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39 \title{Distributed Coverage Optimization Protocol to Improve the Lifetime in Heterogeneous Energy Wireless Sensor Networks}
42 % author names and affiliations
43 % use a multiple column layout for up to three different
45 \author{\IEEEauthorblockN{Ali Kadhum Idrees, Karine Deschinkel, Michel Salomon and Raphael Couturier }
46 \IEEEauthorblockA{FEMTO-ST Institute, UMR CNRS, University of Franche-Comte, Belfort, France \\
47 Email:$\lbrace$ali.idness, karine.deschinkel, michel.salomon,raphael.couturier$\rbrace$@edu.univ-fcomte.fr
49 %\email{\{ali.idness, karine.deschinkel, michel.salomon, raphael.couturier\}@univ-fcomte.fr}
51 %\IEEEauthorblockN{Homer Simpson}
52 %\IEEEauthorblockA{FEMTO-ST Institute, UMR CNRS, University of Franche-Comte, Belfort, France}
54 %\IEEEauthorblockN{James Kirk\\ and Montgomery Scott}
55 %\IEEEauthorblockA{FEMTO-ST Institute, UMR CNRS, University of Franche-Comte, Belfort, France}
64 One of the fundamental challenges in Wireless Sensor Networks (WSNs) is Coverage preservation and extension of network lifetime continuously and effectively during monitoring a certain geographical area.In this paper
65 a distributed coverage optimization protocol to improve the lifetime in in Heterogeneous Energy Wireless Sensor Networks is proposed. The area of interest is divided into subregions using Divide-and-conquer method and an activity scheduling for sensor nodes is planned for each subregion.Our protocol is distributed in each subregion. It divides the network lifetime into activity rounds. In each round a small
66 number of active nodes is selected to ensure coverage.Each round includes four phases: INFO Exchange, Leader election, decision and sensing.Simulation results show that the proposed protocol can prolong the network
67 lifetime and improve network coverage effectively.
72 %\keywords{Area Coverage, Wireless Sensor Networks, lifetime Optimization, Distributed Protocol.}
75 \IEEEpeerreviewmaketitle
78 \section{Introduction}
79 \noindent Recent years have witnessed significant advances in wireless sensor
80 networks which emerge as one of the most promising technologies for
81 the 21st century~\cite{asc02}. In fact, they present huge potential in
82 several domains ranging from health care applications to military
84 A sensor network is composed of a large number of tiny sensing devices deployed in a region of interest. Each device has processing and wireless communication capabilities, which enable to sense its environment, to compute, to store information and to deliver report messages to a base station.
85 %These sensor nodes run on batteries with limited capacities. To achieve a long life of the network, it is important to conserve battery power. Therefore, lifetime optimisation is one of the most critical issues in wireless sensor networks.
86 One of the main design challenges in Wireless Sensor Networks (WSN) is to prolong the system lifetime, while achieving acceptable quality of service for applications. Indeed, sensor nodes
87 have limited resources in terms of memory, energy and computational powers.
90 Since sensor nodes have limited battery life and without being able to replace
91 batteries, especially in remote and hostile environments,
92 it is desirable that a WSN should be deployed with
93 high density and thus redundancy can be exploited to increase
94 the lifetime of the network. In such a high density network, if all sensor nodes
95 were to be activated at the same time, the lifetime would be reduced. Consequently,
96 future software may need to adapt appropriately to achieve acceptable quality of service for applications.
97 In this paper we concentrate on area coverage problem, with the objective of maximizing the network lifetime by using an adaptive scheduling. Area of interest is divided into subregions and an activity scheduling for sensor nodes is planned for each subregion.
98 Our scheduling scheme works in period which includes a discovery phase to exchange information between sensors of the subregion, then a sensor is chosen in suitable manner to carry out a coverage strategy. This coverage strategy involves the resolution of an integer program which provides the activation of the sensors for the $t$ next round.
101 The remainder of the paper is organized as follows.
102 Section~\ref{rw} reviews the related work in the field.
103 Section \ref{pd} is devoted to the scheduling strategy for energy-efficient coverage.
104 Section \ref{cp} gives the coverage model formulation which is used to schedule the activation of sensors.
105 Section \ref{exp} shows the simulation results conducted on OMNET++, that fully demonstrate the usefulness of the proposed approach. Finally, we give concluding remarks in Section~\ref{sec:conclusion}.
107 \section{\uppercase{Related work}}
110 This section is dedicated to the various approaches proposed in the literature for the coverage lifetime maximization problem where the objective is to optimally schedule sensors'activities in order to extend network lifetime in a randomly deployed network. As this problem is subject to a wide range of interpretations, we suggest to recall main definitions and assumptions related to our work.
114 %\item Area Coverage: The main objective is to cover an area. The area coverage requires
115 %that the sensing range of working Active nodes cover the whole targeting area, which means any
116 %point in target area can be covered~\cite{Mihaela02,Raymond03}.
118 %\item Target Coverage: The objective is to cover a set of targets. Target coverage means that the discrete target points can be covered in any time. The sensing range of working Active nodes only monitors a finite number of discrete points in targeting area~\cite{Mihaela02,Raymond03}.
120 %\item Barrier Coverage An objective to determine the maximal support/breach paths that traverse a sensor field. Barrier coverage is expressed as finding one or more routes with starting position and ending position when the targets pass through the area deployed with sensor nodes~\cite{Santosh04,Ai05}.
123 The most discussed coverage problems in literature can be classified into two types \cite{} : area coverage and targets coverage. An area coverage problem is to find a minimum number of sensors to work such that each physical point in the area is monitored by at least a working sensor. Target coverage problem is to cover only a finite number of discrete points called targets.
124 Our work will concentrate on the area coverage by design and implement a strategy which efficiently select the active nodes that must maintain both sensing coverage and network connectivity and in the same time improve the lifetime of the wireless sensor network. But requiring that all physical points are covered may be too strict, specially where the sensor network is not dense.
125 Our approach represents an area covered by a sensor as a set of principle points and tries to maximize the total number of principles points that are covered in each round, while minimizing overcoverage (points covered by multiple active sensors simultaneously).\\
127 Various definitions exist for the lifetime of a sensor network. Main definitions proposed in the literature are related to the remaining energy of the nodes \cite{} or to the percentage of coverage \cite{}. The lifetime of the network is mainly defined as the amount of time that the network can satisfy its coverage objective (the amount of time that the network can cover a given percentage of its area or targets of interest) . In our simulation we assume that the network is alive until all sensor nodes are died and we measure the coverage ratio during the process.
129 {\bf Activity scheduling}\\
130 Activity scheduling is to schedule the activation and deactivation of nodes 'sensor units. The basic objective is to decide which sensors are in which states (active or sleeping mode) and for how long a time such that the application coverage requirement can be guaranteed and network lifetime can be prolonged. Various approaches, including centralized, distributed and localized algorithms, have been proposed for activity scheduling. In the distributed algorithms, each node in the network autonomously makes decisions on whether to turn on or turn off itself only using local neighbor information. In centralized algorithms, a central controller (node or base station) informs every sensor of the time intervals to be activated.
132 {\bf Distributed approaches}
134 Some distributed algorithms have been developed in~\cite{Gallais06,Tian02,Ye03,Zhang05,HeinzelmanCB02}. Distributed algorithms typically operate in roundsf predetermined duration. At the beginning of each round, a sensor exchange information with its neighbors and makes a decision to either turn on or go to sleep for the round. This decision is basically based on simple greedy criteria like the largest uncovered area \cite{Berman05efficientenergy}, maximum uncovered targets \cite{1240799}.
135 In \cite{Tian02}, the sheduling scheme is divided into rounds, where each round has a self-scheduling phase followed by a sensing phase. Each sensor broadcasts a message to its neighbors containing node ID and node location at the beginning of each round. Sensor determines its status by a rule named off-duty eligible rule which tells him to turn off if its sensing area is covered by its neighbors. A back-off scheme is introduced to let each sensor delay the decision process with a random period of time, in order to avoid that nodes make conflicting decisions simultaneously and that a part of the area is no longer covered.
136 \cite{Prasad:2007:DAL:1782174.1782218} propose a model for capturing the dependencies between different cover sets and propose localized heuristic based on this dependency. The algorithm consists of two phases, an initial setup phase during which each sensor calculates and prioritize the covers and a sensing phase during which each sensor first decides its on/off status and then remains on or off for the rest of the duration.
137 Authors in \cite{chin2007} propose a novel distributed heuristic named distributed Energy-efficient Scheduling for k-coverage (DESK) so that the energy consumption among all the sensors is balanced, and network lifetime is maximized while the coverage requirements being maintained. This algorithm works in round, requires only 1-sensing-hop-neigbor information, and a sensor decides its status (active/sleep) based on its perimeter coverage computed through the k-Non-Unit-disk coverage algorithm proposed in \cite{Huang:2003:CPW:941350.941367}.\\
139 Some others approaches do not consider synchronized and predetermined period of time where the sensors are active or not. Each sensor maintains its own timer and its time wake-up is randomized \cite{Ye03} or regulated \cite{cardei05} over time.
140 %A ecrire \cite{Abrams:2004:SKA:984622.984684}p33
143 %The scheduling information is disseminated throughout the network and only sensors in the active state are responsible
144 %for monitoring all targets, while all other nodes are in a low-energy sleep mode. The nodes decide cooperatively which of them will remain in sleep mode for a certain
147 %one way of increasing lifeteime is by turning off redundant nodes to sleep mode to conserve energy while active nodes provide essential coverage, which improves fault tolerance.
149 %In this paper we focus on centralized algorithms because distributed algorithms are outside the scope of our work. Note that centralized coverage algorithms have the advantage of requiring very low processing power from the sensor nodes which have usually limited processing capabilities. Moreover, a recent study conducted in \cite{pc10} concludes that there is a threshold in terms of network size to switch from a localized to a centralized algorithm. Indeed the exchange of messages in large networks may consume a considerable amount of energy in a localized approach compared to a centralized one.
150 {\bf Centralized approaches}\\
151 Power efficient centralized schemes differ according to several criteria \cite{Cardei:2006:ECP:1646656.1646898}, such as the coverage objective (target coverage or area coverage), the node deployment method (random or deterministic) and the heterogeneity of sensor nodes (common sensing range, common battery lifetime). The major approach is to divide/organize the sensors into a suitable number of set covers where each set completely covers an interest region and to activate these set covers successively.
153 First algorithms proposed in the literature consider that the cover sets are disjoint: a sensor node appears in exactly one of the generated cover sets. For instance Slijepcevic and Potkonjak \cite{Slijepcevic01powerefficient} propose an algorithm which allocates sensor nodes in mutually independent sets to monitor an area divided into several fields. Their algorithm constructs a cover set by including in priority the sensor nodes which cover critical fields, that is to say fields that are covered by the smallest number of sensors. The time complexity of their heuristic is $O(n^2)$ where $n$ is the number of sensors. ~\cite{cardei02} present
154 a graph coloring technique to achieve energy savings
155 by organizing the sensor nodes into a maximum number of disjoint
156 dominating sets which are activated successively. The dominating
157 sets do not guarantee the coverage of the whole region of interest.
158 Abrams et al.\cite{Abrams:2004:SKA:984622.984684} design three approximation algorithms for a variation of the set k-cover problem, where the objective is
159 to partition the sensors into covers such that the number of
160 covers that include an area, summed over all areas, is maximized. Their work builds upon previous work in~\cite{Slijepcevic01powerefficient} and the generated cover sets do not provide complete coverage of the monitoring zone.
163 %examine the target coverage problem by disjoint cover sets but relax the requirement that every cover set monitor all the targets and try to maximize the number of times the targets are covered by the partition. They propose various algorithms and establish approximation ratio.
165 In~\cite{Cardei:2005:IWS:1160086.1160098}, the authors propose a heuristic to
166 compute the disjoint set covers (DSC). In order to compute the maximum number of covers, they
167 first transform DSC into a maximum-flow problem ,
168 which is then formulated as a mixed integer programming problem
169 (MIP). Based on the solution of the MIP, they design a heuristic
170 to compute the final number of covers. The results show a slight performance
171 improvement in terms of the number of produced DSC in comparison to~\cite{Slijepcevic01powerefficient} but it incurs
172 higher execution time due to the complexity of the mixed integer programming resolution.
173 %Cardei and Du \cite{Cardei:2005:IWS:1160086.1160098} propose a method to efficiently compute the maximum number of disjoint set covers such that each set can monitor all targets. They first transform the problem into a maximum flow problem which is formulated as a mixed integer programming (MIP). Then their heuristic uses the output of the MIP to compute disjoint set covers. Results show that these heuristic provides a number of set covers slightly larger compared to \cite{Slijepcevic01powerefficient} but with a larger execution time due to the complexity of the mixed integer programming resolution.
174 Zorbas et al. \cite{Zorbas2007} present B\{GOP\}, a centralized coverage algorithm introducing sensor candidate categorisation depending on their coverage status and the notion of critical target to call targets that are associated with a small number of sensors. The total running time of their heuristic is $0(m n^2)$ where $n$ is the number of sensors, and $m$ the number of targets. Compared to algorithm's results of Slijepcevic and Potkonjak \cite{Slijepcevic01powerefficient}, their heuristic produces more cover sets with a slight growth rate in execution time.
175 %More recently Manju and Pujari\cite{Manju2011}
177 In the case of non-disjoint algorithms \cite{Manju2011}, sensors may participate in more than one cover set.
178 In some cases this may prolong the lifetime of the network in comparison to the disjoint cover set algorithms but designing algorithms for non-disjoint cover sets generally incurs a higher order of complexity. Moreover in case of a sensor's failure, non-disjoint scheduling policies are less resilient and less reliable because a sensor may be involved in more than one cover sets. For instance, Cardei et al.~\cite{cardei05bis} present a linear programming (LP) solution
179 and a greedy approach to extend
180 the sensor network lifetime by organizing the sensors into a
181 maximal number of non-disjoint cover sets. Simulation results show that by allowing sensors to
182 participate in multiple sets, the network lifetime
183 increases compared with related work~\cite{Cardei:2005:IWS:1160086.1160098}. In~\cite{berman04}, the authors have formulated the lifetime problem and suggested another (LP) technique to solve this problem. A centralized provably near
184 optimal solution based on the Garg-K\"{o}nemann algorithm~\cite{garg98} is also proposed.
186 {\bf Our contribution}
187 %{decoupage de la region en sous region, selection de noeud leader, formulation %et resolution du probleme de couverture, planification périodique
188 There are three main questions which should be answered to build a scheduling strategy. We give a brief answer to these three questions to describe our approach before going into details in the subsequent sections.
190 \item {\bf How must be planned the
191 phases for information exchange, decision and sensing over time?}
192 Our algorithm partitions the time line into a number of periods. Each period contains 4 phases : information Exchange, Leader Election, Decision, and Sensing. Our work further divides sensing phase into a number of rounds of predetermined length.
193 \item {\bf What are the rules to decide which node has to turn on or off?}
194 Our algorithm tends to limit the overcoverage of points of interest to avoid turning on too much sensors covering the same areas at the same time, and tries to prevent undercoverage. The decision is a good compromise between these two conflicting objectives and is made for the next $T$ rounds of sensing. In our experimentations we will check which value of $T$ is the most appropriate.
195 \item {\bf Which node should make such decision ?}
196 As mentioned in \cite{pc10}, both centralized and distributed algorithms have their own advantages and disadvantages. Centralized coverage algorithms have the advantage of requiring very low processing power from the sensor nodes which have usually limited processing capabilities. Distributed algorithms are very adaptable to the dynamic and scalable nature of sensors network. Authors in \cite{pc10} concludes that there is a threshold in terms of network size to switch from a localized to a centralized algorithm. Indeed the exchange of messages in large networks may consume a considerable amount of energy in a localized approach compared to a centralized one. Our work does not consider only one leader to compute and to broadcast the schedule decision to all the sensors. When the size of network increases, the network is divided in many subregions and the decision is made by a leader in each subregion.
201 \section{\uppercase{Distributed coverage model}}
203 We consider a randomly and uniformly deployed network consisting of static wireless sensors. The wireless sensors are deployed in high density to ensure initially a full coverage of the interested area. We assume that all nodes are homogeneous in terms of communication and processing capabilities and heterogeneous in term of energy. The location information is available to the sensor node either through hardware such as embedded GPS or through location discovery algorithms.
204 The area of interest can be divided using the divide-and-conquer strategy into smaller area called subregions and then our coverage protocol will be implemented in each subregion simultaneously. Our protocol works in rounds fashion as in figure \ref{fig:4}.
205 %Given the interested Area $A$, the wireless sensor nodes set $S=\lbrace s_1,\ldots,s_N \rbrace $ that are deployed randomly and uniformly in this area such that they are ensure a full coverage for A. The Area A is divided into regions $A=\lbrace A^1,\ldots,A^k,\ldots, A^{N_R} \rbrace$. We suppose that each sensor node $s_i$ know its location and its region. We will have a subset $SSET^k =\lbrace s_1,...,s_j,...,s_{N^k} \rbrace $ , where $s_N = s_{N^1} + s_{N^2} +,\ldots,+ s_{N^k} +,\ldots,+s_{N^R}$. Each sensor node $s_i$ has the same initial energy $IE_i$ in the first time and the current residual energy $RE_i$ equal to $IE_i$ in the first time for each $s_i$ in A. \\
209 \includegraphics [width=70mm]{FirstModel.eps}
210 \caption{Multi-Round Coverage Protocol}
214 Each round is divided into 4 phases : INFO Exchange, Leader Election, Decision, and Sensing. For each round there is exactly one set cover responsible for sensing task. This protocol is more reliable against the unexpectedly node failure because it works into rounds,and if the node failure detected before taking the decision, the node will not participate in decision and if the the node failure obtain after the decision the sensing task of the network will be affected temporarily only during the period of sensing until starting new round, since a new set cover will take charge of the sensing task in the next round. The energy consumption and some other constraints can easily be taken into account since the sensors can update and then exchange the information (including their residual energy) at the beginning of each round. However, the preprocessing phase (INFO Exchange, leader Election, Decision) are energy consuming for some nodes even when they not join the network to monitor the area. We describe each phase in more detail.
216 \subsection{\textbf INFO Exchange Phase}
218 Each sensor node $j$ sends its position, remaining energy $RE_j$, number of local neighbours $NBR_j$ to all wireless sensor nodes in its subregion by using INFO packet and listen to the packets sent from other nodes. After that, each node will have information about all the sensor nodes in the subregion. In our model.
220 % the remaining energy corresponds to the time that a sensor can live in the active mode.
223 %\subsection{\textbf Working Phase:}
225 %The working phase works in rounding fashion. Each round include 3 steps described as follow :
227 \subsection{\textbf Leader Election Phase}
228 This step includes choosing the Wireless Sensor Node Leader (WSNL) which will be responsible of executing coverage algorithm to choose the list of active sensor nodes that contribute in covering the subregion.
229 % The sensors in the same region are capable to communicate with each others using a routing protocol provided by the simulator OMNET++ in order to provide multi-hop communication protocol.
230 The WSNL will be chosen based on the number of local neighbours $NBR_j$ of sensor node $s_j$ and it's remaining energy $RE_j$.
231 If we have more than one node has the same $NBR_j$ and $RE_j$, this leads to choose WSNL based on the largest index among them. Each subregion in the area of interest will select its WSNL independently for each round.
234 \subsection{\textbf Decision Phase}
235 The WSNL will execute the GLPK algorithm to select which sensors will be activated in the next rounds to cover the subregion. WSNL will send Active-Sleep packet to each sensor in the subregion based on algorithm's results.
236 %The main goal in this step after choosing the WSNL is to produce the best representative active nodes set that will take the responsibility of covering the whole region $A^k$ with minimum number of sensor nodes to prolong the lifetime in the wireless sensor network. For our problem, in each round we need to select the minimum set of sensor nodes to improve the lifetime of the network and in the same time taking into account covering the region $A^k$ . We need an optimal solution with tradeoff between our two conflicting objectives.
237 %The above region coverage problem can be formulated as a Multi-objective optimization problem and we can use the Binary Particle Swarm Optimization technique to solve it.
240 \subsection{\textbf Sensing Phase}
241 The algorithm will produce the best representative set of the active nodes that will take the mission of coverage preservation in the subregion during the Sensing phase. Since that we use a homogeneous wireless sensor network, we will assume that the cost of keeping a node awake (or sleep) for sensing task is the same for all wireless sensor nodes in the network.
245 %\subsection{Sensing coverage model}
248 %\noindent We try to produce an adaptive scheduling which allows sensors to operate alternatively so as to prolong the network lifetime. For convenience, the notations and assumptions are described first.
249 %The wireless sensor node use the binary disk sensing model by which each sensor node will has a certain sensing range is reserved within a circular disk called radius $R_s$.
250 \noindent We consider a boolean disk coverage model which is the most widely used sensor coverage model in the literature. Each sensor has a constant sensing range $R_s$. All space points within a disk centered at the sensor with the radius of the sensing range is said to be covered by this sensor. We also assume that the communication range is at least twice of the sening range. In fact, Zhang and Zhou ~\cite{Zhang05} prove that if the tranmission range is at least twice of the sensing range, a complete coverage of a convex area implies connectivity amnong the working nodes in the active mode.
251 %To calculate the coverage ratio for the area of interest, we can propose the following coverage model which is called Wireless Sensor Node Area Coverage Model to ensure that all the area within each node sensing range are covered. We can calculate the positions of the points in the circle disc of the sensing range of wireless sensor node based on the Unit Circle in figure~\ref{fig:cluster1}:
256 %%\includegraphics[scale=0.25]{fig1.pdf}\\ %& \includegraphics[scale=0.10]{1.pdf} \\
257 %%(A) Figure 1 & (B) Figure 2
259 %\caption{Unit Circle in radians. }
260 %\label{fig:cluster1}
263 %By using the Unit Circle in figure~\ref{fig:cluster1},
264 %We choose to representEach wireless sensor node will be represented into a selected number of principle points by which we can know if the sensor node is covered or not.
265 % Figure ~\ref{fig:cluster2} shows the selected principle points that represents the area of the sensor node and according to the sensing range of the wireless sensor node.
267 \noindent Instead of working with area coverage, we consider for each sensor a set of points called principal points. And we assume the sensing disk defined by a sensor is covered if all principal points of this sensor are covered.
272 %%\includegraphics[scale=0.25]{fig2.pdf}\\ %& \includegraphics[scale=0.10]{1.pdf} \\
273 %%(A) Figure 1 & (B) Figure 2
275 %\caption{Wireless Sensor Node Area Coverage Model.}
276 %\label{fig:cluster2}
281 \noindent By knowing the position (point center :($p_x,p_y$) of the Wireless sensor node and its $R_s$ , we calculate the principle points directly based on proposed model. We use these principle points (that can be increased or decreased as if it is necessary) as references to ensure that the monitoring area of the region is covered by the selected set of sensors instead of using the all points in the area.
285 % \begin{multicols}{6}
287 %\includegraphics[scale=0.10]{fig21.pdf}\\~ ~ ~(a)
288 %\includegraphics[scale=0.10]{fig22.pdf}\\~ ~ ~(b)
289 \includegraphics[scale=0.2]{principles13.eps}
290 %\includegraphics[scale=0.10]{fig25.pdf}\\~ ~ ~(d)
291 %\includegraphics[scale=0.10]{fig26.pdf}\\~ ~ ~(e)
292 %\includegraphics[scale=0.10]{fig27.pdf}\\~ ~ ~(f)
294 \caption{Wireless Sensor node represented by 13 principle points }
298 \noindent We can calculate the positions of the selected principle points in the circle disk of the sensing range of wireless sensor node in figure ~\ref{fig3} as follow:\\
299 $p_x,p_y$ = point center of wireless sensor node. \\
301 $X_2=( p_x + R_s * (1), p_y + R_s * (0) )$\\
302 $X_3=( p_x + R_s * (-1), p_y + R_s * (0)) $\\
303 $X_4=( p_x + R_s * (0), p_y + R_s * (1) )$\\
304 $X_5=( p_x + R_s * (0), p_y + R_s * (-1 )) $\\
305 $X_6= ( p_x + R_s * (\frac{-\sqrt{2}}{2}), p_y + R_s * (0)) $\\
306 $X_7=( p_x + R_s * (\frac{\sqrt{2}}{2}), p_y + R_s * (0))$\\
307 $X_8=( p_x + R_s * (\frac{-\sqrt{2}}{2}), p_y + R_s * (\frac{-\sqrt{2}}{2})) $\\
308 $X_9=( p_x + R_s * (\frac{\sqrt{2}}{2}), p_y + R_s * (\frac{-\sqrt{2}}{2})) $\\
309 $X_{10}=( p_x + R_s * (\frac{-\sqrt{2}}{2}), p_y + R_s * (\frac{\sqrt{2}}{2})) $\\
310 $X_{11}=( p_x + R_s * (\frac{\sqrt{2}}{2}), p_y + R_s * (\frac{\sqrt{2}}{2})) $\\
311 $X_{12}=( p_x + R_s * (0), p_y + R_s * (\frac{\sqrt{2}}{2})) $\\
312 $X_{13}=( p_x + R_s * (0), p_y + R_s * (\frac{-\sqrt{2}}{2})) $\\
317 \section{\uppercase{Coverage problem formulation}}
319 %We can formulate our optimization problem as energy cost minimization by minimize the number of active sensor nodes and maximizing the coverage rate at the same time in each $A^k$ . This optimization problem can be formulated as follow: Since that we use a homogeneous wireless sensor network, we will assume that the cost of keeping a node awake is the same for all wireless sensor nodes in the network. We can define the decision parameter $X_j$ as in \eqref{eq11}:\\
322 %To satisfy the coverage requirement, the set of the principal points that will represent all the sensor nodes in the monitored region as $PSET= \lbrace P_1,\ldots ,P_p, \ldots , P_{N_P^k} \rbrace $, where $N_P^k = N_{sp} * N^k $ and according to the proposed model in figure ~\ref{fig:cluster2}. These points can be used by the wireless sensor node leader which will be chosen in each region in A to build a new parameter $\alpha_{jp}$ that represents the coverage possibility for each principal point $P_p$ of each wireless sensor node $s_j$ in $A^k$ as in \eqref{eq12}:\\
325 \noindent Our model is based on the model proposed by \cite{pedraza2006} where the objective is to find a maximum number of disjoint cover sets. To accomplish this goal, authors propose a integer program which forces undercoverage and overcoverage of targets to become minimal at the same time. They use variables $x_{s,l}$ to indicate if the sensor $s$ belongs to cover set $l$. In our model, we consider binary variables $X_{j,t}$ which determine the activation of sensor $j$ in round $t$. We replace the constraint guarantying that each sensor is a member of only one cover of the entire set of disjoint covers by a constraint specifying that the sum of energy consumed by the activation of sensor during several rounds is less than or equal to the remaining energy of the sensor. We also consider principle points as targets. \\
326 \noindent For a principle point $p$, let $\alpha_{jp}$ denote the indicator function of whether the point $p$ is covered, that is, \\
328 \alpha_{jp} = \left \{
330 1 & \mbox{if the principal point $p$ is covered} \\
331 & \mbox{by active sensor node $j$} \\
332 0 & \mbox{Otherwise}\\
336 The number of sensors that are covering point $p$ during a round $t$ is equal to $\sum_{j \in J} \alpha_{jp} * X_{j,t}$ where :
340 1& \mbox{if sensor $s_j$ is active during round } t\\
341 0 & \mbox{Otherwise}\\
345 We define the Overcoverage variable $\Theta_{p,t}$ .\\
348 \Theta_{p,t} = \left \{
350 0 & \mbox{if point p is not }\\
351 &\mbox{covered during round } t\\
352 \left( \sum_{j \in J} \alpha_{jp} * X_{j,t} \right)- 1 & \mbox{Otherwise}\\
358 \noindent$\Theta_{p}$ represents the number of active sensor nodes minus one that cover the principle point $p$.\\
359 The Undercoverage variable $U_{p,t}$ of the principle point $p$ is defined as follow :\\
364 1 &\mbox{if point } $p$ \mbox{ is not covered during round } $t$\\
365 0 & \mbox{Otherwise}\\
370 \noindent Our coverage optimization problem can be formulated as follow.\\
371 \begin{equation} \label{eq:ip2r}
374 \min \sum_{p \in P} (w_{\theta,t} \Theta_{p,t} + w_{u,t} U_{p,t})&\\
375 \textrm{subject to :}&\\
376 \sum_{j \in J} \alpha_{jp} X_{j,t} - \Theta_{p,t}+ U_{p,t} =1, &\forall p \in P, \forall t \in T\\
378 \sum_{t \in T} X_{j,t} \leq \frac{RE_j}{e_t} &\forall j \in J \\
380 \Theta_{p,t}\in \mathbb{N} , &\forall p \in P, \forall t \in T\\
381 U_{p,t} \in \{0,1\}, &\forall p \in P, \forall t \in T \\
382 X_{j,t} \in \{0,1\}, &\forall j \in J, \forall t \in T
387 \item $X_{j,t}$ : indicating whether or not sensor $j$ is active in round $t$(1 if yes and 0 if not)
388 \item $\Theta_{p,t}$ : {\it overcoverage}, the number of sensors minus one that are covering point $p$ in round $t$
389 \item $U_{p,t}$ : {\it undercoverage}, indicating whether or not point $p$ is being covered (1 if not covered and 0 if covered) in round $t$
391 The first group of constraints indicates that some point $p$ should be covered by at least one sensor in every round $t$ and, if it is not always the case, overcoverage and undercoverage variables help balance the restriction equation by taking positive values. Second group of contraints ensures for each sensor that the amount of energy consumed during its activation periods will be less than or equal to its remaining energy.
392 There are two main objectives. We limit overcoverage of principle points in order to activate a minimum number of sensors and we prevent that parts of the subregion are not monitored by minimizing undercoverage. The weights $w_{\theta,t}$ and $w_{u,t}$ must be properly chosen so as to guarantee that the maximum number of points are covered during each round.
394 %In equation \eqref{eq15}, there are two main objectives: the first one using the Overcoverage parameter to minimize the number of active sensor nodes in the produced final solution vector $X$ which leads to improve the life time of wireless sensor network. The second goal by using the Undercoverage parameter to maximize the coverage in the region by means of covering each principle point in $SSET^k$.The two objectives are achieved at the same time. The constraint which represented in equation \eqref{eq16} refer to the coverage function for each principle point $P_p$ in $SSET^k$ , where each $P_p$ should be covered by
395 %at least one sensor node in $A^k$. The objective function in \eqref{eq15} involving two main objectives to be optimized simultaneously, where optimal decisions need to be taken in the presence of trade-offs between the two conflicting main objectives in \eqref{eq15} and this refer to that our coverage optimization problem is a multi-objective optimization problem and we can use the BPSO to solve it. The concept of Overcoverage and Undercoverage inspired from ~\cite{Fernan12} but we use it with our model as stated in subsection \ref{Sensing Coverage Model} with some modification to be applied later by BPSO.
396 %\subsection{Notations and assumptions}
399 %\item $m$ : the number of targets
400 %\item $n$ : the number of sensors
401 %\item $K$ : maximal number of cover sets
402 %\item $i$ : index of target ($i=1..m$)
403 %\item $j$ : index of sensor ($j=1..n$)
404 %\item $k$ : index of cover set ($k=1..K$)
405 %\item $T_0$ : initial set of targets
406 %\item $S_0$ : initial set of sensors
407 %\item $T $ : set of targets which are not covered by at least one cover set
408 %\item $S$ : set of available sensors
409 %\item $S_0(i)$ : set of sensors which cover the target $i$
410 %\item $T_0(j)$ : set of targets covered by sensor $j$
411 %\item $C_k$ : cover set of index $k$
412 %\item $T(C_k)$ : set of targets covered by the cover set $k$
413 %\item $NS(i)$ : set of available sensors which cover the target $i$
414 %\item $NC(i)$ : set of cover sets which do not cover the target $i$
415 %\item $|.|$ : cardinality of the set
418 \section{\uppercase{Simulation Results}}
420 In this section, we conducted a series of simulations to evaluate the efficiency of our approach
421 based on the discrete event simulator OMNeT++ (http://www.omnetpp.org/).we conduct simulations for six
422 different densities varying from 50 to 300 nodes. Experimental results were obtained from randomly generated
423 networks in which nodes are deployed over a $ 50\times25(m2) $sensing field. For each network deployment, we
424 assume that the deployed nodes can fully cover the sensing field with the given sensing range. 100 simulation runs are performed with different network topologies. The results presented hereafter are the average of these 100 runs.Simulation ends when there is at least one active node has no connectivity with the network.Our proposed coverage protocol use the Radio energy dissipation model that defined by~\cite{HeinzelmanCB02} as energy consumption model by each wireless sensor node for transmitting and receiving the packets in the network.The energy of each node in the network is initialized randomly within the range 24-60 joules, and each sensor will consumes 0.2 watts during the period of sensing which it is 60 seconds.Each active node will consumes 12 joules during sensing phase and each sleep node will consumes 0.002 joules.Each sensor node will not participate in the next round if it's remaining energy less than 12 joules. In all experiments the parameters are given by $R_s = 5m $ , $ W_{\Theta} =1$ and $W_{\Psi} = P^2$.
425 We evaluate the efficiency of our approach using some performance metrics such as:coverage ratio, number of
426 active nodes ratio, energy saving ratio, number of rounds, network lifetime and execution time of our approach.Coverage ratio measures how much area of a sensor field is covered. In our case, the coverage ratio is regarded as the number of principle points covered among the set of all principle points within the field.In our simulation the sensing field is sub divided into two subregions each one equal to $ 25\times25(m2) $ of the sensing field.
428 \subsection{The impact of the Number of Rounds on Coverage Ratio:}
429 In this experiment, we study the impact of the number of rounds on the coverage ratio and for different sizes for sensor network.For each Sensor network size we will take the average of coverage ratio per round and for 100 simulation.Fig. 3 show the impact of the number of rounds on coverage ratio for different network sizes and for two subregions.
433 % \begin{multicols}{6}
435 %\includegraphics[scale=0.10]{fig21.pdf}\\~ ~ ~(a)
436 %\includegraphics[scale=0.10]{fig22.pdf}\\~ ~ ~(b)
437 \includegraphics[scale=0.5]{CR2R2L_1.eps}\\~ ~ ~(a)
438 \includegraphics[scale=0.5]{CR2R2L_2.eps}\\~ ~ ~(b)
439 %\includegraphics[scale=0.10]{fig26.pdf}\\~ ~ ~(e)
440 %\includegraphics[scale=0.10]{fig27.pdf}\\~ ~ ~(f)
442 \caption{The impact of the Number of Rounds on Coverage Ratio.(a):subregion 1. (b): subregion 2 }
446 As shown Fig. 3 (a) and (b) our protocol can give a full average coverage ratio in the first rounds and then it decreases when the number of rounds increases due to dead nodes.Although some nodes are dead, sensor activity scheduling choose other nodes to ensure the coverage of interest area. Moreover, when we have a dense sensor network, it leads to maintain the full coverage for larger number of rounds.
448 \subsection{The impact of the Number of Rounds on Energy Saving Ratio:}
450 \subsection{The impact of the Number of Rounds on Active Sensor Ratio:}
452 \subsection{The impact of Number of Sensors on Number of Rounds:}
454 \subsection{The impact of Number of Sensors on Network Lifetime:}
456 \subsection{The impact of Number of Sensors on Execution Time:}
458 \subsection{Performance Comparison:}
459 \label{Simulation Results}
462 \section{\uppercase{Conclusions}}
463 \label{sec:conclusion}
464 In this paper, we have addressed the problem of lifetime optimization in wireless sensor networks. This is a very
465 natural and important problem, as sensor nodes
466 have limited resources in terms of memory, energy and computational power.
467 %energy-efficiency is crucial in power-limited wireless sensor network.
468 To cope with this problem,
469 %an efficient centralized energy-aware algorithm is presented and analyzed. Our algorithm seeks to
470 %Energy-efficiency is crucial in power-limited wireless sensor network, since sensors have significant power constraints (battery life). In this paper we have investigated the problem of
476 % use section* for acknowledgement
477 \section*{Acknowledgment}
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