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-%% CHAPTER 06 %%
+%% CHAPTER 06 %%
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\chapter{ Perimeter-based Coverage Optimization to Improve Lifetime in WSNs}
\subsection{Assumptions and Models}
\label{ch6:sec:02:01}
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).
+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
\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: $Dist(u,v)=\sqrt{\vert
+ u_x - v_x \vert^2 + \vert u_y-v_y \vert^2}$,
+
+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.
-
+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
\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!]
\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)
\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}
& \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}
\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}.
\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!]
\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
\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
\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 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
+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)} \\
\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.
+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]
+\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}
