[IEEE 2013 IEEE Symposium on Computers and Communications (ISCC) - Split, Croatia...

Deriving Lower Bounds for Energy Consumption in Wireless Sensor Networks

Adriana Gomes Penaranda, Andre Ricardo Melo Araujo, Fabiola Guerra Nakamura, Eduardo Freire Nakamura Institute of Computing

Federal University of Amazonas

Manaus, Amazonas, Brazil

Email: [email protected]@[email protected]@icomp.ufam.edu.br

Abstract-Wireless Sensor Networks (WSNs) are a special kind of ad hoc networks designed to comprise a high density of sensor nodes. These networks have high traffic of data and waste energy with an unnecessary number of active sensor nodes. In this paper we address the Density Control, Coverage and Connectivity Problem (DCCCP) in WSNs, that consists in activating a subset of sensor nodes, which assure the area coverage and the nodes connectivity, and minimize the energy consumption. We propose Multiperiod and Periodic approaches to solve the DCCCP. The Multiperiod approach divides the expected network lifetime in time periods and calculates, in a global way, a solution for each period, that minimizes the network energy consumption considering all time periods at once. The Multiperiod Approach has a global view of the nodes and the network expected lifetime. The Periodic Approach solves the problem as an static problem, updates the list of available nodes and repeats the procedure. The Periodic Approach has a global view of the nodes but not of the network lifetime, it finds local solutions (considering the periods) that together form a global solution, represented by the sum of all local solutions. These approaches are modeled through Integer Linear Programming (ILP). We compare our optimal solutions with Geographical Adaptive Fidelity (GAF) and Hierarchical Geographical Adaptive Fidelity (HGAF). Given the global aspects of our approaches we expect to derive lower bound for energy consumption for density control algorithms in WSNs

Index Terms-Density control; sensor networks; periodic approach; multi period approach.

I. IN TRODUC TION

Wireless Sensor Networks (WSNs) are a special type of

ad hoc networks whose goal is to collect data from a phe

nomenon and transmit them to an external observer. They are

composed of nodes with energy, processing, and communi

cation constraints, due mostly to their small size and price.

These networks have many application areas, such as tracking

animals; monitoring vehicles; monitoring human organs; and

detecting the presence of enemies in battle fields [1].

High density WSNs are usually designed to assure applica

tions requirements such as area coverage and nodes connectiv

ity. However, a high number of nodes in the monitoring area

can lead to problems such as unnecessary energy consumption,

interferences, and packet collision. The Density Control, Cov

erage and Connectivity Problem (DCCCP), addressed in this

paper, consists in determining a subset of sensor nodes to stay

active while the others are turned off or scheduled to sleep.

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This subset should assure the area coverage and the network

connectivity and minimize the energy consumption [2]-[4].

We use two approaches to address the DCCCP: a Multi

period and a Periodic. The Multiperiod Approach primarily

divides the expected network lifetime in time periods and

calculates, in a global way, a solution for the DCCCP at

each period, respecting the number of periods that a node

can remain active. The Periodic Approach finds the best

solution in a given time and repeats this procedure period

ically. These approaches are modeled through Integer Linear

Programming (ILP) mathematical formulation and solved with

a commercial optimization package. Their objective function

minimizes the energy spent with activation, transmission, and

node maintenance, that is the sum of the energy consumed by

the sensor board, the processor, and the radio. The Density

Control Problem is NP-Hard and given the global aspects

of our approaches we expect use the optimal solutions to

derive lower bounds for energy consumption with application

in sensor networks. We also compare the optimal solutions

with Geographical Adaptive Fidelity (GAF) and Hierarchical

Geographical Adaptive Fidelity (HGAF) algorithms.

The paper is organized as follows. Section II lists some of

best known algorithms to density control in WSNs. Section

III formally defines the DCCCP and presents our approaches.

Section IV shows and analyzes the computational results and

Section V presents our conclusions.

II. RELATED WORK

A. Probing Environment and Adaptive Sleeping

Probing Environment and Adaptive Sleeping (PEAS) is

a distributed protocol that aims to build and maintain the

network operation [5]. In this work sensor nodes have three

states: Working, Sleeping, and Probing. In Working State,

nodes sense the phenomenon of interest and route, send, and,

receive messages. In Probing State, nodes exchange PROBE

and REPLY messages. In Sleeping State, nodes turn off the

radio.

PEAS strategy is composed of two algorithms:

• The Probing Environment detennines which sensor nodes

will be active, based on the node neighborhood.

• The Adaptive Sleeping adjusts the time that sensor nodes

will be sleeping.

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All sensor nodes begins in the sleeping state and remain in

this state for a time exponentially distributed according to a

probability density function.

B. Optimal Geographical Density Control Algorithm

The Algorithm Optimal Geographical Density Control

(OGDC) is a decentralized and localized algorithm that defines

a set of optimal conditions to choose a set of active sensor

nodes. Sensor nodes begin in Undecided state and exchange

messages to choose which will be On or Off [6], [7].

The algorithm is devised under the following assumptions:

• The radio range must be at least twice the sensing range.

According to Zang and Hou (2005) this condition ensures

that the complete area coverage results in the connectivity

of sensor nodes.

• Each sensor node is aware of its own position. This

condition is necessary during the selecting process.

• All sensor nodes must be time synchronized. Because

for each round the sensor nodes should be available at

the right time, so it must be synchronized since the first

round.

C. Geographical Adaptive Fidelity and Hierarchical Geo

graphical Adaptive Fidelity

Geographical Adaptive Fidelity (GAF) is an algorithm that

exploits the high density of sensor nodes to reduce energy

consumption. It identifies sensor nodes that are equivalent,

from a routing perspective, and turns off unnecessary sensor

nodes [8]. Thus keeping a constant level of, what the authors

call, routing fidelity.

GAF identifies equivalent nodes using location information

and virtual grids. Virtual grids are used to determine groups

of equivalent sensor nodes. The algorithm divides the area in

grids and a sensor nodes can be in only one virtual grid. The

virtual grids are defined such as any sensor node of a grid

A can reach any sensor node to an adjacent grid B, and vice

versa. All sensor nodes in a virtual grid are equivalents. Fig.

1 shows an example of virtual grids.

L. - - - � - - - 1. - - - _lor 1 A 0 ............ BI CI: 1 1 ........ ... R I'

G) 0........ r

� _ � _ .0 __ � � _ ��01 . 1- - r· - - -1- _. r - - -1 - - - r - - -1-

Fig. l. Example of virtual grid in GAF [8]

To assure the communication among nodes of adjacent

grids, the grid dimension is calculated in function of the radio

range R. In this case we have:

or (1)

Sensor nodes have three states: Discovery, Active and

Sleeping. In Discovery state, nodes connect the radio and send

a discovery message. These messages are used to find sensor

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nodes that belong to the same grid. The Active state consists in

sensing, routing, sending, and, receiving messages. In Sleeping

state, the nodes turn off the radio.

The number of virtual grids defines the number of sensor

nodes active at each round. GAF can turn on an unnecessary

number of nodes as stated in [9] for it has a large number of

grids.

Inagaki and Ishihara (2009) propose a variation of GAF,

called Hierarchical Geographical Adaptive Fidelity (HGAF).

The idea is reduce the number of grids, thus reducing the

number of active sensor nodes. For this, HGAF increase the

grid size and divide, each grid, in N2 subgrids. The Fig. 2 shows the division. As in GAF, HGAF considers that sensor

nodes in adjacent grids must conununicate with each other, so

each grid needs to turn on one sensor node in a subgrid with

position matching a subgrid of the adjacent grid.

r- 0 . r-;;;: /0

/: IL ------r---0

(a) N=2 (b) N=3

Fig. 2. Grid division in N2 subgrids [9]

To assure the communication, the distance between sensor

nodes in adjacent grids must not be larger than the radio ranger

R. In this case we have:

(2)

Thus, the subgrid size dN and the grid side DN are:

1 dN < R - J(N+l)2+1

When N � 3 the grid size is:

V2DN:S; R (4)

We use GAF and HGAF to demonstrated that the ILP mod

els can be used to derive lower bounds of energy consumption

in WSN. Our focus is the energy consumed with activation,

sensing, processing, and data communication, which means

that the lower bounds we will be able to provide will exclude

the energy consumed to disseminate a density control solution .

III. MULTIPERIOD AND PERIODIC ApPROACHES

In this work, the monitoring area is modeled through the

use of demand points for this concept allows to evaluate the

coverage in a discrete space, it is very useful for modeling

purposes, and permits to quantify the coverage. Formally, we

define the static Density Control, Coverage and Connectivity

Problem as follows:

Given a monitoring area A, a set of sensor nodes S, a set

of sinks nodes M, and a set of demand points D, the problem

of density control consist in assuring, if possible, that every

demand point d E D in the area A is covered by at least one

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sensor nodes s E S and there is, at least, a route between

every active sensor s E S and a sink node m E M. This definition refers to a static DCCCP because the solution

is for a specific instant. Thus, based on the static problem

and to suit better the dynamic aspect of the WSN the density

control problem can be addressed through a periodic or

muItiperiod approaches.

The periodic approach is composed of the static problem

solved periodically. The approach finds the best solution for

a given instant of time, regardless the topology of previous

periods. At the beginning of each round the set of sensor nodes

is updated, and nodes with no energy left are excluded from

the list. This approach does not have an overview of all periods

and it chooses the best solution for the current period without

take into account that in the future there may be large regions

not covered or disconnected.

The multiperiod approach estimates or chooses an expected

lifetime for the network, divides it into periods and sets,

globally, the solution for each period. This is done taking the

topology defined in previous periods into consideration [10].

The solution for each period is chosen with an overview of

the network and periods. This may cause some periods to have

small gaps in coverage in order to reduce coverage fails in the

future. The best solution is the one that minimizes overall

energy consumed and coverage fail.

Both approaches are modeled as Integer Linear Program

ming (ILP) problems and consider that the node coverage

area is a circle of range R, where R is the node sensing

range. If the distance between a demand point and a sensor

node is less than the value R then the node covers this point.

The models assume that the nodes know their location and

have a unique identification number. To quantify the nodes

energy consumption, we define that the application requires

continuous sensing and periodic dissemination, and the traffic

concerns only application data.

For the periodic approach we use the model present by

Menezes [11]. Based on this model we propose a multiperiod

mathematical model defined as follows. The parameters used

in our formulation are:

S set of sensor nodes

D set of demand points

M set of sinks

T set of time periods

A S set of arcs connecting sensor nodes

A m set of arcs connecting sensor nodes to the sink node

Ij set of arcs (i, j) incoming on the sensor node j E S Oi set of arcs (i, j) outgoing the sensor node i E S Gij coverage matrix that has value 1 in the cell (i, j)

if node i E S reaches demand point JED and 0

otherwise

AEi activation energy for node i E S M Ei maintenance energy for node i E S T Eij transmission energy between nodes i and j, {i, j} E

{AS U Am} EH Non-Coverage Penalty.

Let d( i, j) be the distance between the node i and demand

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point j. The coverage matrix Gij is composed by the arcs

(i, j) where d( i, j) is less or equal than the sensing range of

node i. Let d(i,j) be the distance between nodes i and j. The sets AS and Am are composed by the arcs (i, j) where

d( i, j) is less or equal than the communication range of node

i. The Euclidean distance between two sensor nodes is used

to quantify energy transmission. If the distance is more than

sensing range then the sensor nodes do not communicate each

other. The model variables are:

yf

variable that has value 1 if node i covers demand

point j at period t, and 0 otherwise

decision variable that has value 1 if arc (i, j) is in

the path between sensor node I and the sink node m

at period t, and 0 otherwise.

variable that has value 1 if node l is active at period

t, and 0 otherwise

decision variable that has value 1 if sensor node i is

active at period t, and 0 otherwise

variable that has value 1 if the demand point JED is not covered at period t, and 0 otherwise.

The ILP formulation proposed is presented below. The objec

tive function (5) minimizes the network energy consumption.

The optimal solution of the mathematical formulation indicates

the set of nodes that assure the best possible coverage and

the nodes connectivity, at minimal energy cost. The second

term of the objective function penalizes the demand points

not covered.

Min L L(AElWl + MElYl + L TEijZlij) lES tET (i,j)EA8UAm

+ (5)

The set of constraints (6), (7), and (8) deals with the coverage

problem. Constraints (6) specifically assure that at least one

sensor node will cover each demand point. Constraints (7)

indicate that a node can only cover a point if it is active.

Constraints (8) set limits for variables x and h.

L (xfjGij ) + h; ;::: 1, Vj E D, Vt E T (6)

lES

L Xlj � IDlyf,Vl E S, Vt E T (7)

JED

o � x,h � 1 (8)

The set of constraints (9), (10), (11) and (12) is related to

the connectivity problem. Constraints (9) and (10) assure a

path between each active sensor node I E S and the sink node

m E M and constraints (11) and (12) only allow active nodes

to be part of these paths.

L Zfij - L Zfjk = 0, (i,j)Elj(As) (j,k)EOj(AsUA=) Vj E (S\{I}),VI E S,Vt E T (9)

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t '" t t Zlij - � Zljk = -Yl'

(i,j)Eli(As) (j,k)EOj(A8UAm) Vj = I, VI E S, Vt E T

L Zfij::; ISlyf, VI E S, Vi E S, Vt E T jESUM

(10)

(11)

L Zlij ::; ISlyj, VI E S, Vj E S, Vt E T (12)

iES The set of constraints (13) and (14) deals with relationship

between the variables wand y, the period when the node was

active and if the node was or was not active at last period.

wi -yi 2: 0, Vt E S

wf -yf + y;-l 2: 0, VI E S, Vt E T e t 2: 2

(13)

(14)

The set of constraints (1S) and (16) define the minimal number

of nodes that can be active at each period and the maximal

number of periods a node can remain active.

Constraints

boolean.

Lyf 2: A/1fr2 IES

Lyf ::; n

IES (17) define the decision

y,ZE{O,l}

Vt E T (IS)

ViET (16)

variables y and Z as

(17)

The model for the periodic approach looses the index t and

the constraints (13), (14) and (16).

IV. COMPUTATIONAL RESULTS

Consider u.d. (unit of distance) an unit to quantify

distance and u.e. (unit of energy) an unit to quantify energy

consumption, the test parameters are shown as follows:

Monitoring Area: lOu.d. x 10u.d.

Number of demand points 100, equivalent to 1 demand

point per (u.d.)2

Activation Energy 10 u.e.

Maintenance Energy 1.2 u.e.

Non-Coverage Penalty (EH): 1000

To quantify the coverage area we calculate the percentage

as the arithmetic mean of demand points covered for at least

one sensor node for each period. The energy consumption con

siders that a node spends energy with activation, maintenance,

and transmission, as modeled in the first term of Equation

(S). The network has homogeneous nodes, is flat, and with

a sink placed at the superior left corner. The periods have

the same duration of 1 U.t. The nodes position follows an

irregular grid configuration, where the nodes are placed in a

grid and randomly errors are applied to their (x, y) coordinates

as illustrated in Fig. 3. The monitoring area is discretized in

demand points thus generating a regular grid with a distance

of 1 u.d. between demand points. Instances are composed by

the amount of sensor nodes, amount of demand points, energy

transmission matrix (ETij) and connectivity matrix (Glj). The

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energy transmission matrix is formed with Euclidean distances

between sensor nodes. The connectivity matrix is formed with

the Euclidean distances between sensor node and demand

point, if the distance is less or equal than the sensing range

then the sensor node reaches the demand point.

Fig. 3. Irregular Grid

We divided the tests into 3 batteries. In battery 1, we

compare our optimal solutions of the multiperiod and period

approaches, both obtained by solving the mathematical models

with the commercial optimization package CPLEX [12]. In

battery 2, we compare our optimal solution of the periodic

approach with GAF and HGAF Algorithms. In battery 3, we

executed the algorithms and the approaches until the network

had lost the connectivity and coverage.

We generate a network with 2S sensor nodes for battery 1,

169 sensor nodes for battery 2, and one instance with 36 sensor

nodes for battery 3. The sensor nodes parameters are Sensing

Range (SR) equals 3u.d., and maximal Communication Range

(CR) equals 6u.d. and 7u.d .. We designed the tests for 2, 3 and,

6 time periods, and the value of n, i.e., the number of time

periods a node can remain active, are 1, 2, and 2, respectively.

A. Battery 1

The results are shown in Tables I and II. We can notice

that instances FileO_2S_6 and File3_2S_6 in Table I have

the same energy consumption and the same percentage of

coverage with 2 and 3 periods. The difference occurs with 6

periods where the instance FileO_2S_6 consumes less energy

and guarantees more coverage in multiperiod approach, and

the instance File3_2S_6 consumes more energy but provides

more coverage in multiperiod approach, as also occurs with

instances File3_2S_7 and file4_2S_7 in Table II. This occurs

because the not coverage of a demand point is added as

a penalty in the objective function, thus the multiperiod

approach attempts to minimize the amount of demand points

not covered in all periods while trying to minimize energy

consumption, on the other hand the periodic approach finds

the best solution for each period and the penalty for non

coverage is added only if there is no sensor node to reach the

demand point.

We also can highlight instances File1_2S_6 in Table I

and FileO_2S_7, File2_2S_7, File3_2S_7 and file4_2S_7 in

Table II, for 2 and 3 periods, where the energy consumption

of both approaches is near and ensure the same percentage

of coverage. This occurs because the multiperiod approach,

unlike the periodic approach, does not always uses the best

path for each period, thus enabling that better paths can be

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TABLE I 25 SENSOR NODES, SENSING RANGE EQUALS 3U.D. AND

COMMUNICATION RANGE EQUALS 6U.D.

Optimal Solution - Periodic Approach Tolal

Inslance Energy(u.e.)

Time(s) Coverage(% ) Periods Consumption

FileO_25_6 255,75 1,13 100,00% File 1_25_6 243,82 0,68 99,50%

2 File2_25_6 256,79 1,07 98,00% File3_25_6 238,93 0,71 98,50% File4_25_6 300,61 0,60 97,50% FileO_25_6 311,42 1,13 100,00% Filel_25_6 300,65 0,68 99,67%

3 File2_25_6 311,08 1,07 98,67% File3_25_6 290,98 0,71 98,67% File4 25 6 369,14 0,60 98,00%

FileO_25_6 583,58 1,17 96,33% Filel_25_6 563,83 0,73 96,33%

6 File2_25_6 616,81 I,ll 95,67% File3_25_6 522,82 0,76 95,00% File4 25 6 597,37 0,63 90,33%

Optimal Solution - Multiperiod Approach Tolal

Instance Energy(u.e.)


FileO_25_6 255,75 4,47 100,00% Filel_25_6 240,54 4,73 99,50%

2 File2_25_6 241,44 4,64 98,50% File3_25_6 238,93 4,42 98,50% File4 25 6 284,65 5,13 98,00% FileO_25_6 311,42 14,62 100,00% Filel_25_6 297,38 7,53 99,67%

3 File2_25_6 295,73 8,09 99,00% File3_25_6 290,98 8,6 98,67% File4 25 6 267,86 10,81 98,33% FileO_25_6 577,04 440,10 99,00% File 1_25_6 521,44 265,91 97,67%

6 File2_25_6 586,57 3844,58 97,33% File3_25_6 525,62 391,40 96,67% File4 25 6 546,43 480,50 95,33%

chosen in future periods totaling less energy consumption and

without prejudice the coverage.

We noticed that with a greater quantity of time periods,

the muItiperiod approach obtains better results in comparison

with the periodic approach, both in energy consumption and

coverage. We can highlight the instance File2_25_7 to 6

periods, in Table II, where the multiperiod approach guarantees

100% coverage and consumes less energy while the periodic

assures 95% coverage.

For most of the tests the Multiperiod approach has best

results due to global view of the nodes and the expected life

time of the network.

B. Battery 2

In GAF and HGAF, the number of active sensor nodes at

each period is fixed because the number of virtual grid is

fixed and the algorithms activate one sensor node for grid.

For the parameters used in this battery, GAF must active 16

sensor nodes and HGAF must active 9 sensor nodes at each

period. The optimal solution of periodic approach and GAF

and HGAF ensured 100% of coverage for all instances. HGAF

uses N = 2 that means each grid is divide in 4 subgrids.

Graphs 4 shows the results. We can notice that GAF consumes

more energy than HGAF due the number of sensor nodes

required by the algorithm. The periodic approach consumes

less energy than GAF and HGAF in all instances since this

provides the optimal solution for static DCCCP. Thus, the

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TABLE II 25 SENSOR NODES, SENSING RANGE EQUALS 3U.D. AND

COMMUNICATION RANGE EQUALS 7U.D.

Optimal Solution - Periodic Approach Tolal



FileO_25_7 278,25 0,83 99,50% Filel_25_7 296,74 0,79 97,00%

2 File2_25_7 252,68 1,20 100,00% File3_25_7 236,53 0,67 100,00% File4_25_7 242,67 1,05 99,50% FileO_25_7 337,86 0,83 99,67% Filel_25_7 358,77 0,79 98,00%

3 File2_25_7 305,16 1,20 100,00% File3_25_7 284,46 0,67 100,00% File4 25 7 298,05 1,05 99,67%

FileO_25_7 603,68 0,87 96,67% Filel_25_7 608,75 0,83 90,67%

6 File2_25_7 601,63 1,25 95,00% File3_25_7 515,82 0,75 97,33% File4 25 7 550,28 1,12 96,67%

Optimal Solution - Multiperiod Approach Tolal



FileO_25_7 275,77 4,97 99,50% Filel_25_7 294,97 5,22 99,00%



6 File2_25_7 565,34 595,95 100% File3_25_7 536,53 978,99 98,33% File4 25 7 551,80 276,77 98,33%

approach is an interesting alternative to generate lower bounds

for the energy consumption in sensor networks, disregarding

control messages of each approach.

When the node density decreases, GAF and HGAF may not

work properly because some grids may be empty. However,

even in low density networks we can still find solutions with

the periodic approach. We must make a few adjusts such as

excluding demands points not covered to solve the model and

including them afterwards as coverage fail.

1200

1000

800

............................................................................................................... 600

400

200 .. .. .. .. .. .. .. .. .. .. .•. .. Periodic ........ . .

HGAF .......... .. GAF --

OL-----'------'-------'--------' 2 4 5 6

Periods

Fig. 4. Energy Consumption for 169 sensor nodes

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C. Battery 3

We also run tests to verify the last period the network

has coverage and connectivity. For this test we consider

a 36-nodes network with Sensing Range equals 3u.d. and

Communication Range equals 6u.d .. The Graphs 5 shows the

energy consumption and coverage. We notice that GAF and

HGAF spend more energy and remain active less periods than

the model solutions and the multiperiod approach spend less

energy and ensure more coverage than the periodic approach.

300

250

" 0 200 '0 0.

E :I C/O " 150 0 U >, 2.'l '" 100 "

"'-l

50

0 2

100 �.,,�.r.: ...... . "

80

� 60 '" OJ) e '" > 0 40 U

20

0 1 2

3

3

4 5 6 7

Periods

(a) Energy Consumption

4

\ \

\

5 6

Periods

(b) Coverage

7

Multiperiod -Periodic ---------

HGAF · GAF ·

8 9 10 II

Multiperiod -Periodic ---------

HGAF · GAF ·

8 9 10 11

Fig. 5. Energy Consumption

The tests show that the solution which consider the optimal

solution as the sum of all periods offers advantage for the mul

tiperiod approach, because the approach attempts to provide

good solutions for all periods. The periodic approach always

uses the best sensor nodes, which means that the solution of

latter periods could be worse.

V. F INAL REMARKS

This work deals with the Density Control, Coverage, and

Connectivity Problem in Wireless Sensor Network (DCCCP).

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The DCCCP is addressed by using two different approaches:

Multiperiod and Periodic, both modeled with ILP mathemati

cal formulations and solved up to the optimal.

The Multiperiod Approach is a density control scheme

that primarily divides the expected network lifetime in time

periods. The approach calculates, in a global way, a solution

for the density control problem at each period, respecting the

number of periods a node can remain active. The Periodic

Approach is proposed as an alternative to the Multiperiod

Approach and consists in finding the optimal solution for

the DCCCP in a given time and repeating this procedure

periodically. The periodic approach is an alternative because

it can work with higher density networks once it solves one

period at a time.

Given the global aspect of the Multiperiod Approach regard

ing the available nodes and the network lifetime, the optimal

solution provides a network configuration that has the best cov

erage possible with the minimum overall energy consumption.

However, with the right parameters, both approaches could

provide a lower bound for periodic density control schemes.

We are current working in exact methods and heuristics to

solve the models given the high computational cost to achieve

the optimal solution.

GAF and HGAF are well known algorithms to density

control that divide the whole area in grids and each grid

active one sensor node, which means that at each period the

number of nodes active is fixed. Compared to the optimal

solution of Periodic Approach, the algorithms reach the same

results regarding coverage but consume more energy. So we

consider the approach an interesting alternative to generate

lower bounds for energy consumed with application in sensor

networks and to evaluate density control algorithms. We intend

to extent the evaluation for other density control algorithms as

PEAS, A3 and, OGDC.

ACKNOW LEDGEMENT

This work is partially financed by the National Coun

cil for Scientific and Technological Development (CNPq)

and the Amazon State Research Foundation (FAPEAM),

trough the grant 221O.UNI175.3532.03022011 (Projeto Anura

- PRONEX 02312009) and Projeto "Ncleo de ExceBncia em

Desenvolvimento de Sistemas Embarcados para Veculos Areos

No-tripulados e Robs Tticos Mveis" (PRONEX 02312009 -

Decision 173/2010).

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