X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/ThesisAli.git/blobdiff_plain/ec6188c6f907fa2ef54fb5b609af61cc87c22708..127dd5ebfa42d2c6220639b057082517a4502c38:/CHAPITRE_06.tex?ds=sidebyside

diff --git a/CHAPITRE_06.tex b/CHAPITRE_06.tex
index d566e41..33c8fa3 100644
--- a/CHAPITRE_06.tex
+++ b/CHAPITRE_06.tex
@@ -1,6 +1,6 @@
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%                          %%
-%%       CHAPTER 06        %%
+%%       CHAPTER 06         %%
 %%                          %%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \chapter{ Perimeter-based Coverage Optimization to Improve Lifetime in WSNs}
@@ -17,7 +17,7 @@
 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 framework 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 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 taking 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}.
@@ -35,9 +35,10 @@ we considered (in particular the perimeter coverage one), second we describe the
 
 \subsection{Assumptions and Models}
 \label{ch6:sec:02:01}
-The PeCO protocol uses the same assumptions and network model than both DiLCO and 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).
+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).
+Authors \cite{ref133}  proved that a network area  is $k$-covered  (every point in  the area is  covered by  at least $k$~sensors) 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
@@ -56,18 +57,21 @@ arcs.
 \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: $$\alpha =  \arccos \left(\dfrac{Dist(u,v)}{2R_s}
-\right).$$ The arc on the perimeter of~$u$ defined by the angular interval $[\pi
-  - \alpha,\pi + \alpha]$ is said to be perimeter-covered by sensor~$v$.
+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, the euclidean distance between nodes~$u$ and $v$ is computed as follow: $Dist(u,v)=\sqrt{\left( 
+  u_x  - v_x  \right)^2 +  \left( u_y-v_y  \right)^2}$, 
+
+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~$v$.
 
 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.
-
+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.
 
 \begin{figure*}[t!]
 \centering
@@ -107,7 +111,8 @@ above is thus given by the sixth line of the table.
 \end{table}
 
 
-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.
+%In the PeCO  protocol, the scheduling of the sensor  nodes' activities is formulated  as an integer program  based on  coverage intervals. 
+In the PeCO protocol,  the  scheduling  of  the  sensor nodes'  activities  is formulated  with  an mixed-integer  program based on coverage intervals~\cite{ref239}.  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!]
@@ -125,10 +130,7 @@ In the PeCO  protocol, the scheduling of the sensor  nodes' activities is formul
 \subsection{The Main Idea}
 \label{ch6:sec:02:02}
 
-\noindent The  WSN area of interest is, in a  first step, divided  into regular
-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.
+\noindent The  WSN area of interest is, in a  first step, divided  into regular 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. Sensor nodes  are assumed to be deployed  almost uniformly over the  region. The regular subdivision  is made such that the number of hops between  any pairs of sensors inside a subregion is less than or equal to 3.
 
 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)
@@ -178,8 +180,8 @@ protocol applied by a sensor node $s_j$ where $j$ is the node index in the WSN.
       	 \emph{ Use the same previous cover set for current sensing stage}\;
       }
       \Else{
-            \emph{Update $a^j_{ik}$; prepare data for IP~Algorithm}\;
-            \emph{$\left\{\left(X_{1},\dots,X_{k},\dots,X_{A}\right)\right\}$ = Execute Integer Program Algorithm($A$)}\;
+            \emph{Update $a^j_{ik}$; prepare data for MIP~Algorithm}\;
+            \emph{$\left\{\left(X_{1},\dots,X_{k},\dots,X_{A}\right)\right\}$ = Execute MIP Algorithm($A$)}\;
             \emph{A.PreviousSize = A.CurrentSize}\;
            }
       
@@ -198,39 +200,34 @@ protocol applied by a sensor node $s_j$ where $j$ is the node index in the WSN.
 \label{alg:PeCO}
 \end{algorithm}
 
-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
-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
-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.
-
+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 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 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.
+The sensors  inside a same  region cooperate to  elect a leader.   The selection criteria for the leader are (in order  of priority):
+\begin{enumerate}
+\item larger number of neighbors;
+\item larger  remaining energy;
+\item and then  in case  of equality,  larger index.
+\end{enumerate}
+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.
 
 
 \section{Perimeter-based Coverage Problem Formulation}
 \label{ch6:sec:03}
 
 
-\noindent In this  section, the coverage model is  mathematically formulated. We
-start  with a  description of  the notations  that will  be used  throughout the
-section.
+\noindent In  this  section,  the   perimeter-based  coverage  problem  is  mathematically formulated.    It    has    been    proved   to    be    a    NP-hard    problem by \cite{ref239}. Authors  study the coverage of  the perimeter of a  large object requiring  to be monitored.  For the proposed  formulation in this chapter,  the large  object to  be monitored  is the  sensor itself  (or more precisely its sensing area).
+
+The following notations are used  throughout the section.
 
-First, we have the following sets:
+First, the following sets:
 \begin{itemize}
-\item $J$ represents the set of sensor nodes;
-\item $A \subseteq J $ is the subset of alive sensors;
+\item $S$ represents the set of sensor nodes;
+\item $A \subseteq S $ is the subset of alive sensors;
 \item  $I_j$  designates  the  set  of  coverage  intervals  (CI)  obtained  for
-  sensor~$j$, which have been defined according to the  method introduced in section~\ref{ch6:sec:02:01}.
+  sensor~$j$.
 \end{itemize}
-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:
+$I_j$ 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 $a^j_{ik}$ denote  the indicator function of whether  sensor~$k$ is involved in coverage interval~$i$ of sensor~$j$, that is:
 \begin{equation}
 a^j_{ik} = \left \{ 
 \begin{array}{lll}
@@ -238,64 +235,32 @@ a^j_{ik} = \left \{
   	&	\mbox{coverage interval $i$ of sensor $j$}, \\
   0 & \mbox{otherwise.}\\
 \end{array} \right.
-%\label{eq12} 
-\notag
 \end{equation}
 Note that $a^k_{ik}=1$ by definition of the interval.
 
-Second,  we define  several binary  and integer  variables.  Hence,  each binary
-variable $X_{k}$  determines the activation of  sensor $k$ in the  sensing phase
-($X_k=1$ if  the sensor $k$  is active or 0  otherwise).  $M^j_i$ is  an integer
-variable  which  measures  the  undercoverage  for  the  coverage  interval  $i$
-corresponding to  sensor~$j$. In  the same  way, the  overcoverage for  the same
-coverage interval is given by the variable $V^j_i$.
-
-If we decide to sustain a level of coverage equal to $l$ all along the perimeter
-of sensor  $j$, we have  to ensure  that at least  $l$ sensors involved  in each
-coverage  interval $i  \in I_j$  of  sensor $j$  are active.   According to  the
-previous notations, the number of active sensors in the coverage interval $i$ of
-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
-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.
-
-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 J} \sum_{i \in I_j} (\alpha^j_i ~ M^j_i + \beta^j_i ~ V^j_i )&\\
-\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 J\\
-%\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 J\\
-% \label{c2}
-% \Theta_{p}\in \mathbb{N}, &\forall p \in P\\
-% U_{p} \in \{0,1\}, &\forall p \in P\\
-X_{k} \in \{0,1\}, \forall k \in A
-\end{array}
-\right.
-\notag
+Second,  several variables  are defined.   Hence, each  binary variable  $X_{k}$ determines the  activation of sensor  $k$ in the  sensing phase ($X_k=1$  if the sensor $k$ is active or 0 otherwise).   $M^j_i$ is a variable which measures the undercoverage for the coverage interval  $i$ corresponding to sensor~$j$. In the same  way, the  overcoverage for  the  same coverage  interval is  given by  the variable $V^j_i$.
+
+To sustain a  level of coverage equal  to $l$ all along the  perimeter of sensor $j$, at  least $l$  sensors involved in  each coverage interval  $i \in  I_j$ of sensor $j$ have  to be active.  According to the  previous notations, the number of  active sensors  in the  coverage  interval $i$  of  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 coverage  intervals. Therefore
+variables  $M^j_i$ and  $V^j_i$ are  introduced 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.
+
+The coverage optimization problem can then be mathematically expressed as follows:
+\begin{equation}
+  \begin{aligned}
+    \text{Minimize } & \sum_{j \in S} \sum_{i \in I_j} (\alpha^j_i ~ M^j_i + \beta^j_i ~ V^j_i ) \\
+    \text{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}) - V^j_i  \leq l \quad \forall i \in I_j, \forall j \in S \\
+    & X_{k} \in \{0,1\}, \forall k \in A \\
+    & M^j_i, V^j_i \in \mathbb{R}^{+} 
+  \end{aligned}
 \end{equation}
-$\alpha^j_i$ 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
-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
-has  changed (new  leader,  death of  some  sensors). Note  that  the number  of
-constraints in the model is constant  (constraints of coverage expressed for all
-sensors), whereas the number of variables $X_k$ decreases over periods, since we
-consider only alive  sensors (sensors with enough energy to  be alive during one
-sensing phase) in the model.
+
+If a given level of coverage $l$ is  required for one sensor, the sensor is said to be undercovered (respectively overcovered) if the level of coverage of one of its  CI  is  less  (respectively  greater)  than $l$.   If  the  sensor  $j$  is undercovered, there exists at least one of its CI (say $i$) for which the number of active  sensors (denoted by $l^{i}$)  covering this part of  the perimeter is less than $l$ and in this case : $M_{i}^{j}=l-l^{i}$, $V_{i}^{j}=0$. Conversely, if the sensor $j$ is overcovered, there exists  at least one of its CI (say $i$) for which the  number of active sensors (denoted by  $l^{i}$) covering this part of  the  perimeter  is  greater  than  $l$  and  in  this  case:  $M_{i}^{j}=0$, $V_{i}^{j}=l^{i}-l$.
+
+$\alpha^j_i$ 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 region. This kind of mixed-integer program  is inspired from the model developed for   brachytherapy  treatment   planning  for   optimizing  dose   distribution \cite{0031-9155-44-1-012}.  The choice of the values for variables $\alpha$ and $\beta$  should be  made according  to the  needs of  the application.  $\alpha$ should be  large enough  to prevent  undercoverage and so  to reach  the highest
+possible coverage ratio. $\beta$ should  be large enough to prevent overcoverage and so to activate a minimum  number of sensors.  The mixed-integer program must be solved  by the  leader in  each subregion  at the  beginning of  each sensing phase, whenever the environment has changed (new leader, death of some sensors). Note that  the number of  constraints in the  model is constant  (constraints of coverage  expressed for  all sensors),  whereas  the number  of variables  $X_k$ decreases over periods, since only alive  sensors (sensors with enough energy to be alive during one sensing phase) are considered in the model.
+
+
 
 \section{Performance Evaluation and Analysis}
 \label{ch6:sec:04}
@@ -303,54 +268,15 @@ sensing phase) in the model.
 \subsection{Simulation Settings}
 \label{ch6:sec:04:01}
 
-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
-%Parameter & Value  \\ [0.5ex]
-%\hline
-%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
+The WSN  area of interest is  supposed to be divided  into 16~regular subregions. The simulation  parameters are summarized in Table~\ref{tablech4}. 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.
 
 
-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.
+The values  of $\alpha^j_i$ and  $\beta^j_i$ have been  chosen to ensure  a good network coverage  and a longer  WSN lifetime.  Higher  priority is given  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, $\beta^j_i$ is assigned to a value which is slightly lower so as to minimize the  number of active sensor nodes which  contribute in covering the interval. Section~\ref{sec:Impact} investigates more deeply how the values of
+both parameters affect the performance of PeCO protocol.
 
-\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{table}
 
 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}.
 
@@ -361,23 +287,17 @@ With the performance metrics, described in section \ref{ch4:sec:04:04}, we evalu
 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.
-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 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}
 
-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
-slightly decreases the coverage ratio), while the three other protocols activate
-more sensor  nodes. Later, when the  number of periods is  beyond~70, it clearly
-appears that  PeCO provides a better  coverage ratio and keeps  a coverage ratio
-greater  than 50\%  for  longer periods  (15  more compared  to  DiLCO, 40  more
-compared to DESK). The energy saved by  PeCO in the early periods allows later a
-substantial increase of the coverage performance.
+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 and PeCO protocols  put to  sleep status  more redundant  sensors (which slightly decreases the coverage ratio), while the two other protocols activate more sensor  nodes. Later, when the  number of periods is  beyond~70, it clearly
+appears that  PeCO provides a better  coverage ratio and keeps  a coverage ratio greater  than 50\%  for  longer periods  (15  more compared  to  DiLCO, 40  more compared to DESK). The energy saved by  PeCO in the early periods allows later a substantial increase of the coverage performance.
 
 \parskip 0pt    
 \begin{figure}[h!]
@@ -392,15 +312,8 @@ substantial increase of the coverage performance.
 \subsubsection{Active Sensors Ratio}
 \label{ch6:sec:04:02:02}
 
-Having the less active sensor nodes in  each period is essential to minimize the
-energy consumption  and thus to  maximize the network  lifetime.  Figure~\ref{fig444}
-shows the  average active nodes ratio  for 200 deployed nodes.   We observe that
-DESK and  GAF have 30.36  \% and  34.96 \% active  nodes for the  first fourteen
-rounds and  DiLCO and PeCO  protocols compete perfectly  with only 17.92  \% and
-20.16 \% active  nodes during the same  time interval. As the  number of periods
-increases, PeCO protocol  has a lower number of active  nodes in comparison with
-the three other approaches, while keeping a greater coverage ratio as shown in
-Figure \ref{fig333}.
+Having the less active sensor nodes in  each period is essential to minimize the energy consumption  and thus to  maximize the network  lifetime.  Figure~\ref{fig444} shows the  average active nodes ratio  for 200 deployed nodes.   We observe that DESK and  GAF have 30.36  \% and  34.96 \% active  nodes for the  first fourteen rounds, and  DiLCO and PeCO  protocols compete perfectly  with only 17.92  \% and 20.16 \% active  nodes during the same  time interval. As the  number of periods increases, PeCO protocol  has a lower number of active  nodes in comparison with
+the three other approaches, while keeping a greater coverage ratio as shown in Figure \ref{fig333}. \\ 
 
 \begin{figure}[h!]
 \centering
@@ -412,15 +325,8 @@ Figure \ref{fig333}.
 \subsubsection{Energy Consumption}
 \label{ch6:sec:04:02:03}
 
-We studied the effect of the energy  consumed by the WSN during the communication,
-computation, listening, active, and sleep status for different network densities
-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
-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.
+We studied the effect of the energy  consumed by the WSN during the communication, computation, listening, active, and sleep status for different network densities and  the  four  approaches compared.  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  by both figures, PeCO consumes much less energy than the 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. Let  us notice that the energy overhead  when increasing network size is the lowest with PeCO.
 
 \begin{figure}[h!]
   \centering
@@ -438,10 +344,10 @@ One might think that the  resolution of the integer  program is too costly  in e
 \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
+Figures~\ref{fig3LT}(a)  and (b),  $Lifetime95$ and  $Lifetime50$ are  shown for different  network  sizes.   As  can be seen in  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} [p]
+\begin{figure} [h!]
   \centering
   \begin{tabular}{@{}cr@{}}
     \includegraphics[scale=0.8]{Figures/ch6/R/LT95.eps} & \raisebox{4cm}{(a)} \\  
@@ -451,47 +357,53 @@ Figures~\ref{fig3LT}(a)  and (b),  $Lifetime95$ and  $Lifetime50$ are  shown for
   \label{fig3LT}
 \end{figure} 
 
-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 $95\%$
-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} [p]
+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 $95\%$ 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=0.8]{Figures/ch6/R/LTa.eps}
 \caption{Network lifetime for different coverage ratios.}
 \label{figLTALL}
 \end{figure}
 
+\subsubsection{Impact of $\alpha$ and $\beta$ on PeCO's performance}
+\label{sec:Impact}
+
+Table~\ref{my-labelx}  shows network  lifetime results  for different  values of $\alpha$ and $\beta$, and  a network size equal to 200 sensor  nodes. On the one hand, the choice  of $\beta \gg \alpha$ prevents the  overcoverage, and so limit the activation of a large number of  sensors, but as $\alpha$ is low, some areas may be poorly covered.  This explains  the results obtained for {\it Lifetime50} with $\beta \gg \alpha$: a large number  of periods with low coverage ratio.  On the other hand, when we choose $\alpha \gg \beta$, we favor the coverage even if some areas may  be overcovered, so high  coverage ratio is reached,  but a large number  of  sensors are  activated  to  achieve  this goal.   Therefore  network
+lifetime is reduced.   \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 
+The choice $\alpha=0.6$ and $\beta=0.4$  seems to achieve the best compromise  between lifetime and coverage ratio. That  explains why we have  chosen  this  setting  for  the  experiments  presented  in  the  previous subsections.
+
+\begin{table}[h!]
+\centering
+\caption{The impact of $\alpha$ and $\beta$ on PeCO's performance}
+\label{my-labelx}
+\begin{tabular}{|c|c|c|c|}
+\hline
+$\alpha$ & $\beta$ & $Lifetime_{50}$ & $Lifetime_{95}$ \\ \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
+{\bf 0.6} & {\bf 0.4} & {\bf 94} & {\bf 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{table}
+
+
 
- %\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 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 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 mixed-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.
+
+
 %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
