\section{Lifetime Coverage problem formulation}
\label{cp}
-In this section, the coverage model are mathematically formulated, where the objective is to find the maximum number of non-disjoint sets of sensor nodes such that each set cover can assure the coverage for the whole region so as to extend the network lifetime in WSN. Our model will use the segment points which are produced by using the perimeter coverage model~\cite{huang2005coverage} in order to optimize the lifetime coverage in each subregion.
-We defined some parameters, which are related to our optimization model. In our model, we consider binary variables $X_{k}$, which determine the activation of sensor $k$ in the sensing round. We also consider the segment points as targets.
-
-
-\noindent In this paper, let us define some parameters, which are used in our protocol.
-%the set of segment points is denoted by $I$, the set of all sensors in the network by $J$, and the set of alive sensors within $J$ by $K$.
-
-\noindent $J :$ the set of all sensors in the network.\\
-\noindent $K :$ the set of alive sensors within $J$.\\
+In this section, the coverage model is mathematically formulated.
+For convenience, the notations are described first.
+%Then the lifetime problem of sensor network is formulated.
+\noindent $S :$ the set of all sensors in the network.\\
+\noindent $A :$ the set of alive sensors within $S$.\\
%\noindent $I :$ the set of segment points.\\
-\noindent $I_j :$ the set of segment points for sensor $j$.\\
-
-\noindent \begin{equation}
-X_{k} = \left \{
-\begin{array}{l l}
- 1& \mbox{if sensor $k$ is active,} \\
- 0 & \mbox{otherwise.}\\
-\end{array} \right.
-%\label{eq11}
-\notag
-\end{equation}
-
-\noindent $M^j_i (undercoverage): $ integer value $\in \mathbb{N}$ for segment point $i$ of sensor $j$.
-
-\noindent $V^j_i (overcoverage): $ integer value $\in \mathbb{N}$ for segment point $i$ of sensor $j$.
+\noindent $I_j :$ the set of coverage intervals (CI) for sensor $j$.\\
-
-
-\noindent For an segment point $i$, let $a^j_{ik}$ denote the indicator function of whether the sensor $k$ is involved in the segment point $i$ of sensor $j$, that is:
+\noindent For a coverage interval $i$, let $a^j_{ik}$ denote the indicator function of whether the sensor $k$ is involved in the coverage interval $i$ of sensor $j$, that is:
\begin{equation}
a^j_{ik} = \left \{
\begin{array}{lll}
1 & \mbox{If the sensor $k$ is involved in the } \\
- & \mbox{segment point $i$ of sensor $j$}, \\
+ & \mbox{coverage interval $i$ of sensor $j$}, \\
0 & \mbox{Otherwise.}\\
\end{array} \right.
%\label{eq12}
\notag
\end{equation}
+%, where the objective is to find the maximum number of non-disjoint sets of sensor nodes such that each set cover can assure the coverage for the whole region so as to extend the network lifetime in WSN. Our model uses the PCL~\cite{huang2005coverage} in order to optimize the lifetime coverage in each subregion.
+%We defined some parameters, which are related to our optimization model. In our model, we consider binary variables $X_{k}$, which determine the activation of sensor $k$ in the sensing round $k$. .
+We consider binary variables $X_{k}$ ($X_k=1$ if the sensor $k$ is active or 0 otherwise), which determine the activation of sensor $k$ in the sensing phase. We define the integer variable $M^j_i$ which measures the undercoverage for the coverage interval $i$ for sensor $j$. In the same way, we define the integer variable $V^j_i$, which measures the overcoverage for the coverage interval $i$ for sensor $j$. If we decide to sustain a level of coverage equal to $l$ all along the perimeter of the sensor $j$, we have to ensure that at least $l$ sensors involved in each coverage interval $i$ ($i \in I_j$) 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 $\sum_{k \in K} a^j_{ik} X_k$. To extend the network lifetime, the objective is to active 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 satisfy the constraints of coverage. We uses 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 of active sensors. 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 can not be completely satisfied, to reach a coverage level as close as possible that the desired one.
+
+
+
+%A system of linear constraints is imposed to attempt to keep the coverage level in each coverage interval to within specified PCL. Since it is physically impossible to satisfy all constraints simultaneously, each constraint uses a variable to either record when the coverage level is achieved, or to record the deviation from the desired coverage level. These additional variables are embedded into an objective function to be minimized.
+
+%\noindent In this paper, let us define some parameters, which are used in our protocol.
+%the set of segment points is denoted by $I$, the set of all sensors in the network by $J$, and the set of alive sensors within $J$ by $K$.
+
+
+%\noindent \begin{equation}
+%X_{k} = \left \{
+%\begin{array}{l l}
+ % 1& \mbox{if sensor $k$ is active,} \\
+% 0 & \mbox{otherwise.}\\
+%\end{array} \right.
+%\label{eq11}
+%\notag
+%\end{equation}
+
+%\noindent $M^j_i (undercoverage): $ integer value $\in \mathbb{N}$ for segment point $i$ of sensor $j$.
+
+%\noindent $V^j_i (overcoverage): $ integer value $\in \mathbb{N}$ for segment point $i$ of sensor $j$.
+
+
+
+
\noindent Our coverage optimization problem can be mathematically formulated 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 )&\\
+\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 K} ( a^j_{ik} ~ X_{k}) + M^j_i \geq 1 \\
+\sum_{k \in A} ( a^j_{ik} ~ X_{k}) + M^j_i \geq l \forall i \in I_j, \forall j \in S\\
%\label{c1}
-\sum_{k \in K} ( a^j_{ik} ~ X_{k}) - V^j_i \leq 1 \\
+\sum_{k \in A} ( a^j_{ik} ~ X_{k}) - V^j_i \leq l \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 K
+X_{k} \in \{0,1\}, \forall k \in A
\end{array}
\right.
\end{equation}
-The first group of constraints indicates that some segment points $i$
-should be covered by at least one sensor node and, if it is not always the
-case, overcoverage and undercoverage variables help balancing the
-restriction equations by taking positive values. There are two main
-objectives. First, we limit the overcoverage of segment points in order to
-activate a minimum number of sensors. Second, we prevent the absence of monitoring on
-some parts of the subregion by minimizing the undercoverage. The
-weights $\alpha$ and $\beta$ must be properly chosen so as to
-guarantee that the maximum number of segment points are covered during each round.
+
+$\alpha^j_i$ 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 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 \ref{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{\uppercase{PERFORMANCE EVALUATION AND ANALYSIS}}
\end{table}
Simulations with five different node densities going from 100 to 250~nodes were
performed considering each time 25~randomly generated networks, to obtain
-experimental results which are relevant.All simulations are repeated 25 times and the results are averaged. 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.
+experimental results which are relevant. All simulations are repeated 25 times and the results are averaged. 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 it's energy provision reaches a value below the
network lifetime. The Active Sensors Ratio is defined as follows:
\begin{equation*}
\scriptsize
-\mbox{ASR}(\%) = \frac{\sum\limits_{r=1}^R \mbox{$A_r^p$}}{\mbox{$S$}} \times 100 .
+\mbox{ASR}(\%) = \frac{\sum\limits_{r=1}^R \mbox{$A_r$}}{\mbox{$S$}} \times 100 .
\end{equation*}
-Where: $A_r^t$ is the number of active sensors in the subregion $r$ during period $p$ in the current sensing stage, $S$ is the total number of sensors in the network, and $R$ is the total number of the subregions in the network.
+Where: $A_r^t$ is the number of active sensors in the subregion $r$ in the current sensing stage, $S$ is the total number of sensors in the network, and $R$ is the total number of the subregions in the network.
%\end{enumerate}
\subsection{Simulation Results}
-In this section, we present the simulation results of LiCO protocol and the other protocols using a discrete event simulator OMNeT++ \cite{varga} to run different series of simulations. We implemented all protocols precisely on a laptop DELL with Intel Core~i3~2370~M (2.4 GHz) processor (2 cores) and the MIPS (Million Instructions Per Second) rate equal to 35330. To be consistent with the use of a sensor node with Atmels AVR ATmega103L microcontroller (6 MHz) and a MIPS rate equal to 6, the original execution time on the laptop is multiplied by 2944.2 $\left(\frac{35330}{2} \times \frac{1}{6} \right)$ so as to use it by the energy consumption model especially, after the computation and listening.
+In this section, we present the simulation results of LiCO protocol and the other protocols using a discrete event simulator OMNeT++ \cite{varga} to run different series of simulations. We implemented all protocols precisely on a laptop DELL with Intel Core~i3~2370~M (2.4 GHz) processor (2 cores) and the MIPS (Million Instructions Per Second) rate equal to 35330. To be consistent with the use of a sensor node with Atmels AVR ATmega103L microcontroller (6 MHz) and a MIPS rate equal to 6, the original execution time on the laptop is multiplied by 2944.2 $\left(\frac{35330}{2} \times \frac{1}{6} \right)$ so as to use it by the energy consumption model especially, after the computation and listening. Employing the modeling language ????\ref{}, the associated integer program instance is generated 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.
We compared LiCO protocol to three other approaches: the first, called DESK and proposed by ~\cite{ChinhVu} is a fully distributed coverage algorithm; the second, called GAF ~\cite{xu2001geography}, consists in dividing the region
into fixed squares. During the decision phase, in each square, one sensor is
