Energy-aware network configuration for Wireless Sensor Networks

Ada Gogu, Dritan Nace and Yacine Challal

Laboratoire Heudiasyc UMR CNRS 6599

Universite de Technologie de Compiegne

60205 Compiegne Cedex, France

{ada.gogu, dnace, ychallal}@hds.utc.fr

Abstract—This work addresses the problem of designing theoptimal network configuration for Wireless Sensor Network(WSNs). Specifically, given a surveying area our goal is todetermine the parameters of network configuration such thatsome objectives are met. We begin by studying the config-uration for a linear network and then extend our work tothe two dimensional network area. For the linear networkcase we target two objectives: i) minimizing the overall energyconsumption ii) and guarantying the node energy consumptionfairness. Then we discuss the overall energy consumptionminimization problem where the transmission energy takesonly some given discrete values. Next, we study the networkarea configuration problem and show how the computationmethod given for the linear case can be extended to handle it.The solution method for these problems is based on dynamicprogramming algorithm. This method has a low complexitycompared to other methods proposed in the literature andit can be a great help for network designers thanks to itssimplicity in implementation.

Keywords-WSN, Dynamic programming, network configura-tion;

I. INTRODUCTION

WSN technology needs to be carefully managed in order

to meet applications’requirements due to the stringent con-

straints such as energy, bandwidth, memory, etc. Optimiza-

tion techniques and strategies are applied at physical, access

control, network and application layer of this technology

to improve its performance. However a primary concern of

wireless sensor networks is the energy constraint. A careful

design of node deployment scheme can be a very effective

optimization mean for achieving energy conservation and

therefore the network lifetime extension.

In this study, we focus on the design of the optimal

network configuration strategy to meet a given objective.

We will examine a many-to-one wireless sensor network

where information is collected periodically from the sensors

to the base station. Such applications can be data monitoring

for agriculture, weather forecast, etc. Firstly, we consider

a linear network where nodes are placed uniformly. The

whole network will be divided in linear segments that we

will call cells. The configuration problem needs to determine

the number of cells and the size of each of them such

that: i) the total energy consumption is minimized, and ii)

every cell consumes the same energy. Moreover, the system

configuration problem for energy minimization is adapted

for the case where the sensor power transmission takes

discrete values. Next, we study the same problem for a two

dimensional network area.

The rest of the paper is organized as follows. Section II

describes briefly a state of the art for the problem in hand

and revisits some important questions of our previous work.

In section III, we present the linear network model and the

problem definition. Section IV is dedicated to the solution

method for the linear network configuration problem, the

respective algorithms for different optimization criteria and

their results. The two dimensional network configuration

problem, the solution method and results are discussed in

section V. We conclude the paper in section VI.

II. BACKGROUND

A. State of the art

Some similar problems discussed in the literature are

known as node sensor placement and network division for

data gathering sensor networks. The node sensor placement

problem intends to deterministically place the sensors in

order to meet some requirements such as minimization of

energy consummation [3], [4], fault tolerance [5], lifetime

maximization for event-driven networks [2], throughput sta-

bility under attacks [10]. While the network division problem

asks to divide the network in grids or coronas (if the surface

of the network is assumed circular) such that it meets certain

objectives.

In [3] the problem of node placement for minimizing

the total power consumption is formulated as a constrained

multi-variable nonlinear programming problem, which is

solved at optimality. However the considered energy con-

sumption model is very reduced and the authors do not take

in consideration the receiving or idle energy consumption. A

more realistic energy model is taken in account in [2] where

lifetime maximization is achieved through the minimization

of power consumption for each node. Hence, given distance

d from the Base Station (BS) the problem asks to define the

positions and the total number N of nodes which maximize

the network lifetime. They have formulated a multi-variable

nonlinear optimization problem to get the node position for

any given N . Then, they solve numerically this problem

and use the obtained solution to optimize the number Nof nodes. Whereas [5] discusses the problem of relay node

2011 International Conference on Network-Based Information Systems

978-0-7695-4458-8/11 $26.00 © 2011 IEEE

DOI 10.1109/NBiS.2011.14

22

B a s e S t a t i o n B a s e S t a t i o n

a ) b )

S o u r c e S o u r c e

1

2

3

n

2

3

n

1x

x

x

x

S c e n a r i o S c e n a r i o

Πni=1

ki

knd d

Figure 1: The multihop transmission for linear network

placement to ensure the k connectivity between the nodes

in the single and two-tiered network with one or more base

stations. The authors conclude that several (4) variants of the

problem are NP-hard and propose approximation algorithms.

The time complexity is O(n4) where n is the number

of sensor nodes. Despite the heaviness of the proposed

methods, it is shown that allocating nodes properly brings

to performance improvements of both network lifetime and

total power consumption.

Another similar problem concerns the network configura-

tion one. Zhang and Shen in [12] propose a method to divide

the network in coronas, then the coronas in subcoronas and

finally the subcoronas in zones such that the load for every

zone is balanced. The problem of finding the optimal number

of coronas is modeled as an optimization problem and a

simulated annealing algorithm is proposed. Then the division

of coronas in subcoronas and zones is solved iteratively.

This network division scheme assumes that the coronas and

the subcoronas have the same width and the number of

zones is the same for each of them. While in [11], Tsai

et al. propose an algorithm for node deployment strategy to

determine the number of nodes in every zone to handle the

overall generated and relayed traffic for energy balancing.

This method provides a non-uniform random deployment,

with a density which increases closer to BS.

B. Previous work

In our previous work [4], we discussed the problem of

deployment/transmission range assignment scheme in order

to maximize the lifespan of the whole network through total

energy savings. Given a source sensor in a distance d from

the BS, the problem asks to determine the number of hops

and the sensor placement between the sensor and the BS

such that the total energy consumption is minimized. This

model is called a linear network. The energy model that

we use to estimate the energy consumed by the sensors

is proposed in [6]. We note ETX the energy used for

transmitting, ERX for receiving and Eidle for the idle state

as in equation (1).

ETX = (Eelec + Eampdp) · β

ERX = Erec · βEIdle = c · Erec · β

(1)

In these equations β gives the bit rate of the radio and p is

the path loss exponent which varies between 2 and 4. Eelec

is the energy/bit consumed by the transmitter electronics,

Eamp is the energy/bit consumed by the amplifier, Erec

is the energy/bit consumption of the receiving circuitry

and c is a constant. We take the value of c equal to 1

because the energy consumption in idle state is very close

to energy consumption for receiving. For this problem we

examine two scenarios: a) the source sensor node does the

transmitting and all the intermediate sensor nodes act simply

as relaying nodes and b) the intermediate sensors add their

own information before relaying. Figure 1 illustrates the

two cases of the linear network. While the first problem is

quite simple, the second one becomes more complex. Based

on dynamical programming method we found an analytical

formula to solve the second problem which gives the optimal

number of nodes and their respective distances from BS. As

will be shown there is a property which permits us to obtain

this formula.

Let’s suppose that xn is the optimal location of the last

relay node in the linear network where n relay nodes are

employed. Given the energy formula (1), it can be shown

that the value kn = xn/d does not depend on distance d.

Going further, this can be generalized for all intermediate

sensors xp|1 ≤ p ≤ n. The following proposition holds:

Proposition 1. For a given n, the ratios kp = xp/d, 1 ≤p ≤ n, are the same for any distance d.

Proof: We can show this result by mathematical in-

duction on the number of n. Let’s take n = 1. Clearly,

the energy function associated with the transmission through

an intermediate sensor placed at x1 takes its minimum for

derivative set to 0. The energy function is of form

α1(x1)p + α2(d− x1)

p + α3 (2)

where αi are constants. Then, the derivative gives

α1p(x1)p−1−α2p(d−x1)

p−1. It is straightforward that this

function takes 0 for some k1 = x1

dthat does not depend

on d. The reasoning can be followed for some higher nassuming that the result holds for n − 1. Indeed, from the

recurrence assumption we can deduce that also for n the

energy function is similar to the energy function given in

(2) except of using other constants. For this, we decompose

the energy in two parts, the first captures the optimal energy

needed for the transmission through n − 1 sensors from 0to xn and the second from the last sensor xn to d. For

the first part we consider the optimized energy, that is the

23

−60 −40 −20 0 20 40 60−60

−40

−20

0

20

40

60

dR

Base Station

Y0

Sensor node

l

Figure 2: A sensor network with perfect traffic aggregation

first n− 1 sensors are optimally placed for the distance xn.

The recurrence hypothesis allows then to obtain a simplified

formula for the minimum energy used for the transmission

from 0 to xn, which is of form α′

1(xn)p+ c. Then, the total

energy can be expressed as α′

1(xn)p + α

′

2(d − xn)p + α

′

3,

where α′

i are constants. Then, we can apply the same schema

as for n = 1.

The linear case is extended for a two dimensional network,

where we assume to have N nodes placed at a distance dfrom BS, as in Figure 2. Also we assume a perfect aggre-

gation mechanism where each intermediate node transmits

only a certain amount of information despite what it receives.

This problem asks to compute the placement and the number

of relaying sensor nodes such that all the information sensed

by the N sensor nodes may be transmitted with a minimum

energy.

Our method gives optimal solution for the problem, how-

ever the above model remains mainly theoretical because

in real scenarios a node may receive and transmit a certain

traffic basing on network traffic pattern. Nevertheless, it can

be useful to calculate lower bounds of energy consumption.

Our contributions in this paper are twofold. First, we adapt

the mathematical model used for the relay sensor placement

problem to solve the sensor network configuration one which

has certain practical interests. Second, we propose a method

to solve optimally the problem of network configuration

for different cases of traffic load and to reduce its time

complexity.

III. PROBLEM FORMULATION AND LINEAR NETWORK

MODEL

A. Problem statement

We consider a linear network with a length d from the

BS, where the nodes are uniformly distributed, as in Figure

3. The network will be divided in cells. For any cell, there

will be only one node, called cluster head, which receives the

information from the other nodes of the same cell and from

the cluster head of the upstream cell. Then, the received

data will be transmitted towards the cluster head of the

Figure 3: Network model

downstream cell. We employ in this work a aggregation

model proposed by [8] which has the form y = mx + cwhere x and y are the input and output information quantity

respectively, m varies in [0, 1] and c is a constant. More

specifically, in this model we have assumed that the cluster

head will not aggregate its information, and thus m will be

equal to 1. The network model considered here is proposed

in [9]. We assume that each node generate α Erlang of data,

and the radio data rate is β bps. The nodes are uniformly

distributed with a density nd. The ith cell in the network

(Figure 3) will consume an energy ETX for transmitting

the information which is computed according to the equation

(3).

ETX(i) = (Eelec +Eamp(xi − xi−1)p) · xi · nd · α · β (3)

Whereas the energy used by the ith cell for receiving is

expressed by the equation (4).

ERX(i) = Erecxi−1 · nd · α · β (4)

Before a node goes to sleep mode, it can be either in

transmitting, receiving or idle state. Therefore, the energy

consumed in the idle state for a node of the ith cell is:

EIdle(i) = Erec

(

1− 2xi−1 · nd · α)

· β (5)

The network configuration problem consists in determining

the optimal number of cells and the respective length of each

cell such that i) the total energy consumption is minimized

(we will call this Strategy 1) and ii) every cell in the network

consumes the same energy or in other words we need to

ensure the energy fairness between the cells (Strategy 2).

So far we have assumed that the node transmission energy

depends on its communication distance. Since real sensors

have discrete levels of power transmission, we consider the

case of minimizing the total energy consumption of nodes

in the linear network using only some given discrete energy

values.

IV. SOLUTION METHOD AND NUMERICAL RESULTS FOR

THE LINEAR NETWORK MODEL

Our goal is to determine the parameters of network

configuration such that a given objective is met. Specifically,

24

given the linear network size, we need to find the optimal

cells number and their respective distances which guarantee

a lower bound of energy consumption (in IV-A1), the energy

balancing between cells (in IV-A2) or energy consumption

minimization for discrete energy levels (in IV-A3). As it can

be seen, the number of possible combinations is exponential

for any of these problems, therefore we propose a method

based on dynamic programming. Dynamic programming is

a sequential approach [1] for optimizing a given objective

function. The problem is thus broken down into stages and

the aim at every stage is to select the optimal decision so that

the objective is optimized over the total number of stages.

A. Optimization Criteria

1) Total energy minimization: Here the objective is to

minimize the total energy consumption for the linear net-

work presented in III-A. Our problem is separated into stages

determined by the number of cells. At some stage n we

determine the minimum energy necessary to transmit the

data by using the energy values obtained from the previous

stage. Hence, the optimal solution for an instance of a

problem with n cells and a given distance d can be seen as

equivalent to the optimal solution for some distance x < dand n − 1 hops, plus a final hop from this point (x) to the

BS. Algorithm 1 describes these steps. The parameter ∆

Algorithm 1: Strategy 1

Input: Distance d; node density nd

Output: Optimal number of network division n;

Optimal cell sizes x; Minimal energy E.

Initialization : {Eny |y ≤ d}, n = 1;

do {For all 1 ≤ y ≤ d calculate:

En+1y = min0<x<y

{

En−1(x) + ∆}

n → n+ 1} while(En−1

d ≤ End )

display n,E

given in Algorithm 1 is calculated according to formula (6).

∆ = (Eelec + Eamp(x− y)p) · y · nd · α · β+Erec · nd · α · β · x+ Erec · (1− 2 · x · α) · β

(6)

Note that Eny represents the total energy (optimized) con-

sumption where the network length is y and it is divided

into n cells.

2) Energy fairness: Nodes in WSN are often prone to

failures due to energy depletion. This phenomena happens

more frequently between the nodes close to the base station,

which is known as the hole problem [7]. The failure of

such nodes may culminate into disconnections, network

partitioning, and eventually stopping the whole network

operation. In order to avoid this, we want to define the

network configuration parameters such that each node in the

network consumes the same energy despite the amount of

received and generated data. Assuming that the network will

be divided into k cells, the total energy consumption for cell

i is given by (7).

E = (Eelec + Eamp(xi − xi−1)p) · xi · nd · α · β

+ Erec · nd · α · β · xi−1

+ Erec · (1− 2 · xi−1 · α) · β(7)

It is then possible to formulate the problem through a system

of non linear equations. Instead, we propose the Algorithm

2 also based on dynamic programming method to solve this

problem.

Algorithm 2: Strategy 2

Input: Distance d; node density nd.

Output: Optimal number of network division n;

optimal cell sizes x; Minimal energy E.

Initialization : {Eny |y ≤ d}, n = 1;

do {For all 1 ≤ y ≤ d calculate:

Calculate

En+1y |y ≤ d = min0<x<y

{

En(x)− E(y − x)}

n → n+ 1

} while(En−1

d ≤ End )

display n,E

In this case, Eny represents the energy consumed by one

cell1 when the network of a length y is divided in n cells.

E(y − x) = (Eelec + Eamp(y − x)p) · y · nd · α · β+Erec · nd · α · β · x+Erec · (1− 2 · x · α) · β)

(8)

The initialization phase of the algorithm requires O(d/δy)computation to construct the vector En

y , where δy is the step

of distance discretization. The calculation procedure will be

repeated n times and costs at most O(d/δy)2. Finally, the

computation complexity of the algorithm is O(n · (d/δy)2).3) Energy minimization with discrete transmission levels:

In the above models, the transmission sensor energy is

computed according to the exact distance of transmission.

This model does not reflect the reality as node transmission

energy takes values in a given discrete interval. In this

section, we show how to modify Algorithm 1 to handle this

case.

B. Numerical results

We have tested our algorithms for the optimal strategy

1 and 2 and compared their results with the best case of

uniform cell division. The uniform cell division divides the

1every cell will consume the same energy

25

Algorithm 3: Energy minimization with discrete trans-

mission levels

Input: Distance d, node density nd, the list of energy

power levels E = {e1, e2, ...en}, the

corresponding list of the respective distances

L = {l1, l2, ..lm}.

Output: Optimal number of network division n,

optimal cell sizes x, Minimal energy EInitialization : {En

y |y ≤ d}, n = 1

do {For all 1 ≤ y ≤ d calculate:

En+1y = min 0<x<y

⌈y−x⌉∈L

{

Enx + x · Erec · nd · α · β +

Erec · (1− 2 ·x ·α ·nd) · β+E(⌈y − x⌉) · y ·α · β ·nd

}

n → n+ 1} while(En−1

d ≤ End )

display n,E

network in number n of equal cell size and the best case

corresponds to the n that minimizes the energy value. The

results of total energy consumption for the three cases with

respect to the distance are shown in fig 4. For this simulation

we set the simulation parameters as detailed in table I.

Table I: Simulation parameters for the linear network

Type Parameter Value

NetworkNetwork Length 107 ∼ 400m

Node distribution Uniformly

Node density 0.16

ApplicationData generation rate β 485 bps

Data traffic per node α 0.003 Erlang

Compression ratio m = 1 | c = α · β

Energy

Eelec 50nJ/bit

Eamp 100pJ/bit/m2

Erec 50nJ/bit

p 2

L = {l1, l2, ·lm} [10, 20, . . . 80] in meters

E = {e1, e2, ·em} [0.06, 0.09 . . . 0.54] · 10−6J/bit

Method Distance discretization δy = 1

We notice that strategies 1 and 2 outperform the uniform

one saving 31% of the energy or in other words increasing

130% the network lifetime. On the other part, we notice

that Strategy 1 gives indeed a lower bound for total energy

consummation but it is quite near to this obtained from

the strategy 2. So, for e.x. in 400m it saves 3% in energy

compared with strategy 2.

For a range of distances we studied the optimal number

of cells provided by the two algorithms and the results are

100 150 200 250 300 350 4000

1

2

3

x 10−4

distance D from the Base St.

Glo

ba

l E

ne

rgy E

(J)

Energy consummation

Strategy 2

Strategy 1

Uniform energy

Figure 4: Energy consumption

0 50 100 150 200 250 300 350 4001

2

3

4

5

6

7

Network length (/m)

Nu

mb

er

of

ce

lls

Optimal number of cells versus distance

Strategy−2

Strategy−1

Figure 5: Optimal number of cell division versus distance

shown in Figure 5. In fact, we observe that the number of

cells for the two strategies changes only for some small

interval of distances, however the cells division is not

necessarily the same even for the same number of cells.

The table II gives the cells division beginning from the

one closest to the Base Station. We notice that the cells

division for the distance d = 200m is quite different even

when the optimal number of cells is 3 for both. Table III

presents the energy consumption for the two surveyed cases

of d = 200m and d = 400m from which we can distinguish

the energy differences between the two strategies.

As expected, for the energy discretization problem the

sensors spent more energy. For this simulation we assume

that each node can transmit to distances L = l1, l2, . . . lmusing the respective energies E = e1, e2, . . . em. Figure

6 shows that the continuous energy value still remains an

acceptable approximation of energy behavior.

26

cell sizesStrategy 1 Strategy 2

d=200m d=400m d=200m d=400m

cell 1 57 52 42 39

cell 2 65 53 48 41

cell 3 78 59 110 44

cell 4 66 47

cell 5 77 52

cell 6 93 60

cell 7 117

Table II: The optimal configuration for the linear network

Energy (J)

Strategy 1d=200m 9.2842 · 10−5

d=400m 2.6981 · 10−4

Strategy 2d=200m 1.0058 · 10−4

d=400m 2.7893e · 10−4

Table III: Energy

V. TWO-DIMENSIONAL SENSOR NETWORK

A. Problem statement

The second problem considered here is the two-

dimensional sensor network configuration one. We assume

that the nodes are uniformly distributed in a circular moni-

toring area. There is only one sink positioned in the center

of the zone.

The sensors will transmit their information according

to a many-to-one traffic pattern using a multihop scheme.

Minimizing total energy consumption can be achieved by op-

timally dividing the area in concentric coronas C1, C2, ...Cn

centered at the sink and further, dividing every corona in

a given number of zones. Figure 7 gives an example of

network division with 3 coronas containing respectively in

100 150 200 250 300 350 4000

0.5

1

1.5

2

2.5

3

3.5x 10

−4

Distance (/m)

En

erg

y (

/J)

Energy Discretization

Discrete Overall Energy

Overall Energy

Figure 6: Energy disretization

−60 −40 −20 0 20 40 60−50

−40

−30

−20

−10

0

10

20

30

40

50

Figure 7: Two dimensional network

1, 4 and 8 zones. Moreover, the length of every corona

can be varied. For each zone it will be only one sensor

that will receive the information from other sensors of the

zone, aggregate and transmit it to the next zone closer to

the BS. This sensor will be called a cluster head of the zone

and every node of the zone may play this role. Notice that

we can easily shift from this model to the sensor specific

sensor placement, which asks to determine the cluster head

position, by adapting the radio transmission range affected

to each zone. According to the aggregation model proposed

in the section III-A we will analyze the case2 with m = 0.

Indeed, if we suppose that the sensor information will not

be aggregated by the cluster heads the problem would be

reduced to the linear network case because the sensor will

select the shortest path to transmit the information. Finally,

for the network model as described above, the questions are:

• What is the optimal number of coronas that achieves

the energy minimization of the whole network? What

are their respective radii?

• What is the optimal number of zones that a corona is

divided in?

Here we assume that it will be a cluster head for each

zone that will aggregate the traffic and will transmit

toward the cluster head of the zone closer to the BS.

B. Minimum energy consumption

Unlike the linear network case, the problem for a two

dimensional network with traffic aggregation becomes more

complex to handle. We opt for a method based on parameter

discretization of the radius and the number of zones for each

corona respectively, and apply the dynamic programming

principle. Hence, for a given sensor network radius R with

uniform node distribution nd and a constant traffic α · βgenerated by each sensor, we apply the Algorithm 4.

2A node will totally aggregate the information and transmit only a fixedamount of information

27

Algorithm 4: Network area configuration

Input: Node density nd, Network area radius R,

Generated traffic per node α · βOutput: Minimum energy E; Optimal number of

coronas n; Coronas’ radius x; Number of

zones for each corona P .

Initialization : {End |d ≤ R}, n = 1

do {Calculate En+1

d,N matrix

En+1

d,N = minx≤d,P≤N

{

Enx,P + P · f1(d, x,N, P )

+f2(d, x,N, P )}

(9)

set Ond,N = x, P

n → n+ 1

} while(En−1

R,N0≤ En

R,N0)

display n,E

According to this algorithm we begin by computing the

first vector E1 containing values of E1d |d ≤ R using the

formula (10).

E1d |d ≤ R = (Eelec + Eamp · d

p) · nd · π · d2 · α · β (10)

This corresponds to the case when nodes transmit directly

to the BS. In the next step, the number of coronas is two

and we calculate all {E2d,N |d ≤ R,N ≤ N0} values. The

parameters used in equation (9) are computed as follows:

f1(d, x,N, P ) = (Eelec+Eamp·Rp(d, x,N, P )+Erec)·α·B

(11)

where R(d, x,N, P ) is such that:

R2(d, x,N, P ) = d2 + x2 − 2 · d · x · cos((NP

− 1) · π

N)

This is the energy used by the cluster head of each zone of

the nth corona to transmit the information towards the next

cluster head closer to the BS.

f2(d, x, n, p) = ((Eelec + Eamp ·Rp(d, x,N) + Erec) · nd

·π · (d2 − x2) · α · β(12)

with R(d, x,N) such that:

R2(d, x,N) = d2 + x2 − 2 · d · x · cos(π

N)

Equation (12) gives the energy consumption spent by all the

sensors in a corona due to the intra zone transmissions3.

We continue like this with {End,N |d ≤ R,N ≤ N0} for

subsequent values of parameter n. We stop the calculations

when we reach some n such that EnR,N0

≤ En+1

R,N0. The

3Intra zone transmissions include the receiving and transmitting opera-tions inside a zone between the cluster head and the other sensors in thezone.

solution can be easily built by backtracking the stored

values Ond,N . At the end of the algorithm we obtain the

optimal number of coronas to achieve a minimum energy

consumption, as well as the number of zones for each

intermediate corona and the corresponding radius.

C. Numerical results

For the simulations, the nodes are uniformly deployed in

the circular area and generate a constant traffic. The cluster

head of each zone aggregates all the information that it

receives and transmits a constant traffic to the closest cluster

head. Table IV lists the system configuration parameters in

details. For these parameters, some experimental results of

the Algorithm 4 are presented in table V. As we can observe,

for each corona the algorithm determines its respective

radius and number of zones, which both decrease uniformly.

Table IV: Simulation parameters

Type Parameter Value

NetworkArea radius 100m

Node distribution Uniformly

Node density 0.02

ApplicationData generation rate β 100 bps

Data traffic per node α 0.003 Erlang

Compression ratio m = 0 | c = α · β

Energy

Eelec 50nJ/bit

Eamp 100pJ/bit/m2

Eelec 50nJ/bit

p 2

Cell sizesAlgorithm results

Radius of each Number of zonescorona from the BS

corona 1 100 27

corona 2 90 24

corona 3 80 21

corona 4 70 18

corona 5 60 15

corona 6 50 13

corona 7 40 10

corona 8 30 7

corona 9 20

Table V: Optimal configuration for the network area

The energy consumption of the whole network for differ-

ent network radius and node density changes is presented

in Figure 8. We notice that small changes in node density

28

50 100 150 200 250 3000

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

−4

Area Radius (/m)

En

erg

y (

/J)

Node density = 0.04 m2

= 0.028 m2

= 0.02m2

Figure 8: Energy versus network area radius

may greatly influence the energy consumption of the net-

work. Moreover, we observe that the network configuration

solution, as the one depicted in table V, is quite sensitive

toward these changes.

VI. CONCLUSIONS

In this paper we have analyzed the network configuration

problem for many-to-one WSN. For a given network where

the nodes are uniformly distributed, the problem requires

to determine the optimal number of cell division, and their

respective cell sizes such that some energetic objectives are

met. We show that appropriate configuration of WSN, which

addresses the cell organization and the respective power

assignments, leads to considerable savings in energy. Our

algorithms which intend to i) minimize the total energy

consumption and ii) assure the energy node consumption

fairness are based on dynamic programming approach. We

also show how this method can be adapted even for the

discrete case of energy consumption minimization problem.

Next, the linear network case is extended to the two dimen-

sional network one. In general, we observe that dynamic

programming is an effective method to handle the trade-offs

between the parameters for some variants of the network

configuration problem. It provides optimal solution of the

problem for a given objective function and reduce the com-

putation complexity compared with other methods proposed

in literature. Moreover, the method can be adapted for other

similar energy models consumption without increasing the

computational complexity of the solution. In the future, we

will extend our work to adapt and implement this method

for the two dimensional sensor network with randomly node

distribution and test it in real scenarios.

REFERENCES

[1] R. Bellman (1957). Dynamic Programming. PrincetonUniversity Press, ISBN 0486428095, 2003.

[2] Y. Chen, C.N. Chuah, and Q. Zhao. Sensor placement formaximizing lifetime for unit cost in wireless sensor networks.IEEE MILCOM, 2:1097–1102, 2005.

[3] P. Cheng, C.N. Chuah, and X. Liu. Energy-aware nodeplacement in wireless sensor networks. IEEE GLOBECOM,5:3210–3214, 2004.

[4] A. Gogu, D. Nace, and Y. Challal. A note on joint optimaltransmission range assignment and deployment for wirelesssensor networks. IEEE Networks, pages 1–6, 2010.

[5] B. Hao, H. Tang, and G. Xue. Fault-tolerant relay nodeplacement in wireless sensor networks. HPSR, pages 246–250, 2004.

[6] W. R. Heinzelman, A. P. Chandrakasan, and H.Balakrishnan.Energy efficient communication protocol for wireless mi-crosensor networks. Proceeding of the 33rd Hawaii Inter-national Conference on System Sciences, 2000.

[7] J. Li and P. Mohapatra. Analytical modeling and mitigationtechniques for the energy hole problem in sensor networks.Pervasive and Mobile Computing, pages 233–254, 2007.

[8] V.P. Mhatre, C. Rosenberg, D. Kofman R. Mazmudar, andN. Shrof. Design guidelignes for Wireless Sensor Networks:Communication, Clustering and Aggregation. Ad Hoc Net-works, 2:45–63, 2004.

[9] Q.Gao, K.J. Blow, D.J. Holding, I.W.Marshall, and X.H.Peng. Radio range adjustment for energy efficient wirelesssensor networks. Elsevier, Ad Hoc Networks, pages 75–82,2006.

[10] X. Wang, W. Gu, S. Chellappan, K. Schosek, and D. Xuan.Lifetime optimization of sensor networks under physicalattacks. IEEE ICC, 5:3295–3301, 2005.

[11] Y.Tsai, K. Yang, and S. Yeh. Non-uniform node deploymentfor lifetime extension in large-scale randomly distributedwireless sensor networks. AINA Proceedings, pages 517–524,2008.

[12] H. Zhang and H. Shen. Balancing energy consumption tomaximize network lifetime in data-gathering sensor networks.IEEE Trans. on Par. and Distrib. Sys., 20:1526–1539, 2009.

29