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 \begin{document}
 
 \begin{document}

-
-\title{Energy-Efficient Activity Scheduling in Heterogeneous Energy 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-Comt\'e, Belfort, France \\
-Email:$\lbrace$ali.idness, karine.deschinkel, michel.salomon,raphael.couturier$\rbrace$@femto-st.fr}
-%\email{\{ali.idness, karine.deschinkel, michel.salomon, raphael.couturier\}@univ-fcomte.fr}
-%\and
-%\IEEEauthorblockN{Homer Simpson}
-%\IEEEauthorblockA{FEMTO-ST Institute, UMR CNRS, University of Franche-Comte, Belfort, France}
-%\and
-%\IEEEauthorblockN{James Kirk\\ and Montgomery Scott}
-%\IEEEauthorblockA{FEMTO-ST Institute, UMR CNRS, University of Franche-Comte, Belfort, France}
-}
-
+%
+% paper title
+% can use linebreaks \\ within to get better formatting as desired
+\title{Coverage and Lifetime Optimization \\
+in Heterogeneous Energy Wireless Sensor Networks}
+
+\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}}

 
 \maketitle
 
 
 \maketitle
 

-

 \begin{abstract}
 One of  the fundamental challenges in Wireless  Sensor Networks (WSNs)
 \begin{abstract}
 One of  the fundamental challenges in Wireless  Sensor Networks (WSNs)

-is  coverage   preservation, while  extending   the  network  lifetime
-continuously  and effectively  during  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 scheduling  of sensor node
-activity  is  planned for  each  subregion.   The proposed  scheduling
-considers  activity  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
+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  in four phases:  (i)~Information Exchange, (ii)~Leader

 Election, (iii)~Decision,  and (iv)~Sensing.  The  decision process is
 Election, (iii)~Decision,  and (iv)~Sensing.  The  decision process is

-carried  out  by  a  leader  node which  solves  an  integer  program.
+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}
 
 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}
+Wireless   Sensor   Networks,   Area   Coverage,   Network   lifetime,
+Optimization, Scheduling.
+\end{IEEEkeywords}
+%\keywords{Area Coverage, Network lifetime, Optimization, Distributed Protocol}

  
  

- \IEEEpeerreviewmaketitle
+\IEEEpeerreviewmaketitle

 
 \section{Introduction}
 
 \section{Introduction}

-\noindent Recent years have witnessed significant advances in wireless
+
+%\indent  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{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. Thelimited  energy of sensors represents  the main challenge
+%in the  WSNs design~\cite{Sudip03}, 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.
+
+Recent   years  have  witnessed   significant  advances   in  wireless

 communications and embedded micro-sensing MEMS technologies which have
 communications and embedded micro-sensing MEMS technologies which have

-made  emerge wireless  ensor networks  as  one of  the most  promising
-technologies~\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 is to
-prolong the  network lifetime,  while achieving acceptable  quality of
-service for applications.  Indeed, sensor nodes have limited resources
-in terms of memory, energy and computational power.
-
-%\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 round 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 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}}
+led to the emergence of Wireless  Sensor Networks (WSNs) as one of the
+most promising  technologies \cite{Akyildiz02}. 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 it to sense its environment, to compute, to
+store information  and to  deliver report messages  to a  base station
+\cite{Sudip03}.  One of  the main design issues in  WSNs is to prolong
+the network  lifetime, while  achieving acceptable quality  of service
+for  applications. Indeed,  sensors  nodes have  limited resources  in
+terms of memory, energy and computational power.
+
+Since sensor nodes have limited battery life and since it is impossible to
+replace batteries,  especially in remote and  hostile environments, it
+is desirable that  a WSN should be deployed  with high density because
+spatial redundancy can  then 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. 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}.
+Obviously, the deactivation of nodes  is only relevant if the coverage
+of the monitored  area is not affected. In  this paper, we concentrate
+on  the area coverage  problem \cite{Nayak04},  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}
 \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}
+\indent In this section, we only review some recent works dealing with
+the coverage lifetime maximization  problem, where the objective is to
+optimally  schedule  sensors'  activities  in  order  to  extend  WSNs
+lifetime.
+
+In \cite{chin2007}, the author proposed a novel distributed heuristic,
+called Distributed Energy-efficient  Scheduling for k-coverage (DESK),
+which  ensures  that  the  energy  consumption among  the  sensors  is
+balanced and the lifetime  maximized while the coverage requirement is
+maintained.   This  heuristic works  in  rounds,  requires only  1-hop
+neighbor information,  and each sensor  decides its status  (active or
+sleep)   based   on  the   perimeter   coverage   model  proposed   in
+\cite{Huang:2003:CPW:941350.941367}.    More    recently,   Shibo   et
+al. \cite{Shibo}  expressed the coverage  problem as a  minimum weight
+submodular  set cover  problem  and proposed  a Distributed  Truncated
+Greedy Algorithm (DTGA) to solve it. They take in particular advantage
+from  both temporal and  spatial correlations  between data  sensed by
+different sensors.
+
+The  works  presented  in  \cite{Bang,  Zhixin, Zhang}  focus  on  the
+definition  of  coverage-aware,  distributed  energy-efficient  and
+distributed clustering  methods respectively.  They aim  to extend the
+network  lifetime  while ensuring  the  coverage.   S.   Misra et  al.
+\cite{Misra05} proposed a localized algorithm which conserves energy and
+coverage  by  activating  the  subset  of  sensors  with  the  minimum
+overlapping area. It preserves  the network connectivity thanks to the
+formation of the  network backbone. J.~A.~Torkestani \cite{Torkestani}
+designed a Learning  Automata-based Energy-Efficient Coverage protocol
+(LAEEC)  to construct  a Degree-constrained  Connected  Dominating Set
+(DCDS)  in   WSNs.   He  showed   that  the  correct  choice   of  the
+degree-constraint  of DCDS  balances the  network load  on  the active
+nodes and leads to enhance the coverage and network lifetime.
+ 
+The main  contribution of our approach addresses  three main questions
+to build a scheduling strategy.\\

 %\begin{itemize}
 %\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
-%point in target area can be covered~\cite{Mihaela02,Raymond03}.
-
-%\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}. 
-
-%\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}.
+%\item 
+{\indent \bf  How must the  phases for information  exchange, decision
+  and sensing be  planned over time?}  Our algorithm  divides the timeline into 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?}  A  leader node should
+make such  a decision. 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}
 
 %\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 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).\\
-{\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 nodes have been drained of their energy or the sensor network disconnected 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 neighbour 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 rounds for 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 scheduling 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 neighbours 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 neighbours. 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-neighbour 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. 
-%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.
-
- %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. 
-
-
-%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.
-%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 périodique
-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.
-\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 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 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. 
-
-\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.
-\end{itemize}
-
-
-
- \section{\uppercase{Activity scheduling}}
+\section{Activity scheduling}

 \label{pd}
 \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. \\ 
+
+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}.

 
 \begin{figure}[ht!]
 \centering
 
 \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} 
 
 \end{figure} 
 

-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 sensing task. This protocol is more reliable against the  unexpectedly node failure because it works into rounds, and if the node failure is detected before taking the decision, the node will not participate in decision and if the node failure occurs 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 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.
-All the sensor nodes cooperates to select WSNL. After the phase of information exchange , each sensor have all information about the other nodes in the same subregion, after that each node will execute the leader election procedure to determine who is the leader?(each node will know who is the leader) where all the nodes in the same subregion will select the same leader based on the received information from all other nodes in the same subregion.The leader will be selected as follow: It will select the node that have maximum number of neighbour  as a leader.If there are more than one node have the same maximum number of neighbours , it will take all these nodes and then select the node that have larger remaining energy. If there are more than one node have the same maximum number of neighbours and the same remaining energy then it will select the node  with larger index among them. Each subregion in the area of interest will select its own WSNL independently for each round. 
-
-\subsection{\textbf Decision Phase}
-The WSNL will solve an integer program (see section \ref{cp}) to select which sensors will be activated in the next round 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. 
 %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}
-       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 sleep) for sensing task is the same for all wireless sensor nodes in the network. Each sensor received Active-Sleep packet will go to awake or sleep for a time equal to the period of sensing until starting new round.  
-

 
 

+\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 $R_s$.
 
 %\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 sensing range. In fact, Zhang and Zhou ~\cite{Zhang05} prove that if the transmission range is at least twice of the sensing range, 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}:
+\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 $R_c \geq 2R_s$ ~\cite{Zhang05}. 
+
+

 
 %\begin{figure}[h!]
 %\centering
 
 %\begin{figure}[h!]
 %\centering


@@ -272,8 +326,10 @@ The WSNL will solve an integer program (see section \ref{cp}) to select which se
 %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.
 
 %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.
 

-\noindent Instead of working with area coverage, we consider for each sensor a set of points called primary points. And we assume the sensing disk defined by a sensor is covered  if all primary points of this sensor are covered.
-
+\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}
 %\begin{figure}[h!]
 %\centering
 %\begin{tabular}{cc}


@@ -283,28 +339,17 @@ The WSNL will solve an integer program (see section \ref{cp}) to select which se
 %\caption{Wireless Sensor Node Area Coverage Model.}
 %\label{fig:cluster2}
 %\end{figure}
 %\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 primary points directly based on proposed model. We use these primary 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 primary points }
-\label{fig3}
-\end{figure}
-
-\noindent We can calculate the positions of the selected primary 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)) $\\
 $X_1=(p_x,p_y)$ \\ 
 $X_2=( p_x + R_s * (1), p_y + R_s * (0) )$\\           
 $X_3=( p_x + R_s * (-1), p_y + R_s * (0)) $\\


@@ -317,64 +362,83 @@ $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_{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{Sensor node represented by 13 primary points}
+\label{fig2}
+\end{figure}

 
 

-\section{\uppercase{Coverage problem formulation}}
+\section{Coverage problem formulation}

 \label{cp}
 \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}:\\

 
 

+\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 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 binary variables $x_{s,l}$ to indicate if the sensor $s$ belongs to cover set $l$.  In our model, we consider binary variables $X_{j}$ which determine the activation of sensor $j$ in 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, \\
+\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 primary point $p$ is covered} \\
 \begin{equation}
 \alpha_{jp} = \left \{ 
 \begin{array}{l l}
   1 & \mbox{if the primary point $p$ is covered} \\

- & \mbox{by active sensor node $j$} \\
-  0 & \mbox{Otherwise}\\
+ & \mbox{by sensor node $j$}, \\
+  0 & \mbox{otherwise.}\\

 \end{array} \right.
 %\label{eq12} 
 \end{equation}
 \end{array} \right.
 %\label{eq12} 
 \end{equation}

-The number of sensors that are covering point $p$ is equal to $\sum_{j \in J} \alpha_{jp} * X_{j}$ 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} = \left \{ 
 \begin{array}{l l}
 \begin{equation}
 X_{j} = \left \{ 
 \begin{array}{l l}

-  1& \mbox{if sensor $j$  is active} \\
-  0 &  \mbox{otherwise}\\
+  1& \mbox{if sensor $j$  is active,} \\
+  0 &  \mbox{otherwise.}\\

 \end{array} \right.
 %\label{eq11} 
 \end{equation}
 \end{array} \right.
 %\label{eq11} 
 \end{equation}

-We define the Overcoverage variable $\Theta_{p}$ .\\
-
+We define the Overcoverage variable $\Theta_{p}$ as:

 \begin{equation}
  \Theta_{p} = \left \{ 
 \begin{array}{l l}
 \begin{equation}
  \Theta_{p} = \left \{ 
 \begin{array}{l l}

-  0            & \mbox{if point p is not covered}\\
-  \left( \sum_{j \in J} \alpha_{jp} * X_{j} \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}
 \end{array} \right.
 \label{eq13} 
 \end{equation}

-
-
-\noindent$\Theta_{p}$ 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 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} = \left \{ 
 \begin{array}{l l}
 \begin{equation}
 U_{p} = \left \{ 
 \begin{array}{l l}

-  1 &\mbox{if point } $p$ \mbox{ is not covered} \\
-  0 & \mbox{otherwise}\\
+  1 &\mbox{if the primary point $p$ is not covered,} \\
+  0 & \mbox{otherwise.}\\

 \end{array} \right.
 \label{eq14} 
 \end{equation}
 
 \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}
 \begin{equation} \label{eq:ip2r}
 \left \{
 \begin{array}{ll}


@@ -391,173 +455,249 @@ X_{j} \in \{0,1\}, &\forall j \in J
 \right.
 \end{equation}
 \begin{itemize}
 \right.
 \end{equation}
 \begin{itemize}

-\item $X_{j}$ : indicating whether or not sensor $j$ is active in the round (1 if yes and 0 if not)
-\item $\Theta_{p}$ : {\it overcoverage}, the number of sensors minus one that are covering point $p$
-\item $U_{p}$ : {\it undercoverage}, indicating whether or not point $p$ is being covered (1 if not covered and 0 if covered) 
+\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}
 \end{itemize}

-The first group of constraints indicates that some point $p$ should be covered by at least one sensor 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 primary 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 $w_{\theta}$ and $w_{U}$ 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 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}
+\section{Simulation results}
+\label{exp}

 
 

-%\begin{itemize}
-%\item $m$ : the number of targets
-%\item $n$ : the number of sensors
-%\item $K$ : maximal number of cover sets
-%\item $i$ : index of target ($i=1..m$)
-%\item $j$ : index of sensor ($j=1..n$)
-%\item $k$ : index of cover set ($k=1..K$)
-%\item $T_0$ : initial set of targets
-%\item $S_0$ : initial set of sensors
-%\item $T $ : set of targets which are not covered by at least one cover set
-%\item $S$ : set of available sensors
-%\item $S_0(i)$ : set of sensors which cover the target $i$
-%\item $T_0(j)$ : set of targets covered by sensor $j$
-%\item $C_k$ : cover set of index $k$
-%\item $T(C_k)$ : set of targets covered by the cover set $k$
-%\item $NS(i)$ : set of  available sensors which cover the target $i$
-%\item $NC(i)$ : set of cover sets which do not cover the target $i$
-%\item $|.|$ : cardinality of the set
+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.

 
 

-%\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 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\times25(m^2) $sensing field. For each network deployment, we
-assume that the deployed nodes can fully cover the sensing field with the given sensing range. 10 simulation runs are performed with different network topologies. The results presented hereafter are the average of these 10 runs. Simulation ends when there all the nodes are dead or the sensor network becomes disconnected (some nodes may not be able to sent to a base station an event they sense) . 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 sensing period  of 60 seconds. Each active node will consumes 12 joules during sensing phase and each sleep node will consume 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 $w_{U} = |P^2|$.
-We evaluate the efficiency of our approach using some performance metrics such as : coverage ratio, number of
-active nodes ratio, energy saving ratio, Energy Consumption,network lifetime, execution time and the number of stopped simulation runs.
-Our approach is called Strategy (with Two Leaders) will be compared with two approaches: The first one called Strategy (with One Leader) that use the same our approach method but it implemented in $ 50\times25(m^2) $sensing field with one leader. The second method called Simple Heuristic which consists in dividing uniformly the region into squares $(5 * 5)m^2$ . During the  pre-sensing phase, in each square, a sensor is  chosen randomly,  and will be turned on for the next round.
-In our simulation the sensing field is subdivided into two subregions each one equal to $ 25\times25)m^2 $ of the sensing field.
-
-\subsection{The impact of the Number of Rounds on Coverage Ratio:} 
-In this experiment,the Coverage ratio measures how much 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. Fig. \ref{fig3} shows the impact of the number of rounds on the average coverage ratio for 150 deployed nodes for the three approaches.
-The comparison shows that the Strategy (with One Leader) gives the same or better than the Strategy (with Two Leaders) for rounds from 1 to 14. after that the Strategy (with One Leader) will suffer from the sensor network becomes disconnected more than Strategy (with Two Leaders) leads to give the advantage for Strategy (with Two Leaders) that will give better coverage and more network lifetime for the rest last rounds because the Strategy (with Two Leaders) subdivide the region into 2 subregions and if on of the two subregion becomes disconnected, it not leads to stop the the responsibility of coverage optimization in the other subregion.

 \parskip 0pt 
 \begin{figure}[h!]
 \parskip 0pt 
 \begin{figure}[h!]

-%\centering
-% \begin{multicols}{6}

 \centering
 \centering

-\includegraphics[scale=0.5]{TheCoverageRatio150.pdf} %\\~ ~ ~(a)
-
-\caption{The impact of the Number of Rounds on Coverage Ratio for 150 deployed nodes }
+\includegraphics[scale=0.37]{CR1.eps} %\\~ ~ ~(a)
+\caption{The impact of the number of rounds on the coverage ratio for 150 deployed nodes}

 \label{fig3}
 \end{figure} 
 
 \label{fig3}
 \end{figure} 
 

-\begin{figure}[h!]
-%\centering
-% \begin{multicols}{6}
-\centering
-%\includegraphics[scale=0.10]{fig21.pdf}\\~ ~ ~(a)
-%\includegraphics[scale=0.10]{fig22.pdf}\\~ ~ ~(b)
-\includegraphics[scale=0.5]{TheCoverageRatio250.pdf} %\\~ ~ ~(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 for 250 deployed nodes }
-\label{fig4}
-\end{figure}
-
-Fig. \ref{fig4} represents the average coverage ratio provided by
-Strategy (with Two Leaders),Strategy (with One Leader) and Simple Heuristic for 250 deployed nodes while
-varying the number of rounds. We made the same observation as in Fig. \ref{fig3}, i.e.  Strategy (with One Leader) guarantee a good coverage in the beginning the same or equal to Strategy (with Two Leaders) then when the number of rounds increases, the coverage ratio decreases due to network becomes disconnected. Meanwhile,the Strategy (with Two Leaders) ensures better coverage despite the variation in rounds number.\\
-\parskip 0pt 
-As shown in Fig. \ref{fig3} and Fig. \ref{fig4},the Strategy (with Two Leaders) gives a full average coverage ratio or more than 90\% 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 in each subregion choose other nodes to ensure the coverage. 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 Active Sensor 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. The Active Sensor Ratio(\%) = ((the no. of active sensor during this round) /  (Total number of sensors in the network for the subregion)) * 100. Fig. \ref{fig5} and \ref{fig6} shows the average number of active nodes ratio versus rounds for 150 and 250 deployed nodes respectively.
+\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!]
 
 \begin{figure}[h!]

-%\centering
-% \begin{multicols}{6}
-\centering
-\includegraphics[scale=0.5]{TheActiveSensorRatio150.pdf} %\\~ ~ ~(a)
-\caption{The impact of the Number of Rounds on Active Sensor Ratio for 150 deployed nodes }
-\label{fig5}
-\end{figure} 
-\begin{figure}[h!]
-%\centering
-% \begin{multicols}{6}

 \centering
 \centering

-\includegraphics[scale=0.46]{TheActiveSensorRatio250.pdf} %\\~ ~ ~(a)
-\caption{The impact of the Number of Rounds on Active Sensor Ratio for 250 deployed nodes }
-\label{fig6}
+\includegraphics[scale=0.37]{ASR1.eps} %\\~ ~ ~(a)
+\caption{The impact of the number of rounds on the active sensors ratio for 150 deployed nodes }
+\label{fig4}

 \end{figure} 
 
 \end{figure} 
 

-The results in Fig. \ref{fig5} shows the superior of the Strategy (with Two Leaders) and the Strategy (with One Leader) on Simple Heuristic. The Strategy (with One Leader) uses minimum number of active nodes than the Strategy (with Two Leaders) until the last rounds because it uses central control on all the sensing field , the advantage of the Strategy (with Two Leaders) approach is that even if a network is disconnected in a one subregion the other one usually continues the optimization and this leads to extend the lifetime of the network.
-In Fig. \ref{fig6}, we see the same observation that we saw on fig. \ref{fig5} but in the last rounds the Strategy (with Two Leaders) will be the better because the effect of disconnected network increases and it becomes near or equal to the Strategy (with One Leader) leads to give minimum average of active nodes.
-
-\subsection{The impact of the Number of Rounds on Energy Saving Ratio:} 
-
-In this experiment, the Energy saving ratio ESR(\%) =( (No. of alive sensors in the network during this round) / (total number  of sensors in the network for the subregion)) * 100. Fig. \ref{fig5} and \ref{fig6} shows the average number of active nodes ratio versus rounds for all three approach and for 150 and 250 deployed nodes respectively.
+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{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
 
 \begin{figure}[h!]
 %\centering
 % \begin{multicols}{6}
 \centering

-\includegraphics[scale=0.5]{TheEnergySavingRatio150.pdf} %\\~ ~ ~(a)
-\caption{The impact of the Number of Rounds on Energy Saving Ratio for 150 deployed nodes }
+\includegraphics[scale=0.37]{ESR1.eps} %\\~ ~ ~(a)
+\caption{The impact of the number of rounds on the energy saving ratio for 150 deployed nodes}

 \label{fig5}
 \end{figure} 
 
 \label{fig5}
 \end{figure} 
 

+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!]
 \begin{figure}[h!]

-%\centering
-% \begin{multicols}{6}

 \centering
 \centering

-\includegraphics[scale=0.46]{TheEnergySavingRatio250.pdf} %\\~ ~ ~(a)
-\caption{The impact of the Number of Rounds on Energy Saving Ratio for 150 deployed nodes }
+\includegraphics[scale=0.36]{SR1.eps} 
+\caption{The percentage of stopped simulation runs compared to the number of rounds for 150 deployed nodes }

 \label{fig6}
 \end{figure} 
 
 \label{fig6}
 \end{figure} 
 

-Simulation results explain the efficiency of the Strategy (with Two Leaders) to save the energy during larger number of rounds that seems to be very near the central approach (Strategy (with One Leader))which it be better in general.
-
-\subsection{The Network Lifetime:} 
-we can define the network lifetime as the time until all nodes have been drained of their energy or the sensor network disconnected. In Fig. \ref{fig7}, the network lifetime for different network sizes and for the three approaches is illustrated. 
+\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!]
 
 \begin{figure}[h!]

-%\centering
-% \begin{multicols}{6}

 \centering
 \centering

-\includegraphics[scale=0.46]{TheNetworkLifetime.pdf} %\\~ ~ ~(a)
-\caption{The Network Lifetime }
+\includegraphics[scale=0.37]{EC1.eps} 
+\caption{The energy consumption}

 \label{fig7}
 \end{figure} 
 
 \label{fig7}
 \end{figure} 
 

-As shown in fig. \ref{fig7}, the network lifetime increase when the size of the network increase because the efficient of our approach in choosing the best representative nodes that will cover all the subregion and let the others nodes to sleep to be used later in the next rounds. Comparison shows that our approach is better than the other two methods in improving the network lifetime and this shows us by distributing the algorithm in each node in the network and subdivide the sensing field into many subregions to be managed independently and simultaneously can be more powerful against the sensor network disconnected in some subregions leads to maximize the lifetime of the network obviously.
-
-\subsection{The Energy Consumption:} 
-In this experiment, we study the effect of the multi-hop communication protocol on the performance of our approach and compare it with other two approaches. The average energy consumption calculated based only on the energy consumed by transmitting and receiving packets by all sensor nodes in the network and during the lifetime of the network for 10 simulation runs divided by the average number of rounds and for different sizes of the network.We took the total energy consumption in both subregions for Strategy (with Two Leaders) and Simple heuristic. Fig. \ref{fig8} illustrates the Energy Consumption for different network sizes and for the three approaches. 
-
-\begin{figure}[h!]
-\centering
-\includegraphics[scale=0.46]{TheEnergyConsumption.pdf} 
-\caption{The Energy Consumption  }
-\label{fig8}
-\end{figure} 
-
-The results show that our approach consume less energy during the lifetime of the network by using multi-hop communication protocol in comparison with other two approaches especially Strategy (with One Leader) in spite of our approach make the network live for a longer time.
-
-\subsection{The impact of Number of Sensors on Execution Time:}
-It is important to have time efficient algorithm to be executed in sensor node because the limited resources in the sensor node. We took the total Execution Time of algorithms in both subregions for Strategy (with Two Leaders) and Simple heuristic. Table \ref{table1} exhibits the execution time for three approaches using different number of sensors.
+\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]
 
 \begin{table}[ht]

-\caption{The Execution Time(s) vs The Number of Sensors }
+\caption{EXECUTION TIME(S) VS. NUMBER OF SENSORS}

 % title of Table
 \centering
 
 % used for centering table
 % title of Table
 \centering
 
 % used for centering table

-\begin{tabular}{|c| c| c| c |}
+\begin{tabular}{|c|c|c|c|}

 % centered columns (4 columns)
 % centered columns (4 columns)

-\hline\hline
+      \hline

 %inserts double horizontal lines
 %inserts double horizontal lines

-Sensors Number & Strategy & Strategy  & Simple Heuristic \\ [0.5ex]
- & (with Two Leaders) & (with One Leader)  \\ [0.5ex]
+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
 %Case & Strategy (with Two Leaders) & Strategy (with One Leader) & Simple Heuristic \\ [0.5ex]
 % inserts table
 %heading


@@ -572,8 +712,7 @@ Sensors Number & Strategy & Strategy  & Simple Heuristic \\ [0.5ex]
 \hline
 200 & 4.54 & 169.469 & 0.0046 \\
 \hline
 \hline
 200 & 4.54 & 169.469 & 0.0046 \\
 \hline

-250 & 12.252 & 1581.163 & 0.0056 \\ [1ex]   
-
+250 & 12.252 & 1581.163 & 0.0056 \\

 % [1ex] adds vertical space
 \hline
 %inserts single line
 % [1ex] adds vertical space
 \hline
 %inserts single line


@@ -582,219 +721,69 @@ Sensors Number & Strategy & Strategy  & Simple Heuristic \\ [0.5ex]
 % is used to refer this table in the text
 \end{table}
 
 % is used to refer this table in the text
 \end{table}
 

-Table  \ref{table1} shows the efficiency of our algorithm to produce the solution with good execution time though the Simple Heuristic gives the solution in less time but our approach ensure a better coverage for the region with energy saving and less energy consumption led to extend the lifetime of the network.
-
-\subsection{The Number of Stopped Simulation Runs :}
-In this study, we will shows the number of stopped simulation runs (the disconnected network) per round for 150 and 250 deployed nodes and for three approaches. Fig. \ref{fig9} and \ref{fig10} illustrate the number of stopped simulation runs per round for 150 and 250 deployed nodes respectively.
+\subsection{The network lifetime}

 
 

-\begin{figure}[h!]
-\centering
-\includegraphics[scale=0.50]{TheNumberofStoppedSimulationRuns150.pdf} 
-\caption{The Number of Stopped Simulation Runs against Rounds for 150 deployed nodes }
-\label{fig9}
-\end{figure} 
+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!]
 
 \begin{figure}[h!]

+%\centering
+% \begin{multicols}{6}

 \centering
 \centering

-\includegraphics[scale=0.50]{TheNumberofStoppedSimulationRuns250.pdf} 
-\caption{The Number of Stopped Simulation Runs against Rounds for 250 deployed nodes }
-\label{fig10}
+\includegraphics[scale=0.37]{LT1.eps} %\\~ ~ ~(a)
+\caption{The network lifetime }
+\label{fig8}

 \end{figure} 
 
 \end{figure} 
 

-The results explain the powerful of our approach against the Stopped Simulation Runs in comparison with other two approaches especially in the last rounds from the network lifetime and this will participate in extending the life time of the network.
-
-\label{Simulation Results}
-
-
-\section{\uppercase{Conclusions and Future Works}}
+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 work}

 \label{sec:conclusion}
 \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. To cope with this problem, The field of sensing divided into smaller subregion using the concept of divide-and-conquer method and then multi-rounds coverage protocol will optimize the lifetime in each subregion.The suggested protocol joins 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 lifetime and take the responsibility of covering the subregion. The network lifetime in each subregion is divided into rounds, each round consists of four phases information exchange, Leader Election, Decision with optimization, and sensing.Our simulation results show that the proposed protocol outperforms or very near some other methods in terms of lifetime, coverage ratio, Active sensor Ratio, energy saving , energy consumption,  execution time, and the number of stopped simulation runs. In future, we will study and prepare the one round coverage protocol by which all active sensor schedules will be prepared in one round using intelligent optimization methods such as swarms optimization or Evolutionary algorithms.
-

 
 

+In this paper,  we have addressed the problem of  the coverage and the
+lifetime optimization in WSNs. 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. Results from  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 a  coverage protocol which  computes all  active sensor
+schedules in only one step for many rounds,  using optimization  methods
+such as  swarms optimization or evolutionary algorithms.

 % use section* for acknowledgement
 % use section* for acknowledgement

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-
+%\section*{Acknowledgment}

 
 

+\bibliographystyle{IEEEtran}
+\bibliography{bare_conf}

 
 

-% that's all folks

 \end{document}
 
 
 \end{document}