the layers nearer the sink drain out their energy much more

quickly in comparison to nodes deployed in layers farther

away from the sink. This justifies our claim that PDFND is

relatively more energy balanced compared to all the competing

schemes NNDS, NDGD and NDUD.

Now for all the schemes if we compare the results of ideal

and realistic scenario, it is observed that the ER (realistic) in

all the cases is higher compared to ER (ideal). The additional

energy usage for realistic scenario is due to the implementation

of MAC protocol. Another important observation is in realistic

TABLE I

PARAMETER VALUES USED IN SIMULATION

Parameters Values

Initial energy ( )0ε 50 J

elece 50 nJ/bit

ampe 10 pJ/bit/m2

Communication range of a node ( )cR 160m

Sensing range of a node ( )sR 80m

Width of each annuli (r) 160m

Data generation rate (ρ) 0.1 bits/sec

Gaussian standard deviation (σ) 70

Network size 5~10 layers


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12

scenario, ER nearer the sink is less compared to the ER away

from the sink. As CSMA/CA is used in nodes away from the

sink, unlike TDMA, number of collisions, however infrequent,

is non-zero and this justifies the above result. In numerical

value, for PDFND irrespective of network sizes, the average

increase in ER, in realistic scenario compared to the ideal

scenario are 19% and 20%, for the nodes nearer the sink and

far away from the sink respectively. Similarly these values are

20% and 22% for NNDS, 22% and 24% for NDGD, 5% and

10% for NDUD.

(2) Avg RE per node (Average residual energy per node)

Figure 6 illustrates the comparison considering avg RE per

node as a performance metric. We observe that node

deployment using NDGD or NDUD results in relatively abrupt

change in avg RE per node in each layer and this nature

remains independent of network size. For example, in NDGD

(Fig. 6) nodes in each of the two layers viz. layer-5 and layer-

6, have drained off completely, though the nodes of other

layers in the network have sufficient energy for carrying out

normal network operation, causing the phenomenon known as

energy hole. Similarly, in NDUD, the energy in nodes of layer-

1 has drained off completely though the nodes of other layers

in the network have adequate energy for normal network

operation. So, both NDGD and NDUD suffer from the energy

hole problem. In NNDS, the plots upto certain layers starting

from the nearest layer from the sink are relatively flat

compared to the results in rest of the layers and that implies

energy wastage caused by imbalance in energy consumption

among the layers. Therefore, NNDS also suffer from energy

imbalance problem affecting network lifetime. However, the

PDFND plot is almost a straight line indicating that all the

nodes in each layer exhaust energy almost completely ending

the network lifetime. For example, in case of ideal scenario it

leaves less than 0.2 nJ energy for 5-layer network whereas for

10-layer network it is 0.32 nJ. Similarly, in case of realistic

scenario, it leaves less than 0.25 nJ energy for 5-layer network

whereas for 10-layer network it is 0.29 nJ. Therefore, we can

say that PDFND is energy balanced and utilizes energy, the

scarcest resource, more efficiently than the other deployment

schemes.

b) Network lifetime [6]

In this section network lifetime is evaluated for various

network sizes.

The graphs illustrated in Fig. 7 represent the network

lifetime for two different network sizes. For ideal scenario, it is

observed that the network lifetime of PDFND is 18.28%,

48.40% and 350% more than that of NNDS, NDGD and

NDUD respectively for 5-layer network. For 10-layer network

it is 19.83%, 42.30% and 380% more than that of NNDS,

NDGD and NDUD respectively. It is also observed that with

increase in network size network lifetime decreases, e.g. for 5-

layer network it is 816.21 mins whereas for 10-layer network it

is 683.06 mins. This is due to the fact that with increase in

network size, the nodes in the innermost layer need to relay

increased volume of data from the outer layers thereby causing

0.5

1.5

2.5

3.5

4.5

5.5

6.5

7.5

8.5

9.5

1 2 3 4 5Layer nember

ER

(m

J/se

c)

P DFND (Ideal) P DFND (Real)

NNDS (Ideal) NNDS (Real)

NDGD (Ideal) NDGD (Real)

NDUD (Ideal) NDUD (Real)

0.5

2.5

4.5

6.5

8.5

10.5

12.5

14.5

16.5

1 2 3 4 5 6 7 8 9 10Layer number

ER

(m

J/se

c)



NDGD (Ideal) NDGD (Real)NDUD (Ideal) NDUD (Real)

(a) 5-layer network. (b) 10-layer network.

Fig. 5. Energy consumption rate per node.

816.15

816.17

816.19

816.21

816.23

816.25


Net

work

lif

etim

e (m

ins) For λ' (Simulated)

For λ (Analytical)

683.09

683.094

683.098

683.102

683.106

683.11

1 2 3 4 5 6 7 8 9 10

Layer number

Net

wo

rk l

ifet

ime

(min

s)

For λ' (Simulated)

For λ (Analytical)


Fig. 4. Network lifetime for various network sizes (analytical vs simulated).


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13

higher energy consumption. Moreover, in PDFND the flat

nature of the plot ensures that in all the layers, network

lifetime terminates in more or less same time as compared to

NNDS, NDGD and NDUD. This ensures that energy in

PDFND is balanced to a greater extent than all the competent

schemes.

Now if we compare the simulation results of network

lifetime both for ideal and realistic scenarios, network lifetime

is reduced in realistic scenario, as there is additional energy

consumption due to the implementation of MAC protocol.

Further, in realistic scenario, irrespective of network sizes, the

reduction in network lifetime is less near the sink compared to

other parts of the network. To be more specific, in PDFND

when reduction is 19% near the sink, it is 20% in rest of the

network. Similarly in NNDS, NDGD and NDUD these values

are 20% & 22%, 22% & 24% and 4% & 10% respectively. As

CSMA/CA is used in the entire network area except near the

sink, due to collision and retransmission additional energy is

consumed compared to near the sink where TDMA is used.

From the above observations, it is also revealed that the impact

caused by inclusion of realistic scenario on network lifetime is

the highest in NDGD and the least in NDUD.

Although a subset of results on energy balancing and

network lifetime was presented in [6], here the entire result set

is compared (Fig. 5, 6, 7) with one additional competing

scheme NNDS [13]. Moreover, for all the competing schemes

including ours, an additional set of results are plotted using

real MAC.

c) Coverage and connectivity

In Fig. 8, we plot coverage density in all the layers for 5-

layered network. Our primary observation is that except the

scheme NDUD, in all the other schemes i.e. PDFND, NDGD

and NNDS, coverage density reduces in layers as the distance

of layers from the sink increases fulfilling the objective of

deploying more nodes near the sink. The next observation is

NDGD gives more overlapping sensing coverage in layer-2,

and NDUD in layer-5 but NDGD fails to give any overlapping

sensing coverage (see section VI.A.3) in layer-5. To be more

specific, in layer-5, coverage density is less than one implying

NDGD’s incapability of providing coverage. On the other

hand, NDUD provides uniform coverage density in all the

layers but that does not provide energy balancing requirement.

0

5

10

15

20

25

30


Cov

erag

e d

ensi

ty

PDFND

NDGD

NNDS

NDUD

Fig. 8. Coverage density for 5-layed network area.

-1

5

11

17

23

29

35

41

47

53

59

65

71

12345Layer number

Av

g.

RE

per

no

de

(J)

P DFND (Ideal)

P DFND (Real)

NNDS (Ideal)

NNDS (Real)

NDGD (Ideal)

NDGD (Real)

NDUD (Ideal)

NDUD (Real)

-1

5

11

17

23

29

35

41

47

53

59

65

12345678910

Layer number

Avg

. R

E p

er n

od

e (J

)

P DFND (Ideal)

P DFND (Real)

NNDS (Ideal)

NNDS (Real)

NDGD (Ideal)

NDGD (Real)

NDUD (Ideal)

NDUD (Real)


Fig. 6. Average residual energy (RE) per node.

50

200

350

500

650

800

950

1100

1250


Net

work

lif

etim

e (m

ins)





50

150

250

350

450

550

650

750

850

950

1050

1 2 3 4 5 6 7 8 9 10Layer number

Net

wo

rk l

ifet

ime

(min

s)






Fig. 7. Network lifetime for various network sizes.


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14

However, NNDS provides coverage density as per the

requirement of energy balancing. Finally, we claim that our

scheme PDFND not only provides coverage density as per the

requirement of energy balancing but also provides higher

coverage density almost in all the layers compared to the most

competing scheme NNDS.

VII. CONCLUSION AND FUTURE WORK

In this work we have proposed a node deployment scheme

for multi-hop WSNs using a PDF defined by us. The target of

the scheme is to achieve energy balancing and enhancing

network lifetime while maintaining coverage and connectivity.

To start with, we have analyzed network lifetime and

identified node density as a parameter which has significant

influence on network lifetime. Then, theoretical formulation of

node density for balanced energy consumption is presented.

Based on the analysis of network lifetime we have designed a

PDF targeting its implementation in lifetime-enhancing node

distribution in WSNs. Intrinsic characteristics of the PDF and

its suitability for modeling the network architecture of this

work are discussed. A node deployment algorithm is also

developed based on the proposed PDF to implement the

scheme. Further, we have provided theoretical formulation of

coverage-connectivity, energy balancing, network lifetime and

have derived certain constraints, involving some important

network parameters, to be satisfied to achieve the target. We

claim that our scheme successfully achieves the target. The

claims are substantiated by performing both qualitative and

quantitative analyses. Finally, the results of quantitative

analysis are compared with three existing works [13], [18],

[29] on node deployment and that clearly demonstrates our

scheme’s dominance over the existing works.

As a future extension of our work, the deployment strategy

may be made more realistic by considering 3-D environment.

Moreover, the scheme may be analyzed with a target to obtain

optimal node density by considering various QoS parameters.

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