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-
 \documentclass[conference]{IEEEtran}
 
-
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 \else
@@ -10,15 +8,14 @@
 
 \hyphenation{op-tical net-works semi-conduc-tor}
 
-\usepackage{float}
+\usepackage{float} 
 \usepackage{epsfig}
-\usepackage{subfigure}
 \usepackage{calc}
  \usepackage{times,amssymb,amsmath,latexsym}
 \usepackage{graphics}
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-\usepackage{txfonts}
+%\usepackage{txfonts}
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@@ -32,20 +29,25 @@
 \usepackage{epsfig}
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-
+\usepackage{times}
+\usepackage{graphicx,epstopdf}
+\epstopdfsetup{suffix=}
+\DeclareGraphicsExtensions{.ps}
+\DeclareGraphicsRule{.ps}{pdf}{.pdf}{`ps2pdf -dEPSCrop -dNOSAFER #1 \noexpand\OutputFile}
 
 \begin{document}
 
-\title{Distributed Coverage Optimization Protocol to Improve the Lifetime in Heterogeneous Energy Wireless Sensor Networks}
-
+%\title{ Coverage and Lifetime Optimization in Heterogeneous Energy Wireless Sensor Networks} 
+\title{Coverage and Lifetime Optimization in Heterogeneous Energy Wireless Sensor Networks}
+%Activity Scheduling for Coverage and Lifetime Optimization in  Wireless Sensor Networks}
 
 % author names and affiliations
 % use a multiple column layout for up to three different
 % affiliations
-\author{\IEEEauthorblockN{Ali Kadhum Idrees, Karine Deschinkel, Michel Salomon and Raphael Couturier }
-\IEEEauthorblockA{FEMTO-ST Institute, UMR CNRS, University of Franche-Comte, Belfort, France \\
-Email:$\lbrace$ali.idness, karine.deschinkel, michel.salomon,raphael.couturier$\rbrace$@edu.univ-fcomte.fr 
-}
+\author{\IEEEauthorblockN{Ali Kadhum Idrees, Karine Deschinkel, Michel Salomon, and Rapha\"el Couturier}
+\IEEEauthorblockA{FEMTO-ST Institute, UMR 6174 CNRS \\ 
+University of Franche-Comt\'e, Belfort, France \\
+Email: ali.idness@edu.univ-fcomte.fr, $\lbrace$karine.deschinkel, michel.salomon, raphael.couturier$\rbrace$@univ-fcomte.fr}
 %\email{\{ali.idness, karine.deschinkel, michel.salomon, raphael.couturier\}@univ-fcomte.fr}
 %\and
 %\IEEEauthorblockN{Homer Simpson}
@@ -55,61 +57,77 @@ Email:$\lbrace$ali.idness, karine.deschinkel, michel.salomon,raphael.couturier$\
 %\IEEEauthorblockA{FEMTO-ST Institute, UMR CNRS, University of Franche-Comte, Belfort, France}
 }
 
-
 \maketitle
 
-
 \begin{abstract}
-%\boldmath
-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 
-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
-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
-lifetime and improve network coverage effectively.
-
-
+One of  the fundamental challenges in Wireless  Sensor Networks (WSNs)
+is  the coverage  preservation  and  the extension  of  the  network  lifetime
+continuously  and  effectively  when  monitoring a  certain  area  (or
+region) of interest. In this paper, a coverage optimization protocol to
+improve the lifetime in  heterogeneous energy wireless sensor networks
+is proposed.   The area of  interest is first divided  into subregions
+using a  divide-and-conquer method and then the scheduling  of sensor node
+activity  is  planned for  each  subregion.   The proposed  scheduling
+considers  rounds during  which  a small  number  of nodes,  remaining
+active  for  sensing, is  selected  to  ensure  coverage.  Each  round
+consists  of   four  phases:  (i)~Information   Exchange,  (ii)~Leader
+Election, (iii)~Decision,  and (iv)~Sensing.  The  decision process is
+carried  out  by  a  leader  node, which  solves  an  integer  program.
+Simulation  results show that  the proposed  approach can  prolong the
+network lifetime and improve the coverage performance.
 \end{abstract}
 
- %\keywords{Area Coverage, Wireless Sensor Networks, lifetime Optimization, Distributed Protocol.}
-
+\begin{IEEEkeywords}
+Area Coverage, Network lifetime, Optimization, Scheduling, Distributed Protocol.
+\end{IEEEkeywords}
+%\keywords{Area Coverage, Network lifetime, Optimization, Distributed Protocol}
  
- \IEEEpeerreviewmaketitle
-
+\IEEEpeerreviewmaketitle
 
 \section{Introduction}
-\noindent Recent years have witnessed significant advances in wireless sensor
-networks which emerge as one of the most promising technologies for
-the 21st century~\cite{asc02}. In fact, they present huge potential in
-several domains ranging from health care applications to military
-applications. 
-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. 
-%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.
-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
-have limited resources in terms of memory, energy and computational powers. 
-
-%\medskip
-Since sensor nodes have limited battery life and without being able to replace 
-batteries, especially in remote and hostile environments,
-it is desirable that a WSN should be deployed with 
-high density and thus redundancy can be exploited to increase 
-the lifetime of the network. In such a high density network, if all sensor nodes 
-were to be activated at the same time, the lifetime would be reduced. Consequently, 
-future software may need to adapt appropriately to achieve acceptable quality of service for applications.  
-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.
-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.
-
-
-The remainder  of the paper is  organized as follows.
-Section~\ref{rw} reviews the related work in the field. 
-Section \ref{pd} is devoted to the scheduling strategy for energy-efficient coverage. 
-Section \ref{cp} gives the coverage model formulation which is used to schedule the activation of sensors.
-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}.
-
-\section{\uppercase{Related work}}
+
+\noindent The fast developments in the low-cost sensor devices and wireless communications have allowed the emergence the WSNs. WSN includes a large number of small , limited-power sensors  that can sense, process and transmit
+ data over a wireless communication . They communicate with each other by using multi-hop wireless communications , cooperate together to monitor the area of interest, and the measured data can be reported
+ to a monitoring center
+called, sink, for analysis it~\cite{Ammari01, Sudip03}. There are several applications used the WSN including health, home, environmental, military,and industrial applications~\cite{Akyildiz02}.
+The coverage  problem is one of the fundamental challenges in WSNs~\cite{Nayak04} that consists in monitoring efficiently and  continuously the area of interest. The limited energy of sensors represents the main challenge in the WSNs design~\cite{Ammari01}, where it is difficult to replace and/or
+ recharge their batteries because the the area of interest nature (such as hostile environments) and the cost. So, it is necessary that a WSN deployed with high density  because spatial redundancy can then be exploited to increase the lifetime of the network . However, turn on all the sensor nodes, which monitor the same region at the same time leads to decrease the lifetime of the network. To extend the lifetime of the network, the main idea is to take advantage of the overlapping sensing regions of some sensor nodes to save energy by turning off some of them during the sensing phase~\cite{Misra05}. WSNs require energy-efficient solutions to improve the network lifetime that is constrained by the limited power of each sensor node ~\cite{Akyildiz02}. 
+In  this  paper,  we  concentrate on  the area
+coverage  problem,  with  the  objective  of  maximizing  the  network
+lifetime  by using  an adaptive  scheduling. The  area of  interest is
+divided into subregions and an activity scheduling for sensor nodes is
+planned for  each subregion. 
+ In fact, the nodes in a  subregion can be seen as a cluster where
+  each node sends  sensing data to the cluster head  or the sink node.
+  Furthermore, the activities in a subregion/cluster can continue even
+  if another cluster stops due to too many node failures.
+Our scheduling  scheme considers rounds,  where a round starts  with a
+discovery  phase  to  exchange  information  between  sensors  of  the
+subregion,  in order to  choose in  a suitable manner  a sensor  node to
+carry  out a coverage  strategy. This  coverage strategy  involves the
+solving of  an integer  program, which provides  the activation  of the
+sensors for the sensing phase of the current round.
+
+The remainder of the paper is organized as follows.  The next section
+% Section~\ref{rw}
+reviews the related work in the field.  Section~\ref{pd} is devoted to
+the    scheduling     strategy    for    energy-efficient    coverage.
+Section~\ref{cp} gives the coverage model formulation, which is used to
+schedule  the  activation  of  sensors.  Section~\ref{exp}  shows  the
+simulation  results obtained  using  the discrete  event simulator OMNeT++  \cite{varga}. They  fully demonstrate  the usefulness  of the
+proposed  approach.   Finally, we  give  concluding  remarks and  some
+suggestions for future works in Section~\ref{sec:conclusion}.
+
+\section{Related works}
 \label{rw}
-\noindent 
-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. 
 
-{\bf Coverage}
+\noindent 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 have chosen
+to recall the main definitions and assumptions related to our work.
+
 %\begin{itemize}
 %\item Area Coverage: The main objective is to cover an area. The area coverage requires
 %that the sensing range of working Active nodes cover the whole targeting area, which means any
@@ -119,27 +137,101 @@ This section is dedicated to the various approaches proposed in the literature f
 
 %\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}.
 %\end{itemize}
-
-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. 
- 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. 
-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).\\
-{\bf Lifetime}\\
-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.
-
-{\bf Activity scheduling}\\
-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.
-
-{\bf Distributed approaches}
-
-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}. 
-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.
-\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. 
-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}.\\
-
-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. 
+\subsection{Coverage} 
+%{\bf Coverage}
+
+The most  discussed coverage problems in literature  can be classified
+into two types \cite{ma10}: area coverage (also called full or blanket
+coverage) and target coverage.  An  area coverage problem is to find a
+minimum number of sensors to work, such that each physical point in the
+area is within the sensing range  of at least one working sensor node.
+Target coverage problem  is to cover only a  finite number of discrete
+points  called targets.   This type  of coverage  has  mainly military
+applications. Our work will concentrate on the area coverage by design
+and implementation of a  strategy, which efficiently selects the active
+nodes   that  must   maintain  both   sensing  coverage   and  network
+connectivity and at the same time improve the lifetime of the wireless
+sensor  network.   But,  requiring  that  all physical  points  of  the
+considered region are covered may  be too strict, especially where the
+sensor network is not dense.   Our approach represents an area covered
+by a sensor as a set of primary points and tries to maximize the total
+number  of  primary points  that  are  covered  in each  round,  while
+minimizing  overcoverage (points  covered by  multiple  active sensors
+simultaneously).
+
+\subsection{Lifetime} 
+%{\bf Lifetime}
+
+Various   definitions   exist   for   the   lifetime   of   a   sensor
+network~\cite{die09}.  The main definitions proposed in the literature are
+related to the  remaining energy of the nodes or  to the coverage percentage. 
+The lifetime of the  network is mainly defined as the amount
+of  time during which  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 this work, we assume that the network
+is alive  until all  nodes have  been drained of  their energy  or the
+sensor network becomes disconnected, and we measure the coverage ratio
+during the WSN lifetime.  Network connectivity is important because an
+active sensor node without  connectivity towards a base station cannot
+transmit information on an event in the area that it monitors.
+
+\subsection{Activity scheduling} 
+%{\bf Activity scheduling}
+
+Activity scheduling is to  schedule the activation and deactivation of
+sensor nodes.  The  basic objective is to decide  which sensors are in
+what states (active or sleeping mode)  and for how long, so that the
+application  coverage requirement  can be  guaranteed and  the network
+lifetime can be  prolonged. Various approaches, including centralized,
+distributed, and localized algorithms, have been proposed for activity
+scheduling.  In  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  (a node or  base station) informs every  sensors of
+the time intervals to be activated.
+
+\subsection{Distributed approaches}
+%{\bf Distributed approaches}
+
+Some      distributed     algorithms      have      been     developed
+in~\cite{Gallais06,Tian02,Ye03,Zhang05,HeinzelmanCB02}  to perform the
+scheduling.   Distributed algorithms typically  operate in  rounds for
+a predetermined  duration. At  the  beginning of  each  round, a  sensor
+exchanges information with its neighbors and makes a decision to either
+remain turned  on or to  go to sleep  for the round. This  decision is
+basically made on simple greedy criteria like  the largest uncovered
+area   \cite{Berman05efficientenergy},   maximum   uncovered   targets
+\cite{1240799}.   In \cite{Tian02}, the  scheduling scheme  is divided
+into rounds, where each round  has a self-scheduling phase followed by
+a sensing phase.  Each sensor  broadcasts a message containing the node ID
+and the node location  to its neighbors at the beginning  of each round. A
+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
+simultaneous conflicting decisions between nodes and  lack of coverage on any area.
+\cite{Prasad:2007:DAL:1782174.1782218}  defines a model  for capturing
+the dependencies  between different cover sets  and proposes localized
+heuristic  based on this  dependency.  The  algorithm consists  of two
+phases, an initial  setup phase during which each  sensor computes and
+prioritizes 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.  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 requirement  is being  maintained.  This algorithm  works in
+round, requires only  1-sensing-hop-neighbor 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}.
+
+Some other approaches do  not consider a synchronized and predetermined
+period  of time  where the  sensors are  active or  not.  Indeed, each
+sensor  maintains its  own timer  and its  wake-up time is randomized
+\cite{Ye03} or regulated \cite{cardei05} over time.
 %A ecrire \cite{Abrams:2004:SKA:984622.984684}p33
 
-
 %The scheduling information is disseminated throughout the network and only sensors in the active state are responsible
 %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
 %period of time.
@@ -147,107 +239,228 @@ Some others approaches  do not consider synchronized and predetermined period of
  %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. 
 
 %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. 
-{\bf Centralized approaches}\\
-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. 
-
-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
-a graph coloring technique to achieve energy savings 
-by organizing the sensor nodes into a maximum number of disjoint 
-dominating sets which are activated successively. The dominating 
-sets do not guarantee the coverage of the whole region of interest. 
-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
-to partition the sensors into covers such that the number of
-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. 
 
+\subsection{Centralized approaches}
+%{\bf Centralized approaches}
+
+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.
+
+The 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 builds  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. In~\cite{cardei02}, a graph coloring
+technique is described to achieve energy  savings by  organizing the  sensor nodes
+into a maximum number of  disjoint dominating sets, which are activated
+successively. The dominating sets do not guarantee the coverage of the
+whole        region        of        interest.        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 to partition the sensors into covers such that the number
+of covers that  includes 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.
 
 %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.
 
-In~\cite{Cardei:2005:IWS:1160086.1160098}, the authors propose a heuristic to
-compute the disjoint set covers (DSC). In order to compute the maximum number of covers, they
-first transform DSC into a maximum-flow problem ,
-which is then formulated as a mixed integer programming problem
-(MIP). Based on the solution of the MIP, they design a heuristic
-to compute the final number of covers. The results show a slight performance
-improvement in terms of the number of produced DSC in comparison to~\cite{Slijepcevic01powerefficient} but it incurs
-higher execution time due to the complexity of the mixed integer programming resolution.
- %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.
-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.
+In~\cite{Cardei:2005:IWS:1160086.1160098},   the  authors   propose  a
+heuristic  to compute  the  disjoint  set covers  (DSC).  In order  to
+compute the maximum number of  covers, they first transform DSC into a
+maximum-flow problem, which  is then formulated  as a  mixed integer
+programming  problem (MIP).  Based on  the solution  of the  MIP, they
+design a heuristic to compute  the final number of covers. The results
+show  a slight  performance  improvement  in terms  of  the number  of
+produced  DSC in comparison  to~\cite{Slijepcevic01powerefficient}, but
+it incurs  higher execution  time due to  the complexity of  the mixed
+integer      programming     solving.       %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  this  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.
+Zorbas  et  al.  \cite{Zorbas2007}  present  B\{GOP\},  a  centralized
+coverage   algorithm  introducing   sensor   candidate  categorization
+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.
 %More recently Manju and Pujari\cite{Manju2011}
 
-In the case of non-disjoint algorithms \cite{Manju2011}, sensors may participate in more than one cover set. 
-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
-and a greedy approach to extend
-the sensor network lifetime by organizing the sensors into a
-maximal number of non-disjoint cover sets. Simulation results show that by allowing sensors to 
-participate in multiple sets, the network lifetime
-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
-optimal solution based on the Garg-K\"{o}nemann algorithm~\cite{garg98} is also proposed.
-
-{\bf Our contribution}
-%{decoupage de la region en sous region, selection de noeud leader, formulation %et resolution du probleme de couverture, planification pÃ©riodique
-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.
+In the  case of non-disjoint algorithms  \cite{Manju2011}, sensors may
+participate  in more  than  one cover  set.   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 induces 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 and a greedy  approach to
+extend the  sensor network lifetime  by organizing the sensors  into a
+maximal  number of  non-disjoint cover  sets. Simulation  results show
+that by allowing sensors to  participate in multiple sets, the network
+lifetime         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  solution  based      on      the     Garg-K\"{o}nemann
+algorithm~\cite{garg98}, provably near
+the optimal solution,    is also proposed.
+
+\subsection{Our contribution}
+%{\bf Our contribution}
+
+There are  three main questions, which should be addressed  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.
 \begin{itemize}
-\item {\bf How must be planned the 
-phases for information exchange, decision and sensing over time?}
-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.
-\item {\bf What are the rules to decide which node has to turn on or off?}
-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.
-\item {\bf Which node should make such decision ?}
-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.
+\item {\bf How must the  phases for information exchange, decision and
+  sensing be planned over time?}   Our algorithm divides the time line
+  into a number  of rounds. Each round contains  4 phases: Information
+  Exchange, Leader Election, Decision, and Sensing.
+
+\item {\bf What are the rules to decide which node has to be turned on
+  or off?}  Our algorithm tends to limit the overcoverage of points of
+  interest  to avoid  turning on  too many 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.
+
+\item {\bf Which  node should make such a  decision?}  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} conclude
+  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 centralized approach  compared to a distributed one. Our
+  work does not  consider only one leader to  compute and to broadcast
+  the scheduling decision  to all the sensors.  When  the network size
+  increases,  the network  is  divided into  many  subregions and  the
+  decision is made by a leader in each subregion.
 \end{itemize}
 
+\section{Activity scheduling}
+\label{pd}
 
+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 provision.
+The  location  information is  available  to  the  sensor node  either
+through hardware  such as embedded  GPS or through  location discovery
+algorithms.   The   area  of  interest   can  be  divided   using  the
+divide-and-conquer strategy  into smaller areas  called subregions and
+then  our coverage  protocol  will be  implemented  in each  subregion
+simultaneously.   Our protocol  works in  rounds fashion  as  shown in
+figure~\ref{fig1}.
 
- \section{\uppercase{Distributed coverage model}}
-\label{pd}
-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.
-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}. 
 %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. \\ 
 
 \begin{figure}[ht!]
 \centering
-\includegraphics [width=70mm]{FirstModel.eps}
-\caption{Multi-Round Coverage Protocol}
-\label{fig:4}
+\includegraphics[width=85mm]{FirstModel.eps} % 70mm
+\caption{Multi-round coverage protocol}
+\label{fig1}
 \end{figure} 
 
-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.
-
-\subsection{\textbf INFO Exchange Phase}
-
-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.
-
-% the remaining energy corresponds to the time that a sensor can live in the active mode.
-
+Each round  is divided  into 4 phases  : Information  (INFO) Exchange,
+Leader  Election, Decision,  and  Sensing.  For  each  round there  is
+exactly one set cover responsible  for the sensing task.  This protocol is
+more reliable  against an unexpected node failure  because it works
+in rounds.   On the  one hand,  if a node  failure is  detected before
+making the decision, the node will not participate to this phase, and,
+on the other hand, if the  node failure occurs after the decision, the
+sensing task of the network  will be temporarily affected: only during
+the period of sensing until a  new round starts, 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  their
+information (including their residual energy) at the beginning of each
+round.  However,   the  pre-sensing  phases   (INFO  Exchange,  Leader
+Election,  Decision) are energy  consuming for  some nodes,  even when
+they do not  join the network to monitor the  area. Below, we describe
+each phase in more details.
+
+\subsection{Information exchange phase}
+
+Each sensor node $j$ sends  its position, remaining energy $RE_j$, and
+the number of local neighbours  $NBR_j$ to all wireless sensor nodes in
+its subregion by using an INFO  packet and then listens 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,  the
+remaining energy corresponds to the time that a sensor can live in the
+active mode.
 
 %\subsection{\textbf Working Phase:}
 
 %The working phase works in rounding fashion. Each round include 3 steps described as follow :
 
-\subsection{\textbf Leader Election Phase}
-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.
-% 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.
-The WSNL will be chosen based on the number of local neighbours $NBR_j$  of sensor node $s_j$ and it's remaining energy $RE_j$.
-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. 
-
-
-\subsection{\textbf Decision Phase}
-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. 
+\subsection{Leader election phase}
+This  step includes choosing  the Wireless  Sensor Node  Leader (WSNL),
+which  will  be  responsible  for executing  the coverage  algorithm.  Each
+subregion  in  the   area  of  interest  will  select   its  own  WSNL
+independently  for each  round.  All the  sensor  nodes cooperate  to
+select WSNL.  The nodes in the  same subregion will  select the leader
+based on  the received  information from all  other nodes in  the same
+subregion.  The selection criteria  in order  of priority  are: larger
+number  of neighbours,  larger remaining  energy, and  then in  case of
+equality, larger index.
+
+\subsection{Decision phase}
+The  WSNL will  solve an  integer  program (see  section~\ref{cp})  to
+select which sensors will be  activated in the following sensing phase
+to cover  the subregion.  WSNL will send  Active-Sleep packet  to each
+sensor in the subregion based on the algorithm's results.
 %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.
 %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. 
-\\
-
-\subsection{\textbf Sensing Phase}
- 	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.  
-
 
+\subsection{Sensing phase}
+Active  sensors  in the  round  will  execute  their sensing  task  to
+preserve maximal  coverage in the  region of interest. We  will assume
+that the cost  of keeping a node awake (or asleep)  for sensing task is
+the same  for all wireless sensor  nodes in the  network.  Each sensor
+will receive  an Active-Sleep  packet from WSNL  informing it  to stay
+awake or to go to sleep  for a time  equal to  the period of  sensing until
+starting a new round.
 
 %\subsection{Sensing coverage model}
 %\label{pd}
 
 %\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.
 %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 $R_s$.
-\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.
+\indent 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    the size of the   sensing  range.   In  fact,   Zhang  and
+Zhou~\cite{Zhang05} proved that if  the transmission range fulfills the
+previous  hypothesis, a  complete coverage  of a  convex  area implies
+connectivity among the working nodes in the active mode.
 %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}:
 
 %\begin{figure}[h!]
@@ -261,11 +474,13 @@ The WSNL will execute the GLPK algorithm to select which sensors will be activat
 %\end{figure}
 
 %By using the Unit Circle in figure~\ref{fig:cluster1}, 
-%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.
-% 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.
-
-\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.
+%We choose to representEach wireless sensor node will be represented into a selected number of primary points by which we can know if the sensor node is covered or not.
+% Figure ~\ref{fig:cluster2} shows the selected primary points that represents the area of the sensor node and according to the sensing range of the wireless sensor node.
 
+\indent Instead of working with the coverage area, we consider for each
+sensor a set of points called  primary points. We also assume that the
+sensing disk defined  by a sensor is covered if  all the primary points of
+this sensor are covered.
 %\begin{figure}[h!]
 %\centering
 %\begin{tabular}{cc}
@@ -275,28 +490,17 @@ The WSNL will execute the GLPK algorithm to select which sensors will be activat
 %\caption{Wireless Sensor Node Area Coverage Model.}
 %\label{fig:cluster2}
 %\end{figure}
-
-
-
-\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.
-
- \begin{figure}[h!]
-%\centering
-% \begin{multicols}{6}
-\centering
-%\includegraphics[scale=0.10]{fig21.pdf}\\~ ~ ~(a)
-%\includegraphics[scale=0.10]{fig22.pdf}\\~ ~ ~(b)
-\includegraphics[scale=0.2]{principles13.eps}
-%\includegraphics[scale=0.10]{fig25.pdf}\\~ ~ ~(d)
-%\includegraphics[scale=0.10]{fig26.pdf}\\~ ~ ~(e)
-%\includegraphics[scale=0.10]{fig27.pdf}\\~ ~ ~(f)
-%\end{multicols} 
-\caption{Wireless Sensor node represented by 13 principle points }
-\label{fig3}
-\end{figure}
-
-\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:\\
-$p_x,p_y$ = point center of wireless sensor node. \\  
+By  knowing the  position (point  center: ($p_x,p_y$))  of  a wireless
+sensor node  and its $R_s$,  we calculate the primary  points directly
+based on the proposed model. We  use these primary points (that can be
+increased or decreased if necessary)  as references to ensure that the
+monitored  region  of interest  is  covered  by  the selected  set  of
+sensors, instead of using all the points in the area.
+
+\indent  We can  calculate  the positions  of  the selected  primary
+points in  the circle disk of  the sensing range of  a wireless sensor
+node (see figure~\ref{fig2}) as follows:\\
+$(p_x,p_y)$ = point center of wireless sensor node\\  
 $X_1=(p_x,p_y)$ \\ 
 $X_2=( p_x + R_s * (1), p_y + R_s * (0) )$\\           
 $X_3=( p_x + R_s * (-1), p_y + R_s * (0)) $\\
@@ -309,89 +513,122 @@ $X_9=( p_x + R_s * (\frac{\sqrt{2}}{2}), p_y + R_s * (\frac{-\sqrt{2}}{2})) $\\
 $X_{10}=( p_x + R_s * (\frac{-\sqrt{2}}{2}), p_y + R_s * (\frac{\sqrt{2}}{2})) $\\
 $X_{11}=( p_x + R_s * (\frac{\sqrt{2}}{2}), p_y + R_s * (\frac{\sqrt{2}}{2})) $\\
 $X_{12}=( p_x + R_s * (0), p_y + R_s * (\frac{\sqrt{2}}{2})) $\\
-$X_{13}=( p_x + R_s * (0), p_y + R_s * (\frac{-\sqrt{2}}{2})) $\\
-
-
+$X_{13}=( p_x + R_s * (0), p_y + R_s * (\frac{-\sqrt{2}}{2})) $.
 
+ \begin{figure}[h!]
+%\centering
+% \begin{multicols}{6}
+\centering
+%\includegraphics[scale=0.10]{fig21.pdf}\\~ ~ ~(a)
+%\includegraphics[scale=0.10]{fig22.pdf}\\~ ~ ~(b)
+\includegraphics[scale=0.25]{principles13.eps}
+%\includegraphics[scale=0.10]{fig25.pdf}\\~ ~ ~(d)
+%\includegraphics[scale=0.10]{fig26.pdf}\\~ ~ ~(e)
+%\includegraphics[scale=0.10]{fig27.pdf}\\~ ~ ~(f)
+%\end{multicols} 
+\caption{Wireless sensor node represented by 13 primary points}
+\label{fig2}
+\end{figure}
 
-\section{\uppercase{Coverage problem formulation}}
+\section{Coverage problem formulation}
 \label{cp}
 %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}:\\
 
 
 %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}:\\
 
-
-\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.  \\
-\noindent For a principle point $p$, let $\alpha_{jp}$ denote the indicator function of whether the point $p$ is covered, that is, \\
+\indent   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  proposed an
+integer program, which forces undercoverage and overcoverage of targets
+to  become  minimal at  the  same  time.   They use  binary  variables
+$x_{jl}$ to indicate  if sensor $j$ belongs to cover  set $l$.  In our
+model,  we  consider  binary  variables $X_{j}$,  which  determine  the
+activation of  sensor $j$ in the  sensing phase of the  round. We also
+consider  primary points  as targets.   The set  of primary  points is
+denoted by $P$ and the set of sensors by $J$.
+
+\noindent  For  a primary  point  $p$,  let  $\alpha_{jp}$ denote  the
+indicator function of whether the point $p$ is covered, that is:
 \begin{equation}
 \alpha_{jp} = \left \{ 
 \begin{array}{l l}
-  1 & \mbox{if the principal point $p$ is covered} \\
- & \mbox{by active sensor node $j$} \\
-  0 & \mbox{Otherwise}\\
+  1 & \mbox{if the primary point $p$ is covered} \\
+ & \mbox{by sensor node $j$}, \\
+  0 & \mbox{otherwise.}\\
 \end{array} \right.
 %\label{eq12} 
 \end{equation}
-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 :
+The number of active sensors that cover the primary point $p$ is equal
+to $\sum_{j \in J} \alpha_{jp} * X_{j}$ where:
 \begin{equation}
-X_{j,t} = \left \{ 
+X_{j} = \left \{ 
 \begin{array}{l l}
-  1& \mbox{if sensor $s_j$  is active during round } t\\
-  0 &  \mbox{Otherwise}\\
+  1& \mbox{if sensor $j$  is active,} \\
+  0 &  \mbox{otherwise.}\\
 \end{array} \right.
 %\label{eq11} 
 \end{equation}
-We define the Overcoverage variable $\Theta_{p,t}$ .\\
-
+We define the Overcoverage variable $\Theta_{p}$ as:
 \begin{equation}
- \Theta_{p,t} = \left \{ 
+ \Theta_{p} = \left \{ 
 \begin{array}{l l}
-  0            & \mbox{if point p is not }\\
-&\mbox{covered during round } t\\
-  \left( \sum_{j \in J} \alpha_{jp} * X_{j,t} \right)- 1 & \mbox{Otherwise}\\
+  0 & \mbox{if the primary point}\\
+    & \mbox{$p$ is not covered,}\\
+  \left( \sum_{j \in J} \alpha_{jp} * X_{j} \right)- 1 & \mbox{otherwise.}\\
 \end{array} \right.
 \label{eq13} 
 \end{equation}
-
-
-\noindent$\Theta_{p}$ represents the number of active sensor nodes minus one that cover the principle point $p$.\\
-The Undercoverage variable $U_{p,t}$ of the principle point $p$ is defined as follow :\\
-
+\noindent More precisely, $\Theta_{p}$ represents the number of active
+sensor  nodes  minus  one  that  cover the  primary  point  $p$.\\
+The Undercoverage variable $U_{p}$ of the primary point $p$ is defined
+by:
 \begin{equation}
-U_{p,t} = \left \{ 
+U_{p} = \left \{ 
 \begin{array}{l l}
-  1 &\mbox{if point } $p$ \mbox{ is not covered during round } $t$\\
-  0 & \mbox{Otherwise}\\
+  1 &\mbox{if the primary point $p$ is not covered,} \\
+  0 & \mbox{otherwise.}\\
 \end{array} \right.
 \label{eq14} 
 \end{equation}
 
-\noindent Our coverage optimization problem can be formulated as follow.\\
+\noindent Our coverage optimization problem can then be formulated as follows\\
 \begin{equation} \label{eq:ip2r}
 \left \{
 \begin{array}{ll}
-\min \sum_{p \in P} (w_{\theta,t} \Theta_{p,t} + w_{u,t} U_{p,t})&\\
+\min \sum_{p \in P} (w_{\theta} \Theta_{p} + w_{U} U_{p})&\\
 \textrm{subject to :}&\\
-\sum_{j \in J}  \alpha_{jp} X_{j,t} - \Theta_{p,t}+ U_{p,t} =1, &\forall p \in P, \forall t \in T\\
+\sum_{j \in J}  \alpha_{jp} X_{j} - \Theta_{p}+ U_{p} =1, &\forall p \in P\\
 %\label{c1} 
-\sum_{t \in T} X_{j,t} \leq \frac{RE_j}{e_t} &\forall j \in J \\
+%\sum_{t \in T} X_{j,t} \leq \frac{RE_j}{e_t} &\forall j \in J \\
 %\label{c2}
-\Theta_{p,t}\in \mathbb{N} , &\forall p \in P, \forall t \in T\\
-U_{p,t} \in \{0,1\}, &\forall p \in P, \forall t \in T \\
-X_{j,t} \in \{0,1\}, &\forall j \in J, \forall t \in T
+\Theta_{p}\in \mathbb{N} , &\forall p \in P\\
+U_{p} \in \{0,1\}, &\forall p \in P \\
+X_{j} \in \{0,1\}, &\forall j \in J
 \end{array}
 \right.
 \end{equation}
 \begin{itemize}
-\item $X_{j,t}$ : indicating whether or not sensor $j$ is active in round $t$(1 if yes and 0 if not)
-\item $\Theta_{p,t}$ : {\it overcoverage}, the number of sensors minus one that are covering point $p$ in round $t$
-\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 $t$
+\item $X_{j}$  : indicates whether or  not the sensor  $j$ is actively
+  sensing in the round (1 if yes and 0 if not);
+\item $\Theta_{p}$  : {\it overcoverage}, the number  of sensors minus
+  one that are covering the primary point $p$;
+\item $U_{p}$  : {\it undercoverage},  indicates whether or  not the primary point
+  $p$ is being covered (1 if not covered and 0 if covered).
 \end{itemize}
-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.
-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. 
+
+The first group  of constraints indicates that some  primary point $p$
+should be covered by at least one  sensor and, if it is not always the
+case,  overcoverage  and  undercoverage  variables  help  balancing  the
+restriction  equations by taking  positive values.  There are  two main         
+objectives.  First, we limit the overcoverage of primary points in order to
+activate a minimum number of sensors.  Second we prevent the absence of monitoring on
+ some parts of the subregion by  minimizing the undercoverage.   The
+weights  $w_\theta$  and  $w_U$  must  be properly  chosen  so  as  to
+guarantee that  the maximum number  of points are covered  during each
+round.
  
-%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
+%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 primary 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 primary point $P_p$ in $SSET^k$ , where each $P_p$ should be covered by
 %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.
 %\subsection{Notations and assumptions}
 
@@ -415,245 +652,329 @@ There are two main objectives. We limit overcoverage of principle points in orde
 %\item $|.|$ : cardinality of the set
 
 %\end{itemize}
-\section{\uppercase{Simulation Results}}
-\label{exp}
-In this section, we conducted a series of simulations to evaluate the efficiency of our approach
-based on the discrete event simulator OMNeT++ (http://www.omnetpp.org/).we conduct simulations for six
-different densities varying from 50 to 300 nodes. Experimental results were obtained from randomly generated
-networks in which nodes are deployed over a $ 50\times25(m2) $sensing field. For each network deployment, we
-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 $W_{\Psi} = P^2$.
-We evaluate the efficiency of our approach using some performance metrics such as:coverage ratio, number of
-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.
 
-\subsection{The impact of the Number of Rounds on Coverage Ratio:} 
-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.
+\section{Simulation results}
+\label{exp}
 
- \begin{figure}[h!]
-%\centering
-% \begin{multicols}{6}
+In this section, we conducted  a series of simulations to evaluate the
+efficiency  and the relevance of  our approach,  using the  discrete event
+simulator  OMNeT++  \cite{varga}. We  performed  simulations for  five
+different densities varying from 50 to 250~nodes. Experimental results
+were  obtained from  randomly generated  networks in  which  nodes are
+deployed over a  $(50 \times 25)~m^2 $ sensing  field. 
+More precisely, the deployment is controlled at a coarse scale in
+  order to ensure that the  deployed nodes can fully cover the sensing
+  field with the given sensing range.
+10~simulation  runs  are performed  with
+different  network  topologies for  each  node  density.  The  results
+presented hereafter  are the  average of these  10 runs.  A simulation
+ends  when  all the  nodes  are dead  or  the  sensor network  becomes
+disconnected (some nodes may not be  able to send, to a base station, an
+event they sense).
+
+Our proposed coverage protocol uses the radio energy dissipation model
+defined by~\cite{HeinzelmanCB02} as  energy consumption model for each
+wireless  sensor node  when  transmitting or  receiving packets.   The
+energy of  each node in a  network is initialized  randomly within the
+range 24-60~joules, and each sensor node will consume 0.2 watts during
+the sensing period, which will last 60 seconds. Thus, an
+active  node will  consume  12~joules during the sensing  phase, while  a
+sleeping  node will  use  0.002  joules.  Each  sensor  node will  not
+participate in the next round if its remaining energy is less than 12
+joules.  In  all  experiments,  the  parameters  are  set  as  follows:
+$R_s=5~m$, $w_{\Theta}=1$, and $w_{U}=|P^2|$.
+
+We  evaluate the  efficiency of  our approach by using  some performance
+metrics such as: coverage ratio,  number of active nodes ratio, energy
+saving  ratio, energy consumption,  network lifetime,  execution time,
+and number of stopped simulation runs.  Our approach called strategy~2
+(with two leaders)  works with two subregions, each  one having a size
+of $(25 \times 25)~m^2$.  Our strategy will be compared with two other
+approaches. The first one,  called strategy~1 (with one leader), works
+as strategy~2, but considers only one region of $(50 \times 25)$ $m^2$
+with only  one leader.  The  other approach, called  Simple Heuristic,
+consists in uniformly dividing   the region into squares  of $(5 \times
+5)~m^2$.   During the  decision phase,  in  each square,  a sensor  is
+randomly  chosen, it  will remain  turned  on for  the coming  sensing
+phase.
+
+\subsection{The impact of the number of rounds on the coverage ratio} 
+
+In this experiment, the coverage ratio measures how much the area of a
+sensor field is  covered. In our case, the  coverage ratio is regarded
+as the number  of primary points covered among the  set of all primary
+points  within the field.  Figure~\ref{fig3} shows  the impact  of the
+number of rounds on the  average coverage ratio for 150 deployed nodes
+for the  three approaches.  It can be  seen that the  three approaches
+give  similar  coverage  ratios  during  the first  rounds.  From  the
+9th~round the  coverage ratio  decreases continuously with  the simple
+heuristic, while the two other strategies provide superior coverage to
+$90\%$ for five more rounds.  Coverage ratio decreases when the number
+of rounds increases  due to dead nodes. Although  some nodes are dead,
+thanks to  strategy~1 or~2,  other nodes are  preserved to  ensure the
+coverage. Moreover, when  we have a dense sensor  network, it leads to
+maintain the full coverage for a larger number of rounds. Strategy~2 is
+slightly more efficient than strategy 1, because strategy~2 subdivides
+the region into 2~subregions and  if one of the two subregions becomes
+disconnected,  the coverage   may  be  still  ensured   in  the  remaining
+subregion.
+
+\parskip 0pt 
+\begin{figure}[h!]
 \centering
-%\includegraphics[scale=0.10]{fig21.pdf}\\~ ~ ~(a)
-%\includegraphics[scale=0.10]{fig22.pdf}\\~ ~ ~(b)
-\includegraphics[scale=0.5]{CR2R2L_1.eps}\\~ ~ ~(a)
-\includegraphics[scale=0.5]{CR2R2L_2.eps}\\~ ~ ~(b)
-%\includegraphics[scale=0.10]{fig26.pdf}\\~ ~ ~(e)
-%\includegraphics[scale=0.10]{fig27.pdf}\\~ ~ ~(f)
-%\end{multicols} 
-\caption{The impact of the Number of Rounds on Coverage Ratio.(a):subregion 1. (b): subregion 2 }
+\includegraphics[scale=0.5]{TheCoverageRatio150g.eps} %\\~ ~ ~(a)
+\caption{The impact of the number of rounds on the coverage ratio for 150 deployed nodes}
 \label{fig3}
 \end{figure} 
 
-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.
-
-\subsection{The impact of the Number of Rounds on Energy Saving Ratio:} 
-
-\subsection{The impact of the Number of Rounds on Active Sensor Ratio:} 
+\subsection{The impact of the number of rounds on the active sensors ratio} 
+
+It is important to have as few active nodes as possible in each round,
+in  order to  minimize  the communication  overhead  and maximize  the
+network lifetime.  This point is  assessed through the  Active Sensors
+Ratio (ASR), which is defined as follows:
+\begin{equation*}
+\scriptsize
+\mbox{ASR}(\%) = \frac{\mbox{Number of active sensors 
+during the current sensing phase}}{\mbox{Total number of sensors in the network
+for the region}} \times 100.
+\end{equation*}
+Figure~\ref{fig4} shows  the average active nodes ratio versus rounds
+for 150 deployed nodes.
+
+\begin{figure}[h!]
+\centering
+\includegraphics[scale=0.5]{TheActiveSensorRatio150g.eps} %\\~ ~ ~(a)
+\caption{The impact of the number of rounds on the active sensors ratio for 150 deployed nodes }
+\label{fig4}
+\end{figure} 
 
-\subsection{The impact of Number of Sensors on Number of Rounds:} 
+The  results presented  in figure~\ref{fig4}  show the  superiority of
+both proposed  strategies, the strategy  with two leaders and  the one
+with a  single leader,  in comparison with  the simple  heuristic. The
+strategy with one leader uses less active nodes than the strategy with
+two leaders until the last  rounds, because it uses central control on
+the whole sensing field.  The  advantage of the strategy~2 approach is
+that even if a network is disconnected in one subregion, the other one
+usually  continues  the optimization  process,  and  this extends  the
+lifetime of the network.
+
+\subsection{The impact of the number of rounds on the energy saving ratio} 
+
+In this experiment, we consider a performance metric linked to energy.
+This metric, called Energy Saving Ratio (ESR), is defined by:
+\begin{equation*}
+\scriptsize
+\mbox{ESR}(\%) = \frac{\mbox{Number of alive sensors during this round}}
+{\mbox{Total number of sensors in the network for the region}} \times 100.
+\end{equation*}
+The  longer the ratio  is,  the more  redundant sensor  nodes are
+switched off, and consequently  the longer the  network may  live.
+Figure~\ref{fig5} shows the average  Energy Saving Ratio versus rounds
+for all three approaches and for 150 deployed nodes.
+
+\begin{figure}[h!]
+%\centering
+% \begin{multicols}{6}
+\centering
+\includegraphics[scale=0.5]{TheEnergySavingRatio150g.eps} %\\~ ~ ~(a)
+\caption{The impact of the number of rounds on the energy saving ratio for 150 deployed nodes}
+\label{fig5}
+\end{figure} 
 
-\subsection{The impact of Number of Sensors on Network Lifetime:} 
+The simulation  results show that our strategies  allow to efficiently
+save energy by  turning off some sensors during  the sensing phase. As
+expected, the strategy with one leader is usually slightly better than
+the second  strategy, because the  global optimization permits  to turn
+off more  sensors. Indeed,  when there are  two subregions  more nodes
+remain awake  near the border shared  by them. Note that  again as the
+number of  rounds increases  the two leaders'  strategy becomes  the most
+performing one, since it takes longer  to have the two subregion networks
+simultaneously disconnected.
+
+\subsection{The percentage of stopped simulation runs}
+
+We  will now  study  the percentage  of  simulations, which  stopped due  to
+network  disconnections per round  for each  of the  three approaches.
+Figure~\ref{fig6} illustrates the percentage of stopped simulation
+runs per  round for 150 deployed  nodes.  It can be  observed that the
+simple heuristic is  the approach, which  stops first because  the nodes
+are   randomly chosen.   Among  the  two   proposed  strategies,  the
+centralized  one  first  exhibits  network  disconnections.   Thus,  as
+explained previously, in case  of the strategy with several subregions
+the  optimization effectively  continues as  long  as a  network in  a
+subregion   is  still   connected.   This   longer   partial  coverage
+optimization participates in extending the network lifetime.
+
+\begin{figure}[h!]
+\centering
+\includegraphics[scale=0.5]{TheNumberofStoppedSimulationRuns150g.eps} 
+\caption{The percentage of stopped simulation runs compared to the number of rounds for 150 deployed nodes }
+\label{fig6}
+\end{figure} 
 
-\subsection{The impact of Number of Sensors on Execution Time:}
+\subsection{The energy consumption}
+
+In this experiment, we study the effect of the multi-hop communication
+protocol  on the  performance of  the  strategy with  two leaders  and
+compare  it  with  the  other  two  approaches.   The  average  energy
+consumption  resulting  from  wireless  communications  is  calculated
+by taking into account the  energy spent by  all the nodes when  transmitting and
+receiving  packets during  the network  lifetime. This  average value,
+which  is obtained  for 10~simulation  runs,  is then  divided by  the
+average number of rounds to define a metric allowing a fair comparison
+between networks having different densities.
+
+Figure~\ref{fig7} illustrates the energy consumption for the different
+network  sizes and  the three  approaches. The  results show  that the
+strategy  with  two  leaders  is  the  most  competitive  from  the energy
+consumption point  of view.  A  centralized method, like  the strategy
+with  one  leader, has  a  high energy  consumption  due  to  many
+communications.   In fact,  a distributed  method greatly  reduces the
+number  of communications thanks  to the  partitioning of  the initial
+network in several independent subnetworks. Let us notice that even if
+a  centralized  method  consumes  far  more  energy  than  the  simple
+heuristic, since the energy cost of communications during a round is a
+small  part   of  the   energy  spent  in   the  sensing   phase,  the
+communications have a small impact on the network lifetime.
+
+\begin{figure}[h!]
+\centering
+\includegraphics[scale=0.5]{TheEnergyConsumptiong.eps} 
+\caption{The energy consumption}
+\label{fig7}
+\end{figure} 
 
-\subsection{Performance Comparison:}
-\label{Simulation Results}
+\subsection{The impact of the number of sensors on execution time}
+
+A  sensor  node has  limited  energy  resources  and computing  power,
+therefore it is important that the proposed algorithm has the shortest
+possible execution  time. The energy of  a sensor node  must be mainly
+used   for  the  sensing   phase,  not   for  the   pre-sensing  ones.   
+Table~\ref{table1} gives the average  execution times  in seconds
+on a laptop of the decision phase (solving of the optimization problem)
+during one  round.  They  are given for  the different  approaches and
+various numbers of sensors.  The lack of any optimization explains why
+the heuristic has very  low execution times.  Conversely, the strategy
+with  one  leader, which  requires  to  solve  an optimization  problem
+considering  all  the  nodes  presents  redhibitory  execution  times.
+Moreover, increasing the network size by 50~nodes   multiplies the time
+by  almost a  factor of  10. The  strategy with  two leaders  has more
+suitable times.  We  think that in distributed fashion  the solving of
+the  optimization problem  in a  subregion  can be  tackled by  sensor
+nodes.   Overall,  to  be  able to  deal  with  very  large  networks,  a
+distributed method is clearly required.
+
+\begin{table}[ht]
+\caption{THE EXECUTION TIME(S) VS THE NUMBER OF SENSORS}
+% title of Table
+\centering
 
+% used for centering table
+\begin{tabular}{|c|c|c|c|}
+% centered columns (4 columns)
+      \hline
+%inserts double horizontal lines
+Sensors number & Strategy~2 & Strategy~1  & Simple heuristic \\ [0.5ex]
+ & (with two leaders) & (with one leader) & \\ [0.5ex]
+%Case & Strategy (with Two Leaders) & Strategy (with One Leader) & Simple Heuristic \\ [0.5ex]
+% inserts table
+%heading
+\hline
+% inserts single horizontal line
+50 & 0.097 & 0.189 & 0.001 \\
+% inserting body of the table
+\hline
+100 & 0.419 & 1.972 & 0.0032 \\
+\hline
+150 & 1.295 & 13.098 & 0.0032 \\
+\hline
+200 & 4.54 & 169.469 & 0.0046 \\
+\hline
+250 & 12.252 & 1581.163 & 0.0056 \\
+% [1ex] adds vertical space
+\hline
+%inserts single line
+\end{tabular}
+\label{table1}
+% is used to refer this table in the text
+\end{table}
+
+\subsection{The network lifetime}
+
+Finally, we  have defined the network  lifetime as the  time until all
+nodes  have  been drained  of  their  energy  or each  sensor  network
+monitoring  an area has become  disconnected.  In  figure~\ref{fig8}, the
+network  lifetime for different  network sizes  and for  both strategy
+with two  leaders and the simple  heuristic is illustrated. 
+  We do  not consider  anymore the  centralized strategy  with one
+  leader, because, as shown above, this strategy results  in execution
+  times that quickly become unsuitable for a sensor network.
+
+\begin{figure}[h!]
+%\centering
+% \begin{multicols}{6}
+\centering
+\includegraphics[scale=0.5]{TheNetworkLifetimeg.eps} %\\~ ~ ~(a)
+\caption{The network lifetime }
+\label{fig8}
+\end{figure} 
 
-\section{\uppercase{Conclusions}}
+As  highlighted by figure~\ref{fig8},  the network  lifetime obviously
+increases when  the size  of the network  increases, with  our approach
+that leads to  the larger lifetime improvement.  By  choosing the  best 
+suited nodes, for each round,  to cover the  region of interest  and by
+letting the other ones sleep in order to be used later in next rounds,
+our strategy efficiently prolonges the network lifetime. Comparison shows that
+the larger  the sensor number  is, the more our  strategies outperform
+the simple heuristic.  Strategy~2, which uses two leaders, is the best
+one because it is robust to network disconnection in one subregion. It
+also  means   that  distributing  the  algorithm  in   each  node  and
+subdividing the sensing field  into many subregions, which are managed
+independently and simultaneously, is the most relevant way to maximize
+the lifetime of a network.
+
+\section{Conclusion and future works}
 \label{sec:conclusion}
-In this paper, we have addressed the problem of lifetime optimization in wireless sensor networks. This is a very 
-natural and important problem, as sensor nodes
-have limited resources in terms of memory, energy and computational power.
-%energy-efficiency is crucial in power-limited wireless sensor network. 
-To cope with this problem, 
-%an efficient centralized energy-aware algorithm is presented and analyzed. Our algorithm seeks to 
-%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 
-
-
-
-
 
+In this paper, we have  addressed the problem of the coverage and the lifetime
+optimization  in wireless  sensor networks.   This is  a key  issue as
+sensor nodes  have limited  resources in terms  of memory,  energy and
+computational power. To  cope with this problem, the  field of sensing
+is   divided   into   smaller   subregions  using   the   concept   of
+divide-and-conquer method,  and then a  multi-rounds coverage protocol
+will optimize  coverage and  lifetime performances in  each subregion.
+The  proposed  protocol  combines  two efficient  techniques:  network
+leader election  and sensor activity scheduling,  where the challenges
+include how to select the  most efficient leader in each subregion and
+the best  representative active nodes that will  optimize the network lifetime
+while  taking   the  responsibility  of   covering  the  corresponding
+subregion.   The network lifetime  in each  subregion is  divided into
+rounds, each round consists  of four phases: (i) Information Exchange,
+(ii) Leader Election, (iii) an optimization-based Decision in order to
+select  the  nodes remaining  active  for  the  last phase,  and  (iv)
+Sensing.  The  simulations show the relevance  of the proposed
+protocol in  terms of lifetime, coverage ratio,  active sensors ratio,
+energy saving,  energy consumption, execution time, and  the number of
+stopped simulation  runs due  to network disconnection.   Indeed, when
+dealing with  large and dense wireless sensor  networks, a distributed
+approach like the one we propose  allows to reduce the difficulty of a
+single global optimization problem  by partitioning it in many smaller
+problems, one per subregion, that can be solved more easily.
+
+In  future work, we plan  to study  and propose  a coverage  protocol, which
+computes  all  active  sensor  schedules  in  one time,  using
+optimization  methods  such  as  swarms optimization  or  evolutionary
+algorithms.  The round  will still  consist of 4  phases, but  the
+  decision phase will compute the schedules for several sensing phases,
+  which aggregated together, define a kind of meta-sensing phase.
+The computation of all cover sets in one time is far more
+difficult, but will reduce the communication overhead.
 % use section* for acknowledgement
-\section*{Acknowledgment}
-
-
-
-
-\begin{thebibliography}{1}
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-  J. Wang, C. Niu, and R. Shen,~\emph{Randomized approach for target coverage scheduling in directional sensor network},~ICESS2007,~pp. 379-390,~2007.
-  
-
-
-\bibitem{dw60}
-  G.~B. Dantzig and P.~ Wolfe,~\emph {Decomposition principle for linear programs},~Operations Research,~pp. 101-111,~ 1960.
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-
-\bibitem{asc02}
- I.~ Akyildiz ,~ W.~ Su ,~ Y.~Sankarasubramniam and E. Cayirci,~ emph{A survey on sensor networks},~
- IEEE Comm. Magazine,~pp. 102-114,~2002.
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-
-\bibitem{pc10}
- T.~V.~ Padmavathy and M. Chitra,~emph{Extending the Network Lifetime of Wireless Sensor Networks Using Residual Energy ExtractionâHybrid Scheduling Algorithm},~Int. J. of Communications, Network and System Sciences ,~ VOL.~3,No. ~1,pp.~98-106,~2010.
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- 
-
+%\section*{Acknowledgment}
 
+\bibliographystyle{IEEEtran}
+\bibliography{bare_conf}
 
 % that's all folks
 \end{document}
