\begin{equation}
a^j_{ik} = \left \{
\begin{array}{lll}
- 1 & \mbox{If the sensor $k$ is involved in the } \\
+ 1 & \mbox{if the sensor $k$ is involved in the } \\
& \mbox{coverage interval $i$ of sensor $j$}, \\
0 & \mbox{Otherwise.}\\
\end{array} \right.
\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.
+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 level of coverage for all covergae intervals. 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.
\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 \forall i \in I_j, \forall j \in S\\
+\sum_{k \in A} ( a^j_{ik} ~ X_{k}) + M^j_i \geq l \quad \forall i \in I_j, \forall j \in S\\
%\label{c1}
-\sum_{k \in A} ( a^j_{ik} ~ X_{k}) - V^j_i \leq l \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\\
% \label{c2}
% \Theta_{p}\in \mathbb{N}, &\forall p \in P\\
% U_{p} \in \{0,1\}, &\forall p \in P\\
\end{equation}
-$\alpha^j_i$ and $\beta^j_i$ are nonnegative weights selected according to the
+\noindent $\alpha^j_i$ 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
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
-chosen to remain active during the sensing phase; the third, DiLCO protocol~\cite{Idrees2}, which is improved version on the work in ~\cite{idrees2014coverage}.
+chosen to remain active during the sensing phase; the third, DiLCO protocol~\cite{Idrees2} is an improved version on the work presented in ~\cite{idrees2014coverage}. DNote that the LiCO protocol is based on the same framework as that of DiLCO. For thes two protocols, the division of the region of interest in 16 subregions was chosen since it produces the best results. The difference between the two protocols relies on the use of the integer programming to provide the set of sensors that have to be actived in each sensing phase. Whereas DilCO protocol tries to satisfy the coverage of a set of primary points, LiCO protocol tries to reach a desired level of coverage $l$ for each sensor's perimeter. In the experimentations, we chose a level of coverage equal to 1 ($l=1$).
\subsubsection{\textbf{Coverage Ratio}}
-In this experiment, Figure~\ref{fig333} shows the average coverage ratio for 200 deployed nodes.
+Figure~\ref{fig333} shows the average coverage ratio for 200 deployed nodes obtained with the four methods.
\parskip 0pt
\begin{figure}[h!]
\label{fig333}
\end{figure}
-It is shown that DESK, GAF, and DiLCO-16 provides a little better coverage ratio with 99.99\%, 99.91\%, and 99.02\% against 98.76\% produced by LiCO for the lowest number of rounds. This is due to the fact that DiLCO-16 protocol put in sleep mode redundant sensors using optimization (which lightly decreases the coverage ratio) while there are more nodes are active in the case of DESK and GAF, and a little higher in comparison with the optimization algorithm used by LiCO.
+DESK, GAF, and DiLCO provides a little better coverage ratio with 99.99\%, 99.91\%, and 99.02\% against 98.76\% produced by LiCO for the lowest number of periods. This is due to the fact that DiLCO protocol put in sleep mode redundant sensors using optimization (which lightly decreases the coverage ratio) while there are more active nodes in the case of others methods. But when the number of periods exceeds 70 periods, it clearly appears that LiCO 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).
-Moreover, when the number of rounds increases, coverage ratio produced by DESK and GAF protocols decreases. This is due to dead nodes. However, DiLCO-16 protocol maintains almost a good coverage from the round 31 to the round 63 and it is close to LiCO protocol. This is because it optimizes the coverage and the lifetime in WSN based on the primary points by selecting the best representative sensor nodes for the sensing stage. LiCO protocol put in sleep mode a higher number of redundant sensors starting from the round 19 using the new optimization model. The coverage ratio of LiCO Protocol seems to be better than other approaches starting from the round 64 because the optimization algorithm used by LiCO has been optimized the lifetime coverage based on the perimeter coverage model, so it provided acceptable coverage for a larger number of periods and prolonging the network lifetime based on the perimeter of the sensor nodes in each subregion of WSN. Although some nodes are dead, sensor activity scheduling based optimization of LiCO selected another nodes to ensure the coverage of the area of interest. i.e. DiLCO-16 showed a good coverage in the beginning then LiCO, when the number of periods increases, the coverage ratio decreases due to died sensor nodes. Meanwhile, thanks to sensor activity scheduling based new optimization model, which is used by LiCO protocol to ensure a longer lifetime coverage in comparison with other approaches.
+%When the number of periods increases, coverage ratio produced by DESK and GAF protocols decreases. This is due to dead nodes. However, DiLCO protocol maintains almost a good coverage from the round 31 to the round 63 and it is close to LiCO protocol. The coverage ratio of LiCO protocol is better than other approaches from the period 64.
+
+%because the optimization algorithm used by LiCO has been optimized the lifetime coverage based on the perimeter coverage model, so it provided acceptable coverage for a larger number of periods and prolonging the network lifetime based on the perimeter of the sensor nodes in each subregion of WSN. Although some nodes are dead, sensor activity scheduling based optimization of LiCO selected another nodes to ensure the coverage of the area of interest. i.e. DiLCO-16 showed a good coverage in the beginning then LiCO, when the number of periods increases, the coverage ratio decreases due to died sensor nodes. Meanwhile, thanks to sensor activity scheduling based new optimization model, which is used by LiCO protocol to ensure a longer lifetime coverage in comparison with other approaches.
\subsubsection{\textbf{Active Sensors Ratio}}
-It is important to have as few active nodes as possible in each period, in order to minimize the energy consumption and maximize the network lifetime. Figure~\ref{fig444} shows the average active nodes ratio for 200 deployed nodes.
+Having active nodes as few as possible in each period is essential in order to minimize the energy consumption and so maximize the network lifetime. Figure~\ref{fig444} shows the average active nodes ratio for 200 deployed nodes.
\begin{figure}[h!]
\centering
\label{fig444}
\end{figure}
-From figure~~\ref{fig444}, We observed that DESK and GAF have 30.36 \% and 34.96 \% active nodes for the first fourteen rounds and DiLCO-16 and LiCO protocols competes perfectly with only 17.92 \% and 20.16 \% active nodes for the first 17 rounds. Then as the number of rounds increases our LiCO protocol has a lower number of active nodes in comparison with DiLCO-16, DESK and GAF, especially from the round $19^{th}$ because it optimizes the lifetime coverage into the subregion based on the perimeter coverage model, which made LiCO improves the coverage ratio and for a longer time in comparison with other approaches.
+We observe that DESK and GAF have 30.36 \% and 34.96 \% active nodes for the first fourteen rounds and DiLCO and LiCO protocols compete perfectly with only 17.92 \% and 20.16 \% active nodes during the same time interval. As the number of periods increases, LiCO protocol has a lower number of active nodes in comparison with the three other approaches, while keeping of greater coverage ratio as shown in figure \ref{fig333}.
\subsubsection{\textbf{The Energy Consumption}}
-In this experiment, we study the effect of the energy consumed by the WSN during the communication, computation, listening, active, and sleep modes for different network densities and compare it with other approaches. Figures~\ref{fig3EC95} and ~\ref{fig3EC50} illustrate the energy consumption for different network sizes for $Lifetime95$ and $Lifetime50$.
+We study the effect of the energy consumed by the WSN during the communication, computation, listening, active, and sleep modes for different network densities and compare it for the four approaches. Figures~\ref{fig3EC95} and ~\ref{fig3EC50} illustrate the energy consumption for different network sizes and for $Lifetime95$ and $Lifetime50$.
\begin{figure}[h!]
\centering
\label{fig3EC50}
\end{figure}
-The results show that our LiCO protocol is the most competitive from the energy consumption point of view. As shown in figures Figures~\ref{fig3EC95} and ~\ref{fig3EC50}, LiCO consumes less energy especially when the network size increases because it puts in sleep mode less active sensor number as possible in most periods of the network lifetime. The optimization algorithm, which used by LiCO protocol, was improved the lifetime coverage efficiently based on the perimeter coverage model.
+The results show that our LiCO protocol is the most competitive from the energy consumption point of view. As shown in figures~\ref{fig3EC95} and ~\ref{fig3EC50}, LiCO 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 this optimization program allows to reduce significantly the number of active sensors and so the energy consumption while keeping a good coverage level.
+%The optimization algorithm, which used by LiCO protocol, was improved the lifetime coverage efficiently based on the perimeter coverage model.
- The other approaches have a high energy consumption due to activating a larger number of redundant nodes as well as the energy consumed during the different modes of sensor nodes. In fact, a distributed method on the subregions greatly reduces the number of communications and the time of listening so thanks to the partitioning of the initial network into several independent subnetworks.
+ %The other approaches have a high energy consumption due to activating a larger number of sensors. In fact, a distributed method on the subregions greatly reduces the number of communications and the time of listening so thanks to the partitioning of the initial network into several independent subnetworks.
%\subsubsection{Execution Time}
\subsubsection{\textbf{The Network Lifetime}}
-In this experiment, we are observed the superiority of LiCO and DiLCO-16 protocols against other two approaches in prolonging the network lifetime. In figures~\ref{fig3LT95} and \ref{fig3LT50}, network lifetime, $Lifetime95$ and $Lifetime50$ respectively, are illustrated for different network sizes.
+We observe the superiority of LiCO and DiLCO protocols against other two approaches in prolonging the network lifetime. In figures~\ref{fig3LT95} and \ref{fig3LT50}, network lifetime, $Lifetime95$ and $Lifetime50$ respectively, are illustrated for different network sizes.
\begin{figure}[h!]
\centering
\label{fig3LT50}
\end{figure}
-As highlighted by figures~\ref{fig3LT95} and \ref{fig3LT50}, the network lifetime obviously increases when the size of the network increases, with our LiCO and DiLCO-16 protocols that leads to maximize the lifetime of the network compared with other approaches.
+As highlighted by figures~\ref{fig3LT95} and \ref{fig3LT50}, the network lifetime obviously increases when the size of the network increases, and it is clearly larger with DiLCO and LiCO protocols compared with the two other methods. For instance, for a network of 300 sensors, the coverage ratio is greater than 50\% about two times longer with LiCO compared to DESK method.
-By choosing the best suited nodes, for each round, by optimizing the coverage and lifetime of the network to cover the area of interest and by letting the other ones sleep in order to be used later in next rounds, LiCO protocol efficiently prolonged the network lifetime especially for a coverage ratio greater than $50 \%$, whilst it stayed very near to DiLCO-16 protocol for $95 \%$. Figure~\ref{figLTALL} introduces the comparisons of the lifetime coverage for different coverage ratios between LiCO and DiLCO-16 protocols.
+%By choosing the best suited nodes, for each period, by optimizing the coverage and lifetime of the network to cover the area of interest and by letting the other ones sleep in order to be used later in next rounds, LiCO protocol efficiently prolonged the network lifetime especially for a coverage ratio greater than $50 \%$, whilst it stayed very near to DiLCO-16 protocol for $95 \%$.
+Figure~\ref{figLTALL} introduces the comparisons of the lifetime coverage for different coverage ratios for LiCO and DiLCO protocols.
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.
\begin{figure}[h!]
\label{figLTALL}
\end{figure}
-Comparison shows that LiCO protocol, which are used distributed optimization over the subregions, is the more relevance one for most coverage ratios and WSN sizes because it is robust to network disconnection during the network lifetime as well as it consume less energy in comparison with other approaches. LiCO protocol gave acceptable coverage ratio for a larger number of periods using new optimization algorithm that based on a perimeter coverage model. It also means that distributing the algorithm in each node and subdividing the sensing field into many subregions, which are managed independently and simultaneously, is the most relevant way to maximize the lifetime of a network.
+
+%Comparison shows that LiCO protocol, which are used distributed optimization over the subregions, is the more relevance one for most coverage ratios and WSN sizes because it is robust to network disconnection during the network lifetime as well as it consume less energy in comparison with other approaches. LiCO protocol gave acceptable coverage ratio for a larger number of periods using new optimization algorithm that based on a perimeter coverage model. It also means that distributing the algorithm in each node and subdividing the sensing field into many subregions, which are managed independently and simultaneously, is the most relevant way to maximize the lifetime of a network.
\section{\uppercase{Conclusion and Future Works}}
\label{sec:Conclusion and Future Works}
-In this paper, we have studied the problem of lifetime coverage optimization in
-WSNs. To cope with this problem, the area of interest is divided into a smaller subregions using divide-and-conquer method, and then a LiCO protocol for optimizing the lifetime coverage in each subregion. LiCO protocol combines two efficient techniques: the first, network
-leader election, which executes the perimeter coverage model (only one time), the optimization algorithm, and sending the schedule produced by the optimization algorithm to other nodes in the subregion ; the second, sensor activity scheduling based optimization in which a new lifetime coverage optimization model is proposed. The main challenges include how to select the most efficient leader in each subregion and the best schedule of sensor nodes that will optimize the network lifetime coverage
-in the subregion. The network lifetime coverage in each subregion is divided into
-periods, each period consists of four stages: (i) Information Exchange,
-(ii) Leader Election, (iii) a Decision based new optimization model in order to
-select the nodes remaining active for the last stage, and (iv) Sensing.
-The simulation results show that LiCO is is more energy-efficient than other approaches, with respect to lifetime, coverage ratio, active sensors ratio, and energy consumption. Indeed, when dealing with large and dense WSNs, a distributed optimization approach on the subregions of WSN like the one we are proposed allows to reduce the difficulty of a single global optimization problem by partitioning it in many smaller problems, one per subregion, that can be solved more easily.
-
-Our future work is four-fold: the first, we plan to extend a lifetime coverage optimization problem in order to computes all active sensor schedules in only one step for many periods;
+In this paper we have studied the problem of lifetime coverage optimization in
+WSNs. We designed a protocol LiCO that schedules node activities (wakeup and sleep) with the objective of maintaining a good coverage ratio while maximizing the network lifetime. This protocol is applied on each subregion of the area of interest. It works in periods and is based on the resolution of an integer program to select the subset of sensors operating in active mode 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 carried out severals simulations to evaluate the proposed protocol.
+
+
+%To cope with this problem, the area of interest is divided into a smaller subregions using divide-and-conquer method, and then a LiCO protocol for optimizing the lifetime coverage in each subregion. LiCO protocol combines two efficient techniques: network
+%leader election, which executes the perimeter coverage model (only one time), the optimization algorithm, and sending the schedule produced by the optimization algorithm to other nodes in the subregion ; the second, sensor activity scheduling based optimization in which a new lifetime coverage optimization model is proposed. The main challenges include how to select the most efficient leader in each subregion and the best schedule of sensor nodes that will optimize the network lifetime coverage
+%in the subregion.
+%The network lifetime coverage in each subregion is divided into
+%periods, each period consists of four stages: (i) Information Exchange,
+%(ii) Leader Election, (iii) a Decision based new optimization model in order to
+%select the nodes remaining active for the last stage, and (iv) Sensing.
+The simulation results show that LiCO is is more energy-efficient than other approaches, with respect to lifetime, coverage ratio, active sensors ratio, and energy consumption.
+%Indeed, when dealing with large and dense WSNs, a distributed optimization approach on the subregions of WSN like the one we are proposed allows to reduce the difficulty of a single global optimization problem by partitioning it in many smaller problems, one per subregion, that can be solved more easily.
+
+We plan to extend a lifetime coverage optimization problem in order to computes all active sensor schedules in only one step for many periods;
the second, we focus on extend our protocol and optimization algorithm to take into account the heterogeneous sensors, which do not have the same energy, processing, sensing and communication capabilities;
-the third, we are investigating new optimization model based on the sensing range so as to maximize the lifetime coverage in WSN;
+%the third, we are investigating new optimization model based on the sensing range so as to maximize the lifetime coverage in WSN;
Finally, our final goal is to implement our protocol using a sensor-testbed to evaluate their performance in real world applications.
\section*{\uppercase{Acknowledgements}}
