Update by Ali

[ThesisAli.git] / CHAPITRE_06.tex

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new mode 100644 (file)
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--- a/CHAPITRE_06.tex
+++ b/CHAPITRE_06.tex
@@ -3,57 +3,45 @@
 %%       CHAPTER 06        %%
 %%                          %%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%       CHAPTER 06        %%
 %%                          %%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

-
-\chapter{Perimeter-based Coverage Optimization to Improve Lifetime in Wireless Sensor Networks}
+ \chapter{ Perimeter-based Coverage Optimization to Improve Lifetime in WSNs}

 \label{ch6}
 
 
 \label{ch6}
 
 

-\section{summary}
+\section{Introduction}

 \label{ch6:sec:01}
 
 \label{ch6:sec:01}
 

-The most important problem in a Wireless Sensor Network (WSN) is to optimize the
-use of its limited energy provision, so that it can fulfill its monitoring task
-as long as  possible. Among  known  available approaches  that can  be used  to
-improve  power  management,  lifetime coverage  optimization  provides  activity
-scheduling which ensures sensing coverage while minimizing the energy cost. In
-this paper,  we propose such an approach called Perimeter-based Coverage Optimization
-protocol (PeCO). It is a  hybrid of centralized and distributed methods: the
-region of interest is first subdivided into subregions and our protocol is then
-distributed among sensor nodes in each  subregion.
-The novelty of our approach lies essentially in the formulation of a new
-mathematical optimization  model based on the  perimeter coverage level  to schedule
-sensors' activities.  Extensive simulation experiments have been performed using
-OMNeT++, the  discrete event simulator, to  demonstrate that PeCO  can
-offer longer lifetime coverage for WSNs in comparison with some other protocols.
-
-\section{THE PeCO PROTOCOL DESCRIPTION}
+%The continuous progress in Micro Electro-Mechanical Systems (MEMS) and wireless communication hardware  has given rise to the opportunity to use large networks of tiny sensors, called Wireless Sensor Networks (WSN)~\cite{ref1,ref223}, to fulfill monitoring tasks. The features of a WSN made it suitable for a wide range of application  in areas such as business,  environment, health, industry, military, and so on~\cite{ref4}. These large number of applications have led to different design, management, and operational challenges in WSNs. The challenges become harder with considering into account the main limited capabilities of the sensor nodes such memory, processing, battery life,  bandwidth, and short radio ranges. One important feature that distinguish the WSN from the other types of wireless networks is the provision of the sensing capability for the sensor nodes \cite{ref224}.
+
+%The sensor node consumes some energy both in performing the sensing task and in transmitting the sensed data to the sink. Therefore, it is required to activate as less number as possible of sensor nodes that can monitor the whole area of interest so as to reduce the data volume and extend the network lifetime. The sensing coverage is the most important task of the WSNs since sensing unit of the sensor node is responsible for measuring physical,  chemical, or  biological  phenomena in the sensing field. The main challenge of any sensing coverage problem is to discover the redundant sensor node and turn off those nodes in WSN \cite{ref225}. The redundant sensor node is a node whose sensing area is covered by its active neighbors. In previous works, several approaches are used to find out the redundant node such as Voronoi diagram method, sponsored sector, crossing coverage, and perimeter coverage. 
+
+In this chapter,  we propose an approach called Perimeter-based Coverage Optimization
+protocol (PeCO). 
+%The PeCO protocol merges between two energy efficient mechanisms, which are used the main advantages of the centralized and distributed approaches and avoids the most of their disadvantages. An energy-efficient activity scheduling mechanism based new optimization model is performed by each leader in the subregions. 
+The scheme is similar to the one described in section \ref{ch4:sec:02:03}. But in this approach, the optimization model is based on the perimeter coverage model in order to produce the optimal cover set of active sensors, which are taken the responsibility of sensing during the current period. 
+
+
+The rest of the chapter is organized as follows. The next section is devoted to the PeCO protocol description and section~\ref{ch6:sec:03} focuses on the coverage model formulation which is used  to schedule the activation  of sensor nodes.  Section~\ref{ch6:sec:04} presents simulation results and discusses the comparison with other approaches. Finally, concluding remarks   are  drawn in section~\ref{ch6:sec:05}.
+
+
+
+\section{Description of the PeCO Protocol}

 \label{ch6:sec:02}
 
 \label{ch6:sec:02}
 

-\noindent  In  this  section,  we  describe in  details  our  Lifetime  Coverage
-Optimization protocol.  First we present the  assumptions we made and the models
-we considered (in particular the perimeter coverage one), second we describe the
-background idea of our protocol, and third  we give the outline of the algorithm
-executed by each node.
+%\noindent  In  this  section,  we  describe in  details  our  Lifetime  Coverage Optimization protocol.  
+First we present the  assumptions we made and the models
+we considered (in particular the perimeter coverage one), second we describe the background idea of our protocol, and third  we give the outline of the algorithm executed by each node.

 
 
 
 \subsection{Assumptions and Models}
 \label{ch6:sec:02:01}
 
 
 
 \subsection{Assumptions and Models}
 \label{ch6:sec:02:01}

-PeCO protocol uses the same assumptions and network model that presented in chapter 4, section \ref{ch4:sec:02:01}.
-
-The PeCO protocol  uses the  same perimeter-coverage  model as  Huang and
-Tseng in~\cite{ref133}. It  can be expressed as follows:  a sensor is
-said to be perimeter  covered if all the points on its  perimeter are covered by
-at least  one sensor  other than  itself.  They  proved that  a network  area is
+The PeCO protocol uses the same assumptions and network model than both the DiLCO and the MuDiLCO protocols. All the hypotheses can be found in section \ref{ch4:sec:02:01}.
+The PeCO protocol  uses the  same perimeter-coverage  model as  Huang and Tseng in~\cite{ref133}. It  can be expressed as follows:  a sensor is said to be a perimeter covered if all the points on its  perimeter are covered by at least  one sensor  other than  itself.  They  proved that  a network  area is

 $k$-covered if and only if each sensor in the network is $k$-perimeter-covered (perimeter covered by at least $k$ sensors).
   
 $k$-covered if and only if each sensor in the network is $k$-perimeter-covered (perimeter covered by at least $k$ sensors).
   

-Figure~\ref{pcm2sensors}(a)  shows  the coverage  of  sensor  node~$0$. On  this
-figure, we can  see that sensor~$0$ has  nine neighbors and we  have reported on
-its  perimeter (the  perimeter  of the  disk  covered by  the  sensor) for  each
-neighbor  the  two  points  resulting  from  intersection  of  the  two  sensing
-areas. These points are denoted for  neighbor~$i$ by $iL$ and $iR$, respectively
-for  left and  right from  neighbor  point of  view.  The  resulting couples  of
-intersection points subdivide the perimeter of sensor~$0$ into portions called
+Figure~\ref{pcm2sensors}(a)  shows  the coverage  of  sensor  node~$0$. On  this figure, we can  see that sensor~$0$ has  nine neighbors and we  have reported on
+its  perimeter (the  perimeter  of the  disk  covered by  the  sensor) for  each neighbor  the  two  points  resulting  from  intersection  of  the  two  sensing
+areas. These points are denoted for  neighbor~$i$ by $iL$ and $iR$, respectively for  left and  right from  neighbor  point of  view.  The  resulting couples  of intersection points subdivide the perimeter of sensor~$0$ into portions called

 arcs.
 
 \begin{figure}[ht!]
 arcs.
 
 \begin{figure}[ht!]


@@ -67,33 +55,17 @@ arcs.
   \label{pcm2sensors}
 \end{figure} 
 
   \label{pcm2sensors}
 \end{figure} 
 

-Figure~\ref{pcm2sensors}(b) describes the geometric information used to find the
-locations of the  left and right points of  an arc on the perimeter  of a sensor
-node~$u$ covered by a sensor node~$v$. Node~$v$ is supposed to be located on the
-west  side of  sensor~$u$,  with  the following  respective  coordinates in  the
-sensing area~: $(v_x,v_y)$ and $(u_x,u_y)$. From the previous coordinates we can
-compute the euclidean distance between nodes~$u$ and $v$: $Dist(u,v)=\sqrt{\vert
-  u_x  - v_x  \vert^2 +  \vert u_y-v_y  \vert^2}$, while  the angle~$\alpha$  is
-obtained through  the formula: $$\alpha =  \arccos \left(\dfrac{Dist(u,v)}{2R_s}
+Figure~\ref{pcm2sensors}(b) describes the geometric information used to find the locations of the  left and right points of  an arc on the perimeter  of a sensor node~$u$ covered by a sensor node~$v$. Node~$v$ is supposed to be located on the
+west  side of  sensor~$u$,  with  the following  respective  coordinates in  the sensing area~: $(v_x,v_y)$ and $(u_x,u_y)$. From the previous coordinates we can compute the euclidean distance between nodes~$u$ and $v$: $Dist(u,v)=\sqrt{\vert
+  u_x  - v_x  \vert^2 +  \vert u_y-v_y  \vert^2}$, while  the angle~$\alpha$  is obtained through  the formula: $$\alpha =  \arccos \left(\dfrac{Dist(u,v)}{2R_s}

 \right).$$ The arc on the perimeter of~$u$ defined by the angular interval $[\pi
   - \alpha,\pi + \alpha]$ is said to be perimeter-covered by sensor~$v$.
 
 \right).$$ The arc on the perimeter of~$u$ defined by the angular interval $[\pi
   - \alpha,\pi + \alpha]$ is said to be perimeter-covered by sensor~$v$.
 

-Every couple of intersection points is placed on the angular interval $[0,2\pi]$
-in  a  counterclockwise manner,  leading  to  a  partitioning of  the  interval.
-Figure~\ref{pcm2sensors}(a)  illustrates  the arcs  for  the  nine neighbors  of
-sensor $0$ and  Figure~\ref{expcm} gives the position of  the corresponding arcs
-in  the interval  $[0,2\pi]$. More  precisely, we  can see  that the  points are
-ordered according  to the  measures of  the angles  defined by  their respective
-positions. The intersection points are  then visited one after another, starting
-from the first  intersection point  after  point~zero,  and  the maximum  level  of
-coverage is determined  for each interval defined by two  successive points. The
-maximum  level of  coverage is  equal to  the number  of overlapping  arcs.  For
-example, 
-between~$5L$  and~$6L$ the maximum  level of  coverage is equal  to $3$
-(the value is highlighted in yellow  at the bottom of Figure~\ref{expcm}), which
-means that at most 2~neighbors can cover  the perimeter in addition to node $0$. 
-Table~\ref{my-label} summarizes for each coverage  interval the maximum level of
-coverage and  the sensor  nodes covering the  perimeter.  The  example discussed
+Every couple of intersection points is placed on the angular interval $[0,2\pi]$ in  a  counterclockwise manner,  leading  to  a  partitioning of  the  interval.
+Figure~\ref{pcm2sensors}(a)  illustrates  the arcs  for  the  nine neighbors  of sensor $0$ and  Figure~\ref{expcm} gives the position of the corresponding arcs in  the interval  $[0,2\pi]$. More  precisely, we  can see  that the  points are
+ordered according  to the  measures of  the angles  defined by  their respective positions. The intersection points are  then visited one after another, starting from the first  intersection point  after  point~zero,  and  the maximum  level  of coverage is determined  for each interval defined by two  successive points. The maximum  level of  coverage is  equal to  the number  of overlapping  arcs.  For example, 
+between~$5L$  and~$6L$ the maximum  level of  coverage is equal  to $3$ (the value is highlighted in yellow  at the bottom of Figure~\ref{expcm}), which means that at most 2~neighbors can cover  the perimeter in addition to node $0$. 
+Table~\ref{my-label} summarizes for each coverage  interval the maximum level of coverage and  the sensor  nodes covering the  perimeter.  The  example discussed

 above is thus given by the sixth line of the table.
 
 
 above is thus given by the sixth line of the table.
 
 


@@ -135,13 +107,7 @@ above is thus given by the sixth line of the table.
 \end{table}
 
 
 \end{table}
 
 

-In the PeCO  protocol, the scheduling of the sensor  nodes' activities is formulated  with an
-integer program  based on  coverage intervals. The  formulation of  the coverage
-optimization problem is  detailed in~section~\ref{ch6:sec:03}.  Note that  when a sensor
-node  has a  part of  its sensing  range outside  the WSN  sensing field,  as in
-Figure~\ref{ex4pcm}, the maximum coverage level for  this arc is set to $\infty$
-and  the  corresponding  interval  will  not   be  taken  into  account  by  the
-optimization algorithm.
+In the PeCO  protocol, the scheduling of the sensor  nodes' activities is formulated  as an integer program  based on  coverage intervals. The  formulation of  the coverage optimization problem is  detailed in~section~\ref{ch6:sec:03}.  Note that  when a sensor node  has a  part of  its sensing  range outside  the WSN  sensing field,  as in Figure~\ref{ex4pcm}, the maximum coverage level for  this arc is set to $\infty$ and  the  corresponding  interval  will  not   be  taken  into  account  by  the optimization algorithm.

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


@@ -152,8 +118,9 @@ optimization algorithm.
 \end{figure} 
 
 
 \end{figure} 
 
 

-
-
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% This section deleted %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\iffalse

 
 \subsection{The Main Idea}
 \label{ch6:sec:02:02}
 
 \subsection{The Main Idea}
 \label{ch6:sec:02:02}


@@ -163,33 +130,20 @@ homogeneous subregions  using a divide-and-conquer  algorithm. In a  second step
 our  protocol  will  be  executed  in a distributed way in each subregion
 simultaneously to schedule nodes' activities for one sensing period.
 
 our  protocol  will  be  executed  in a distributed way in each subregion
 simultaneously to schedule nodes' activities for one sensing period.
 

-As  shown in  Figure~\ref{fig2}, node  activity  scheduling is  produced by  our
-protocol in a periodic manner. Each period is divided into 4 stages: Information
-(INFO)  Exchange,  Leader Election,  Decision  (the  result of  an  optimization
-problem),  and  Sensing.   For  each  period there  is  exactly  one  set  cover
-responsible for  the sensing task.  Protocols  based on a periodic  scheme, like
-PeCO, are more  robust against an unexpected  node failure. On the  one hand, if
-a node failure is discovered before  taking the decision, the corresponding sensor
-node will  not be considered  by the optimization  algorithm. On  the other
-hand, if the sensor failure happens after  the decision, the sensing task of the
-network will be temporarily affected: only  during the period of sensing until a
-new period starts, since a new set cover will take charge of the sensing task in
-the next period. 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 period.
-However, the pre-sensing  phases (INFO Exchange, Leader  Election, and Decision)
-are energy consuming, even for nodes that will not join the set cover to monitor
-the area.
+As  shown in  Figure~\ref{fig2}, node  activity  scheduling is  produced by  our protocol in a periodic manner. Each period is divided into 4 stages: Information (INFO)  Exchange,  Leader Election,  Decision  (the  result of  an  optimization problem),  and  Sensing.   For  each  period, there  is  exactly  one  set  cover responsible for  the sensing task.  Protocols  based on a periodic  scheme, like PeCO, are more  robust against an unexpected  node failure. On the  one hand, if a node failure is discovered before  taking the decision, the corresponding sensor
+node will  not be considered  by the optimization  algorithm. On  the other hand, if the sensor failure happens after  the decision, the sensing task of the network will be temporarily affected: only  during the period of sensing until a new period starts, since a new set cover will take charge of the sensing task in the next period. 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 period. However, the pre-sensing  phases (INFO Exchange, Leader  Election, and Decision)
+are energy consuming, even for nodes that will not join the set cover to monitor the area.

 
 \begin{figure}[t!]
 \centering
 
 \begin{figure}[t!]
 \centering

-\includegraphics[width=95.5mm]{Figures/ch6/Model.pdf}  
+\includegraphics[scale=0.80]{Figures/ch6/Model.pdf}  

 \caption{PeCO protocol.}
 \label{fig2}
 \end{figure} 
 
 \caption{PeCO protocol.}
 \label{fig2}
 \end{figure} 
 

-
-
+\fi
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 
 \subsection{PeCO Protocol Algorithm}
 \label{ch6:sec:02:03}
 
 \subsection{PeCO Protocol Algorithm}
 \label{ch6:sec:02:03}


@@ -197,7 +151,7 @@ the area.
 
 \noindent The  pseudocode implementing the protocol on a node is  given below.
 More  precisely,  Algorithm~\ref{alg:PeCO}  gives  a brief  description  of  the
 
 \noindent The  pseudocode implementing the protocol on a node is  given below.
 More  precisely,  Algorithm~\ref{alg:PeCO}  gives  a brief  description  of  the

-protocol applied by a sensor node $s_k$ where $k$ is the node index in the WSN.
+protocol applied by a sensor node $s_j$ where $j$ is the node index in the WSN.

 
 \begin{algorithm}[h!]                
  % \KwIn{all the parameters related to information exchange}
 
 \begin{algorithm}[h!]                
  % \KwIn{all the parameters related to information exchange}


@@ -206,54 +160,54 @@ protocol applied by a sensor node $s_k$ where $k$ is the node index in the WSN.
   %\emph{Initialize the sensor node and determine it's position and subregion} \; 
   
   \If{ $RE_k \geq E_{th}$ }{
   %\emph{Initialize the sensor node and determine it's position and subregion} \; 
   
   \If{ $RE_k \geq E_{th}$ }{

-      \emph{$s_k.status$ = COMMUNICATION}\;
+      \emph{$s_j.status$ = COMMUNICATION}\;

       \emph{Send $INFO()$ packet to other nodes in subregion}\;
       \emph{Wait $INFO()$ packet from other nodes in subregion}\; 
       \emph{Send $INFO()$ packet to other nodes in subregion}\;
       \emph{Wait $INFO()$ packet from other nodes in subregion}\; 

-      \emph{Update K.CurrentSize}\;
+      \emph{Update A.CurrentSize}\;

       \emph{LeaderID = Leader election}\;
       \emph{LeaderID = Leader election}\;

-      \If{$ s_k.ID = LeaderID $}{
-         \emph{$s_k.status$ = COMPUTATION}\;
+      \If{$ s_j.ID = LeaderID $}{
+         \emph{$s_j.status$ = COMPUTATION}\;

          
          

-      \If{$ s_k.ID $ is Not previously selected as a Leader }{
+      \If{$ s_j.ID $ is Not previously selected as a Leader }{

           \emph{ Execute the perimeter coverage model}\;
          % \emph{ Determine the segment points using perimeter coverage model}\;
       }
       
           \emph{ Execute the perimeter coverage model}\;
          % \emph{ Determine the segment points using perimeter coverage model}\;
       }
       

-      \If{$ (s_k.ID $ is the same Previous Leader) And (K.CurrentSize = K.PreviousSize)}{
+      \If{$ (s_j.ID $ is the same Previous Leader) And (A.CurrentSize = A.PreviousSize)}{

       
         \emph{ Use the same previous cover set for current sensing stage}\;
       }
       \Else{
             \emph{Update $a^j_{ik}$; prepare data for IP~Algorithm}\;
       
         \emph{ Use the same previous cover set for current sensing stage}\;
       }
       \Else{
             \emph{Update $a^j_{ik}$; prepare data for IP~Algorithm}\;

-            \emph{$\left\{\left(X_{1},\dots,X_{l},\dots,X_{K}\right)\right\}$ = Execute Integer Program Algorithm($K$)}\;
-            \emph{K.PreviousSize = K.CurrentSize}\;
+            \emph{$\left\{\left(X_{1},\dots,X_{k},\dots,X_{A}\right)\right\}$ = Execute Integer Program Algorithm($A$)}\;
+            \emph{A.PreviousSize = A.CurrentSize}\;

            }
       
            }
       

-        \emph{$s_k.status$ = COMMUNICATION}\;
-        \emph{Send $ActiveSleep()$ to each node $l$ in subregion}\;
-        \emph{Update $RE_k $}\;
+        \emph{$s_j.status$ = COMMUNICATION}\;
+        \emph{Send $ActiveSleep()$ to each node $k$ in subregion}\;
+        \emph{Update $RE_j $}\;

       }          
       \Else{
       }          
       \Else{

-        \emph{$s_k.status$ = LISTENING}\;
+        \emph{$s_j.status$ = LISTENING}\;

         \emph{Wait $ActiveSleep()$ packet from the Leader}\;
         \emph{Wait $ActiveSleep()$ packet from the Leader}\;

-        \emph{Update $RE_k $}\;
+        \emph{Update $RE_j $}\;

       }  
   }
       }  
   }

-  \Else { Exclude $s_k$ from entering in the current sensing stage}
-\caption{PeCO($s_k$)}
+  \Else { Exclude $s_j$ from entering in the current sensing stage}
+\caption{PeCO($s_j$)}

 \label{alg:PeCO}
 \end{algorithm}
 
 \label{alg:PeCO}
 \end{algorithm}
 

-In this  algorithm, K.CurrentSize and K.PreviousSize  respectively represent the
+In this  algorithm, A.CurrentSize and A.PreviousSize  respectively represent the

 current number and  the previous number of living nodes in  the subnetwork of the
 current number and  the previous number of living nodes in  the subnetwork of the

-subregion.  Initially, the sensor node checks its remaining energy $RE_k$, which
+subregion.  Initially, the sensor node checks its remaining energy $RE_j$, which

 must be greater than a threshold $E_{th}$ in order to participate in the current
 period.  Each  sensor node  determines its position  and its subregion  using an
 embedded  GPS or a  location discovery  algorithm. After  that, all  the sensors
 collect position coordinates,  remaining energy, sensor node ID,  and the number
 of their  one-hop live  neighbors during the  information exchange.  The sensors
 inside a same region cooperate to elect a leader. The selection criteria for the
 must be greater than a threshold $E_{th}$ in order to participate in the current
 period.  Each  sensor node  determines its position  and its subregion  using an
 embedded  GPS or a  location discovery  algorithm. After  that, all  the sensors
 collect position coordinates,  remaining energy, sensor node ID,  and the number
 of their  one-hop live  neighbors during the  information exchange.  The sensors
 inside a same region cooperate to elect a leader. The selection criteria for the

-leader, in order of priority,  are: larger numbers of neighbors, larger remaining
+leader, in order of priority,  are larger numbers of neighbors, larger remaining

 energy, and  then in case  of equality, larger  index.  Once chosen,  the leader
 collects information to formulate and  solve the integer program which allows to
 construct the set of active sensors in the sensing stage.
 energy, and  then in case  of equality, larger  index.  Once chosen,  the leader
 collects information to formulate and  solve the integer program which allows to
 construct the set of active sensors in the sensing stage.


@@ -270,14 +224,12 @@ section.
 
 First, we have the following sets:
 \begin{itemize}
 
 First, we have the following sets:
 \begin{itemize}

-\item $S$ represents the set of WSN sensor nodes;
-\item $A \subseteq S $ is the subset of alive sensors;
+\item $J$ represents the set of sensor nodes;
+\item $A \subseteq J $ is the subset of alive sensors;

 \item  $I_j$  designates  the  set  of  coverage  intervals  (CI)  obtained  for
 \item  $I_j$  designates  the  set  of  coverage  intervals  (CI)  obtained  for

-  sensor~$j$.
+  sensor~$j$, which have been defined according to the  method introduced in section~\ref{ch6:sec:02:01}.

 \end{itemize}
 \end{itemize}

-$I_j$ refers to the set of  coverage intervals which have been defined according
-to the  method introduced in  subsection~\ref{ch6:sec:02:01}. For a coverage  interval $i$,
-let $a^j_{ik}$ denotes  the indicator function of whether  sensor~$k$ is involved
+First, for a coverage  interval $i$, let $a^j_{ik}$ denotes  the indicator function of whether  sensor~$k$ is involved

 in coverage interval~$i$ of sensor~$j$, that is:
 \begin{equation}
 a^j_{ik} = \left \{ 
 in coverage interval~$i$ of sensor~$j$, that is:
 \begin{equation}
 a^j_{ik} = \left \{ 


@@ -306,24 +258,23 @@ sensor $j$  is given by  $\sum_{k \in A} a^j_{ik}  X_k$.  To extend  the network
 lifetime,  the objective  is to  activate a  minimal number  of sensors  in each
 period to  ensure the  desired coverage  level. As the  number of  alive sensors
 decreases, it becomes impossible to reach  the desired level of coverage for all
 lifetime,  the objective  is to  activate a  minimal number  of sensors  in each
 period to  ensure the  desired coverage  level. As the  number of  alive sensors
 decreases, it becomes impossible to reach  the desired level of coverage for all

-coverage intervals. Therefore we use variables  $M^j_i$ and $V^j_i$ as a measure
+coverage intervals. Therefore, we use variables  $M^j_i$ and $V^j_i$ as a measure

 of the  deviation between  the desired  number of active  sensors in  a coverage
 interval and  the effective  number. And  we try  to minimize  these deviations,
 first to  force the  activation of  a minimal  number of  sensors to  ensure the
 desired coverage level, and if the desired level cannot be completely satisfied,
 to reach a coverage level as close as possible to the desired one.
 
 of the  deviation between  the desired  number of active  sensors in  a coverage
 interval and  the effective  number. And  we try  to minimize  these deviations,
 first to  force the  activation of  a minimal  number of  sensors to  ensure the
 desired coverage level, and if the desired level cannot be completely satisfied,
 to reach a coverage level as close as possible to the desired one.
 

-

 Our coverage optimization problem can then be mathematically expressed as follows: 
 %Objective:
 \begin{equation} %\label{eq:ip2r}
 \left \{
 \begin{array}{ll}
 Our coverage optimization problem can then be mathematically expressed as follows: 
 %Objective:
 \begin{equation} %\label{eq:ip2r}
 \left \{
 \begin{array}{ll}

-\min \sum_{j \in S} \sum_{i \in I_j} (\alpha^j_i ~ M^j_i + \beta^j_i ~ V^j_i )&\\
+\min \sum_{j \in J} \sum_{i \in I_j} (\alpha^j_i ~ M^j_i + \beta^j_i ~ V^j_i )&\\

 \textrm{subject to :}&\\
 \textrm{subject to :}&\\

-\sum_{k \in A} ( a^j_{ik} ~ X_{k}) + M^j_i  \geq l \quad \forall i \in I_j, \forall j \in S\\
+\sum_{k \in A} ( a^j_{ik} ~ X_{k}) + M^j_i  \geq l \quad \forall i \in I_j, \forall j \in J\\

 %\label{c1} 
 %\label{c1} 

-\sum_{k \in A} ( a^j_{ik} ~ X_{k}) - V^j_i  \leq l \quad \forall i \in I_j, \forall j \in S\\
+\sum_{k \in A} ( a^j_{ik} ~ X_{k}) - V^j_i  \leq l \quad \forall i \in I_j, \forall j \in J\\

 % \label{c2}
 % \Theta_{p}\in \mathbb{N}, &\forall p \in P\\
 % U_{p} \in \{0,1\}, &\forall p \in P\\
 % \label{c2}
 % \Theta_{p}\in \mathbb{N}, &\forall p \in P\\
 % U_{p} \in \{0,1\}, &\forall p \in P\\


@@ -336,7 +287,7 @@ $\alpha^j_i$ and $\beta^j_i$  are nonnegative weights selected  according to the
 relative importance of satisfying the associated level of coverage. For example,
 weights associated with  coverage intervals of a specified part  of a region may
 be  given by a  relatively larger  magnitude than  weights associated  with another
 relative importance of satisfying the associated level of coverage. For example,
 weights associated with  coverage intervals of a specified part  of a region may
 be  given by a  relatively larger  magnitude than  weights associated  with another

-region. This  kind of integer program  is inspired from the  model developed for
+region. This kind of an integer program is inspired from the model developed for

 brachytherapy treatment planning  for optimizing dose  distribution
 \cite{0031-9155-44-1-012}. The integer  program must be solved by  the leader in
 each subregion at the beginning of  each sensing phase, whenever the environment
 brachytherapy treatment planning  for optimizing dose  distribution
 \cite{0031-9155-44-1-012}. The integer  program must be solved by  the leader in
 each subregion at the beginning of  each sensing phase, whenever the environment


@@ -352,82 +303,72 @@ sensing phase) in the model.
 \subsection{Simulation Settings}
 \label{ch6:sec:04:01}
 
 \subsection{Simulation Settings}
 \label{ch6:sec:04:01}
 

-The WSN  area of interest is  supposed to be divided  into 16~regular subregions. %and we use the same energy consumption than in our previous work~\cite{Idrees2}.
-Table~\ref{table3} gives the chosen parameters settings.
-
-\begin{table}[ht]
-\caption{Relevant parameters for network initialization.}
-% title of Table
-\centering
-% used for centering table
-\begin{tabular}{c|c}
-% centered columns (4 columns)
-\hline
-Parameter & Value  \\ [0.5ex]
-   
-\hline
-% inserts single horizontal line
-Sensing field & $(50 \times 25)~m^2 $   \\
-
-WSN size &  100, 150, 200, 250, and 300~nodes   \\
-%\hline
-Initial energy  & in range 500-700~Joules  \\  
+The WSN  area of interest is  supposed to be divided  into 16~regular subregions. The simulation  parameters are summarized in Table~\ref{tablech4}.
+%Table~\ref{table3} gives the chosen parameters settings.
+%\begin{table}[ht]
+%\caption{Relevant parameters for network initialization.}
+%\centering
+%\begin{tabular}{c|c}

 %\hline
 %\hline

-Sensing period & duration of 60 minutes \\
-$E_{th}$ & 36~Joules\\
-$R_s$ & 5~m   \\     
+%Parameter & Value  \\ [0.5ex]

 %\hline
 %\hline

-$\alpha^j_i$ & 0.6   \\
-% [1ex] adds vertical space
-%\hline
-$\beta^j_i$ & 0.4
-%inserts single line
+%Sensing field & $(50 \times 25)~m^2 $   \\
+%WSN size &  100, 150, 200, 250, and 300~nodes   \\
+%Initial energy  & in range 500-700~Joules  \\  
+%Sensing period & duration of 60 minutes \\
+%$E_{th}$ & 36~Joules\\
+%$R_s$ & 5~m   \\     
+%$\alpha^j_i$ & 0.6   \\
+%$\beta^j_i$ & 0.4
+%\end{tabular}
+%\label{table3}
+%\end{table}
+To obtain experimental results which are relevant,  simulations  with  five different node densities going from  100 to 300~nodes were performed considering each time 25~randomly  generated networks. The nodes are deployed  on a field of
+interest of $(50 \times 25)~m^2 $ in such a way that they cover the field with a high coverage ratio. 
+%Each node has an  initial energy level, in Joules, which is randomly drawn in the interval $[500-700]$. If its energy provision reaches a value below  the threshold $E_{th}=36$~Joules,  the minimum energy needed  for a node  to stay  active during  one period,  it will no more  participate in the coverage task. This value corresponds to the energy needed by the sensing phase, obtained by multiplying the energy consumed in active state (9.72 mW) with the time in seconds for one  period (3600 seconds), and  adding the energy  for the pre-sensing phases. According  to the interval of initial energy,  a sensor may be active during at most 20 periods.
+
+
+The values  of $\alpha^j_i$ and  $\beta^j_i$ have been chosen to ensure a good network coverage and a longer WSN lifetime as shown in Table \ref{my-beta-alfa}. We set the values  of $\alpha^j_i$ and  $\beta^j_i$ to 0.6 and 0.4 respectively.  We have given a higher priority to the undercoverage  (by  setting  the  $\alpha^j_i$ with  a  larger  value  than $\beta^j_i$)  so as  to prevent  the non-coverage  for the  interval~$i$ of  the sensor~$j$.  On the  other hand,  we have assigned to
+$\beta^j_i$ a value which is slightly lower so as to minimize the number of active sensor nodes which contribute in covering the interval.
+
+\begin{table}[h]
+\centering
+\caption{The impact of $\alpha^j_i$ and $\beta^j_i$ on PeCO's  performance for 200 deployed nodes}
+\label{my-beta-alfa}
+\begin{tabular}{|c|c|c|c|}
+\hline
+$\alpha^j_i$ & $\beta^j_i$ & $Lifetime_{50}$ & $Lifetime_{95}$ \\ \hline
+0.0 & 1.0 & 151 & 0 \\ \hline
+0.1 & 0.9 & 145 & 0 \\ \hline
+0.2 & 0.8 & 140 & 0 \\ \hline
+0.3 & 0.7 & 134 & 0 \\ \hline
+0.4 & 0.6 & 125 & 0 \\ \hline
+0.5 & 0.5 & 118 & 30 \\ \hline
+0.6 & 0.4 & 94 & 57 \\ \hline
+0.7 & 0.3 & 97 & 49 \\ \hline
+0.8 & 0.2 & 90 & 52 \\ \hline
+0.9 & 0.1 & 77 & 50 \\ \hline
+1.0 & 0.0 & 60 & 44 \\ \hline

 \end{tabular}
 \end{tabular}

-\label{table3}
-% is used to refer this table in the text

 \end{table}
 
 \end{table}
 

-
-To obtain experimental results which are relevant,  simulations  with  five
-different node densities going from  100 to 300~nodes were performed considering
-each time 25~randomly  generated networks. The nodes are deployed  on a field of
-interest of $(50 \times 25)~m^2 $ in such a way that they cover the field with a
-high coverage ratio. Each node has an  initial energy level, in Joules, which is
-randomly drawn in the interval $[500-700]$. If its energy provision reaches a
-value below  the threshold $E_{th}=36$~Joules,  the minimum energy needed  for a
-node  to stay  active during  one period,  it will no more  participate in the
-coverage task. This value corresponds to the energy needed by the sensing phase,
-obtained by multiplying the energy consumed in active state (9.72 mW) with the
-time in seconds for one  period (3600 seconds), and  adding the energy  for the
-pre-sensing phases. According  to the interval of initial energy,  a sensor may
-be active during at most 20 periods.
-
-
-The values  of $\alpha^j_i$ and  $\beta^j_i$ have been  chosen to ensure  a good
-network coverage and a longer WSN lifetime.  We have given a higher priority to
-the  undercoverage  (by  setting  the  $\alpha^j_i$ with  a  larger  value  than
-$\beta^j_i$)  so as  to prevent  the non-coverage  for the  interval~$i$ of  the
-sensor~$j$.  On the  other hand,  we have assigned to
-$\beta^j_i$ a value which is slightly lower so as to minimize the number of active sensor nodes which contribute
-in covering the interval.
-
-We applied the performance metrics, which are described in chapter 4, section \ref{ch4:sec:04:04} in order to evaluate the efficiency of our approach. We used the modeling language and the optimization solver which are mentioned in chapter 4, section \ref{ch4:sec:04:02}. In addition, we employed an energy consumption model, which is presented in chapter 4, section \ref{ch4:sec:04:03}.
+With the performance metrics, described in section \ref{ch4:sec:04:04}, we evaluate the efficiency of our approach. We use the modeling language and the optimization solver which are mentioned in section \ref{ch4:sec:04:02}. In addition, we use the same energy consumption model, as previously, described in section \ref{ch4:sec:04:03}.

 
 
 \subsection{Simulation Results}
 \label{ch6:sec:04:02}
 
 
 
 \subsection{Simulation Results}
 \label{ch6:sec:04:02}
 

-In  order  to  assess and  analyze  the  performance  of  our protocol  we  have implemented PeCO protocol in  OMNeT++~\cite{ref158} simulator.  Besides PeCO, three other protocols,  described in  the next paragraph,  will  be  evaluated for comparison purposes. 
+In  order to  assess and  analyze  the  performance  of  our protocol  we  have implemented PeCO protocol in  OMNeT++~\cite{ref158} simulator.  
+%Besides PeCO, three other protocols,  described in  the next paragraph,  will  be  evaluated for comparison purposes. 

 %The simulations were run  on a laptop DELL with an Intel Core~i3~2370~M (2.4~GHz) processor (2  cores) whose MIPS  (Million Instructions Per Second) rate  is equal to 35330. To  be consistent with the use  of a sensor node based on  Atmels AVR ATmega103L microcontroller (6~MHz) having  a MIPS rate equal to 6, the original execution time  on the laptop is  multiplied by 2944.2 $\left(\frac{35330}{2} \times  \frac{1}{6} \right)$.  The modeling  language for Mathematical Programming (AMPL)~\cite{AMPL} is  employed to generate the integer program instance  in a  standard format, which  is then read  and solved  by the optimization solver  GLPK (GNU  linear Programming Kit  available in  the public domain) \cite{glpk} through a Branch-and-Bound method.
 %The simulations were run  on a laptop DELL with an Intel Core~i3~2370~M (2.4~GHz) processor (2  cores) whose MIPS  (Million Instructions Per Second) rate  is equal to 35330. To  be consistent with the use  of a sensor node based on  Atmels AVR ATmega103L microcontroller (6~MHz) having  a MIPS rate equal to 6, the original execution time  on the laptop is  multiplied by 2944.2 $\left(\frac{35330}{2} \times  \frac{1}{6} \right)$.  The modeling  language for Mathematical Programming (AMPL)~\cite{AMPL} is  employed to generate the integer program instance  in a  standard format, which  is then read  and solved  by the optimization solver  GLPK (GNU  linear Programming Kit  available in  the public domain) \cite{glpk} through a Branch-and-Bound method.

-As said previously, the PeCO is  compared with three other approaches. The first one,  called  DESK,  is  a  fully distributed  coverage  algorithm  proposed  by \cite{DESK}. The second one,  called GAF~\cite{GAF}, consists in dividing  the monitoring  area into  fixed  squares. Then,  during the  decision phase, in each square, one sensor is  chosen to remain active during the sensing phase. The last  one, the DiLCO protocol~\cite{Idrees2}, is  an improved version of a research work we presented in~\cite{ref159}. Let us notice that PeCO and  DiLCO protocols are  based on the  same framework. In  particular, the choice for the simulations of a partitioning in 16~subregions was chosen because it corresponds to the configuration producing the better results for DiLCO. The protocols are distinguished from one another by the formulation  of the integer program providing the set of sensors which have to be activated in each sensing phase. DiLCO protocol tries to satisfy the coverage of a set of primary points, whereas PeCO protocol objective is to reach a desired level of coverage for each sensor perimeter. In our experimentations, we chose a level of coverage equal to one ($l=1$).
+PeCO protocol is  compared with three other approaches. DESK \cite{DESK}, GAF~\cite{GAF}, and DiLCO~\cite{Idrees2} is  an improved version of a research work we presented in~\cite{ref159}, where DiLCO protocol is described in chapter 4. Let us notice that the PeCO and  the DiLCO protocols are  based on the  same scheme. In  particular, the choice for the simulations of a partitioning in 16~subregions was chosen because it corresponds to the configuration producing the better results for DiLCO. The protocols are distinguished from one another by the formulation  of the integer program providing the set of sensors which have to be activated in each sensing phase. DiLCO protocol tries to satisfy the coverage of a set of primary points, whereas PeCO protocol objective is to reach a desired level of coverage for each sensor perimeter. In our experimentations, we chose a level of coverage equal to one ($l=1$).

 
 
 
 \subsubsection{Coverage Ratio}
 \label{ch6:sec:04:02:01}
 
 
 
 
 \subsubsection{Coverage Ratio}
 \label{ch6:sec:04:02:01}
 

-Figure~\ref{fig333}  shows the  average coverage  ratio for  200 deployed  nodes
-obtained with the  four protocols. DESK, GAF, and DiLCO  provide a slightly better
+Figure~\ref{fig333}  shows the  average coverage  ratio for  200 deployed  nodes obtained with the  four protocols. DESK, GAF, and DiLCO  provide a slightly better

 coverage ratio with respectively 99.99\%,  99.91\%, and 99.02\%, compared to the 98.76\%
 produced by  PeCO for the  first periods. This  is due to  the fact that  at the
 beginning the DiLCO protocol  puts to  sleep status  more redundant  sensors (which
 coverage ratio with respectively 99.99\%,  99.91\%, and 99.02\%, compared to the 98.76\%
 produced by  PeCO for the  first periods. This  is due to  the fact that  at the
 beginning the DiLCO protocol  puts to  sleep status  more redundant  sensors (which


@@ -441,7 +382,7 @@ substantial increase of the coverage performance.
 \parskip 0pt    
 \begin{figure}[h!]
 \centering
 \parskip 0pt    
 \begin{figure}[h!]
 \centering

- \includegraphics[scale=0.5] {Figures/ch6/R/CR.eps} 
+ \includegraphics[scale=0.8] {Figures/ch6/R/CR.eps} 

 \caption{Coverage ratio for 200 deployed nodes.}
 \label{fig333}
 \end{figure} 
 \caption{Coverage ratio for 200 deployed nodes.}
 \label{fig333}
 \end{figure} 


@@ -463,12 +404,12 @@ Figure \ref{fig333}.
 
 \begin{figure}[h!]
 \centering
 
 \begin{figure}[h!]
 \centering

-\includegraphics[scale=0.5]{Figures/ch6/R/ASR.eps}  
+\includegraphics[scale=0.8]{Figures/ch6/R/ASR.eps}  

 \caption{Active sensors ratio for 200 deployed nodes.}
 \label{fig444}
 \end{figure} 
 
 \caption{Active sensors ratio for 200 deployed nodes.}
 \label{fig444}
 \end{figure} 
 

-\subsubsection{The Energy Consumption}
+\subsubsection{Energy Consumption}

 \label{ch6:sec:04:02:03}
 
 We studied the effect of the energy  consumed by the WSN during the communication,
 \label{ch6:sec:04:02:03}
 
 We studied the effect of the energy  consumed by the WSN during the communication,


@@ -477,18 +418,15 @@ and  compared  it for  the  four  approaches.  Figures~\ref{fig3EC}(a)  and  (b)
 illustrate  the  energy   consumption  for  different  network   sizes  and  for
 $Lifetime95$ and  $Lifetime50$. The results show  that our PeCO protocol  is the
 most competitive  from the energy  consumption point of  view. As shown  in both
 illustrate  the  energy   consumption  for  different  network   sizes  and  for
 $Lifetime95$ and  $Lifetime50$. The results show  that our PeCO protocol  is the
 most competitive  from the energy  consumption point of  view. As shown  in both

-figures, PeCO consumes much less energy than the three other methods.  One might
-think that the  resolution of the integer  program is too costly  in energy, but
-the  results show  that it  is very  beneficial to  lose a  bit of  time in  the
-selection of  sensors to  activate.  Indeed the  optimization program  allows to
-reduce significantly the number of active  sensors and so the energy consumption
-while keeping a good coverage level.
+figures, PeCO consumes much less energy than the three other methods.  \\ \\ \\ \\ \\ \\
+
+One might think that the  resolution of the integer  program is too costly  in energy, but the  results show  that it  is very  beneficial to  lose a  bit of time in the selection of  sensors to  activate.  Indeed the optimization program  allows to reduce significantly the number of active  sensors and so the energy consumption while keeping a good coverage level.

 
 \begin{figure}[h!]
   \centering
   \begin{tabular}{@{}cr@{}}
 
 \begin{figure}[h!]
   \centering
   \begin{tabular}{@{}cr@{}}

-    \includegraphics[scale=0.475]{Figures/ch6/R/EC95.eps} & \raisebox{2.75cm}{(a)} \\
-    \includegraphics[scale=0.475]{Figures/ch6/R/EC50.eps} & \raisebox{2.75cm}{(b)}
+    \includegraphics[scale=0.8]{Figures/ch6/R/EC95.eps} & \raisebox{4cm}{(a)} \\
+    \includegraphics[scale=0.8]{Figures/ch6/R/EC50.eps} & \raisebox{4cm}{(b)}

   \end{tabular}
   \caption{Energy consumption per period for (a)~$Lifetime_{95}$ and (b)~$Lifetime_{50}$.}
   \label{fig3EC}
   \end{tabular}
   \caption{Energy consumption per period for (a)~$Lifetime_{95}$ and (b)~$Lifetime_{50}$.}
   \label{fig3EC}


@@ -496,27 +434,18 @@ while keeping a good coverage level.
 
 
 
 
 
 

-\subsubsection{The Network Lifetime}
+\subsubsection{Network Lifetime}

 \label{ch6:sec:04:02:04}
 
 \label{ch6:sec:04:02:04}
 

-We observe the superiority of PeCO and DiLCO protocols in comparison with the
-two    other   approaches    in    prolonging   the    network   lifetime.    In
-Figures~\ref{fig3LT}(a)  and (b),  $Lifetime95$ and  $Lifetime50$ are  shown for
-different  network  sizes.   As  highlighted  by  these  figures,  the  lifetime
-increases with the size  of the network, and it is clearly   largest for DiLCO
-and PeCO  protocols.  For instance,  for a  network of 300~sensors  and coverage
-ratio greater than 50\%, we can  see on Figure~\ref{fig3LT}(b) that the lifetime
-is about twice longer with  PeCO compared to DESK protocol.  The performance
-difference    is    more    obvious   in    Figure~\ref{fig3LT}(b)    than    in
-Figure~\ref{fig3LT}(a) because the gain induced  by our protocols increases with
- time, and the lifetime with a coverage  of 50\% is far  longer than with
-95\%.
+We observe the superiority of PeCO and DiLCO protocols in comparison with the two other   approaches in  prolonging the network lifetime. In
+Figures~\ref{fig3LT}(a)  and (b),  $Lifetime95$ and  $Lifetime50$ are  shown for different  network  sizes.   As  highlighted  by  these  figures,  the  lifetime increases with the size  of the network, and it is clearly   largest for the DiLCO and the PeCO  protocols.  For instance,  for a  network of 300~sensors  and coverage ratio greater than 50\%, we can  see on Figure~\ref{fig3LT}(b) that the lifetime is about twice longer with  the PeCO compared to the DESK protocol.  The performance difference    is    more    obvious   in    Figure~\ref{fig3LT}(b) than in Figure~\ref{fig3LT}(a) because the gain induced  by our protocols increases with  time, and the lifetime with a coverage  of 50\% is far  longer than with
+95\%. 

 
 

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

   \centering
   \begin{tabular}{@{}cr@{}}
   \centering
   \begin{tabular}{@{}cr@{}}

-    \includegraphics[scale=0.475]{Figures/ch6/R/LT95.eps} & \raisebox{2.75cm}{(a)} \\  
-    \includegraphics[scale=0.475]{Figures/ch6/R/LT50.eps} & \raisebox{2.75cm}{(b)}
+    \includegraphics[scale=0.8]{Figures/ch6/R/LT95.eps} & \raisebox{4cm}{(a)} \\  
+    \includegraphics[scale=0.8]{Figures/ch6/R/LT50.eps} & \raisebox{4cm}{(b)}

   \end{tabular}
   \caption{Network Lifetime for (a)~$Lifetime_{95}$ and (b)~$Lifetime_{50}$.}
   \label{fig3LT}
   \end{tabular}
   \caption{Network Lifetime for (a)~$Lifetime_{95}$ and (b)~$Lifetime_{50}$.}
   \label{fig3LT}


@@ -526,43 +455,24 @@ Figure~\ref{figLTALL}  compares  the  lifetime  coverage of  our  protocols  for
 different coverage  ratios. We denote by  Protocol/50, Protocol/80, Protocol/85,
 Protocol/90, and  Protocol/95 the amount  of time  during which the  network can
 satisfy an area coverage greater than $50\%$, $80\%$, $85\%$, $90\%$, and $95\%$
 different coverage  ratios. We denote by  Protocol/50, Protocol/80, Protocol/85,
 Protocol/90, and  Protocol/95 the amount  of time  during which the  network can
 satisfy an area coverage greater than $50\%$, $80\%$, $85\%$, $90\%$, and $95\%$

-respectively, where the term Protocol refers to  DiLCO  or PeCO.  Indeed there  are applications
-that do not require a 100\% coverage of  the area to be monitored. PeCO might be
-an interesting  method since  it achieves  a good balance  between a  high level
-coverage ratio and network lifetime. PeCO always outperforms DiLCO for the three
-lower  coverage  ratios,  moreover  the   improvements  grow  with  the  network
-size. DiLCO is better  for coverage ratios near 100\%, but in  that case PeCO is
-not ineffective for the smallest network sizes.
+respectively, where the term Protocol refers to  DiLCO  or PeCO.  Indeed there  are applications that do not require a 100\% coverage of  the area to be monitored. PeCO might be an interesting  method since  it achieves  a good balance  between a  high level coverage ratio and network lifetime. PeCO always outperforms DiLCO for the three lower  coverage  ratios,  moreover  the   improvements  grow  with  the  network size. DiLCO is better  for coverage ratios near 100\%, but in  that case PeCO is not ineffective for the smallest network sizes.

 
 

-\begin{figure}[h!]
-\centering \includegraphics[scale=0.5]{Figures/ch6/R/LTa.eps}
+\begin{figure} [p]
+\centering \includegraphics[scale=0.8]{Figures/ch6/R/LTa.eps}

 \caption{Network lifetime for different coverage ratios.}
 \label{figLTALL}
 \caption{Network lifetime for different coverage ratios.}
 \label{figLTALL}

-\end{figure} 
+\end{figure}

 
 
 
 

+ %\FloatBarrier
+\section{Conclusion} 
+\label{ch6:sec:05}
+
+In this chapter, we have studied the problem of  Perimeter-based Coverage Optimization in WSNs. We have designed  a new protocol, called Perimeter-based  Coverage Optimization, which schedules nodes'  activities (wake up  and sleep  stages) with the  objective of maintaining a  good coverage ratio  while maximizing the network  lifetime. This protocol is  applied in a distributed  way in regular subregions  obtained after partitioning the area of interest in a preliminary step. It works in periods and
+is based on the resolution of an integer program to select the subset of sensors operating in active status for each period. Our work is original because it proposes for  the first  time an  integer program  scheduling the  activation of sensors  based on  their perimeter  coverage level,  instead of  using a  set of targets/points to be covered. We  have carried out  several simulations  to  evaluate the  proposed protocol. The simulation  results  show   that  PeCO  is  more   energy-efficient  than  other approaches, with respect to lifetime,  coverage ratio, active sensors ratio, and energy consumption.
+
+%We plan to extend our framework so that the schedules are planned for multiple sensing periods. We also want  to improve our integer program to  take into account heterogeneous sensors  from both  energy  and node  characteristics point of views. Finally,  it   would  be   interesting  to  implement   our  protocol   using  a sensor-testbed to evaluate it in real world applications.

 
 

-\section{Conclusion}
-\label{ch6:sec:04}

 
 

-In this chapter, we have studied the problem of  Perimeter-based Coverage Optimization in
-WSNs. We have designed  a new protocol, called Perimeter-based  Coverage Optimization, which
-schedules nodes'  activities (wake up  and sleep  stages) with the  objective of
-maintaining a  good coverage ratio  while maximizing the network  lifetime. This
-protocol is  applied in a distributed  way in regular subregions  obtained after
-partitioning the area of interest in a preliminary step. It works in periods and
-is based on the resolution of an integer program to select the subset of sensors
-operating in active status for each period. Our work is original in so far as it
-proposes for  the first  time an  integer program  scheduling the  activation of
-sensors  based on  their perimeter  coverage level,  instead of  using a  set of
-targets/points to be covered. We  have carried out  several simulations  to  evaluate the  proposed protocol. The simulation  results  show   that  PeCO  is  more   energy-efficient  than  other approaches, with respect to lifetime,  coverage ratio, active sensors ratio, and
-energy consumption.
-
-We plan to extend our framework so that the schedules are planned for multiple
-sensing periods.

 %in order to compute all active sensor schedules in only one step for many periods;
 %in order to compute all active sensor schedules in only one step for many periods;

-We also want  to improve our integer program to  take into account heterogeneous
-sensors  from both  energy  and node  characteristics point of views.
-%the third, we are investigating new optimization model based on the sensing range so as to maximize the lifetime coverage in WSN;
-Finally,  it   would  be   interesting  to  implement   our  protocol   using  a
-sensor-testbed to evaluate it in real world applications.
+%the third, we are investigating new optimization model based on the sensing range so as to maximize the lifetime coverage in WSN;
\ No newline at end of file