+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%% %%
+%% CHAPTER 06 %%
+%% %%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+\chapter{Perimeter-based Coverage Optimization to Improve Lifetime in Wireless Sensor Networks}
+\label{ch6}
+
+
+\section{summary}
+\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}
+\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.
+
+
+
+\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
+$k$-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
+arcs.
+
+\begin{figure}[ht!]
+ \centering
+ \begin{tabular}{@{}cr@{}}
+ \includegraphics[width=95mm]{Figures/ch6/pcm.jpg} & \raisebox{3.25cm}{(a)} \\
+ \includegraphics[width=95mm]{Figures/ch6/twosensors.jpg} & \raisebox{2.75cm}{(b)}
+ \end{tabular}
+ \caption{(a) Perimeter coverage of sensor node 0 and (b) finding the arc of
+ $u$'s perimeter covered by $v$.}
+ \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: $$\alpha = \arccos \left(\dfrac{Dist(u,v)}{2R_s}
+\right).$$ 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.
+
+
+\begin{figure*}[t!]
+\centering
+\includegraphics[width=150.5mm]{Figures/ch6/expcm2.jpg}
+\caption{Maximum coverage levels for perimeter of sensor node $0$.}
+\label{expcm}
+\end{figure*}
+
+
+ \begin{table}[h!]
+ \caption{Coverage intervals and contributing sensors for sensor node 0.}
+ \centering
+\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
+\hline
+\begin{tabular}[c]{@{}c@{}}Left \\ point \\ angle~$\alpha$ \end{tabular} & \begin{tabular}[c]{@{}c@{}}Interval \\ left \\ point\end{tabular} & \begin{tabular}[c]{@{}c@{}}Interval \\ right \\ point\end{tabular} & \begin{tabular}[c]{@{}c@{}}Maximum \\ coverage\\ level\end{tabular} & \multicolumn{5}{c|}{\begin{tabular}[c]{@{}c@{}}Set of sensors\\ involved \\ in coverage interval\end{tabular}} \\ \hline
+0.0291 & 1L & 2L & 4 & 0 & 1 & 3 & 4 & \\ \hline
+0.104 & 2L & 3R & 5 & 0 & 1 & 3 & 4 & 2 \\ \hline
+0.3168 & 3R & 4R & 4 & 0 & 1 & 4 & 2 & \\ \hline
+0.6752 & 4R & 1R & 3 & 0 & 1 & 2 & & \\ \hline
+1.8127 & 1R & 5L & 2 & 0 & 2 & & & \\ \hline
+1.9228 & 5L & 6L & 3 & 0 & 2 & 5 & & \\ \hline
+2.3959 & 6L & 2R & 4 & 0 & 2 & 5 & 6 & \\ \hline
+2.4258 & 2R & 7L & 3 & 0 & 5 & 6 & & \\ \hline
+2.7868 & 7L & 8L & 4 & 0 & 5 & 6 & 7 & \\ \hline
+2.8358 & 8L & 5R & 5 & 0 & 5 & 6 & 7 & 8 \\ \hline
+2.9184 & 5R & 7R & 4 & 0 & 6 & 7 & 8 & \\ \hline
+3.3301 & 7R & 9R & 3 & 0 & 6 & 8 & & \\ \hline
+3.9464 & 9R & 6R & 4 & 0 & 6 & 8 & 9 & \\ \hline
+4.767 & 6R & 3L & 3 & 0 & 8 & 9 & & \\ \hline
+4.8425 & 3L & 8R & 4 & 0 & 3 & 8 & 9 & \\ \hline
+4.9072 & 8R & 4L & 3 & 0 & 3 & 9 & & \\ \hline
+5.3804 & 4L & 9R & 4 & 0 & 3 & 4 & 9 & \\ \hline
+5.9157 & 9R & 1L & 3 & 0 & 3 & 4 & & \\ \hline
+\end{tabular}
+
+\label{my-label}
+\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.
+
+
+\begin{figure}[h!]
+\centering
+\includegraphics[width=95.5mm]{Figures/ch6/ex4pcm.jpg}
+\caption{Sensing range outside the WSN's area of interest.}
+\label{ex4pcm}
+\end{figure}
+
+
+
+
+
+\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.
+
+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
+\includegraphics[width=95.5mm]{Figures/ch6/Model.pdf}
+\caption{PeCO protocol.}
+\label{fig2}
+\end{figure}
+
+
+
+
+\subsection{PeCO Protocol Algorithm}
+\label{ch6:sec:02:03}
+
+
+\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.
+
+\begin{algorithm}[h!]
+ % \KwIn{all the parameters related to information exchange}
+% \KwOut{$winer-node$ (: the id of the winner sensor node, which is the leader of current round)}
+ \BlankLine
+ %\emph{Initialize the sensor node and determine it's position and subregion} \;
+
+ \If{ $RE_k \geq E_{th}$ }{
+ \emph{$s_k.status$ = COMMUNICATION}\;
+ \emph{Send $INFO()$ packet to other nodes in subregion}\;
+ \emph{Wait $INFO()$ packet from other nodes in subregion}\;
+ \emph{Update K.CurrentSize}\;
+ \emph{LeaderID = Leader election}\;
+ \If{$ s_k.ID = LeaderID $}{
+ \emph{$s_k.status$ = COMPUTATION}\;
+
+ \If{$ s_k.ID $ is Not previously selected as a Leader }{
+ \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)}{
+
+ \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{$s_k.status$ = COMMUNICATION}\;
+ \emph{Send $ActiveSleep()$ to each node $l$ in subregion}\;
+ \emph{Update $RE_k $}\;
+ }
+ \Else{
+ \emph{$s_k.status$ = LISTENING}\;
+ \emph{Wait $ActiveSleep()$ packet from the Leader}\;
+ \emph{Update $RE_k $}\;
+ }
+ }
+ \Else { Exclude $s_k$ from entering in the current sensing stage}
+\caption{PeCO($s_k$)}
+\label{alg:PeCO}
+\end{algorithm}
+
+In this algorithm, K.CurrentSize and K.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 $RE_k$, 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.
+
+
+
+\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.
+
+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 $I_j$ designates the set of coverage intervals (CI) obtained for
+ sensor~$j$.
+\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
+in coverage interval~$i$ of sensor~$j$, that is:
+\begin{equation}
+a^j_{ik} = \left \{
+\begin{array}{lll}
+ 1 & \mbox{if sensor $k$ is involved in the } \\
+ & \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 S} \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 S\\
+%\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\\
+% \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
+\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 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.
+
+\section{Performance Evaluation and Analysis}
+\label{ch6:sec:04}
+
+\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 \\
+%\hline
+Sensing period & duration of 60 minutes \\
+$E_{th}$ & 36~Joules\\
+$R_s$ & 5~m \\
+%\hline
+$\alpha^j_i$ & 0.6 \\
+% [1ex] adds vertical space
+%\hline
+$\beta^j_i$ & 0.4
+%inserts single line
+\end{tabular}
+\label{table3}
+% is used to refer this table in the text
+\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}.
+
+
+\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.
+%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 ($l=1$).
+
+
+
+\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.
+
+\parskip 0pt
+\begin{figure}[h!]
+\centering
+ \includegraphics[scale=0.5] {Figures/ch6/R/CR.eps}
+\caption{Coverage ratio for 200 deployed nodes.}
+\label{fig333}
+\end{figure}
+
+
+
+\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}.
+
+\begin{figure}[h!]
+\centering
+\includegraphics[scale=0.5]{Figures/ch6/R/ASR.eps}
+\caption{Active sensors ratio for 200 deployed nodes.}
+\label{fig444}
+\end{figure}
+
+\subsubsection{The 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.
+
+\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)}
+ \end{tabular}
+ \caption{Energy consumption per period for (a)~$Lifetime_{95}$ and (b)~$Lifetime_{50}$.}
+ \label{fig3EC}
+\end{figure}
+
+
+
+\subsubsection{The Network Lifetime}
+\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\%.
+
+\begin{figure}[h!]
+ \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)}
+ \end{tabular}
+ \caption{Network Lifetime for (a)~$Lifetime_{95}$ and (b)~$Lifetime_{50}$.}
+ \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}[h!]
+\centering \includegraphics[scale=0.5]{Figures/ch6/R/LTa.eps}
+\caption{Network lifetime for different coverage ratios.}
+\label{figLTALL}
+\end{figure}
+
+
+
+\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;
+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.
