%% \author[label1,label2]{}
%% \address[label1]{}
%% \address[label2]{}
-\author{Ali Kadhum Idrees, Karine Deschinkel, \\
-Michel Salomon, and Rapha\"el Couturier}
+%\author{Ali Kadhum Idrees, Karine Deschinkel, \\
+%Michel Salomon, and Rapha\"el Couturier}
+
%\thanks{are members in the AND team - DISC department - FEMTO-ST Institute, University of Franche-Comt\'e, Belfort, France.
% e-mail: ali.idness@edu.univ-fcomte.fr, $\lbrace$karine.deschinkel, michel.salomon, raphael.couturier$\rbrace$@univ-fcomte.fr.}% <-this % stops a space
%\thanks{}% <-this % stops a space
-\address{FEMTO-ST Institute, University of Franche-Comt\'e, Belfort, France. \\
-e-mail: ali.idness@edu.univ-fcomte.fr, \\
-$\lbrace$karine.deschinkel, michel.salomon, raphael.couturier$\rbrace$@univ-fcomte.fr.}
+%\address{FEMTO-ST Institute, University of Franche-Comt\'e, Belfort, France. \\
+%e-mail: ali.idness@edu.univ-fcomte.fr, \\
+%$\lbrace$karine.deschinkel, michel.salomon, raphael.couturier$\rbrace$@univ-fcomte.fr.}
+
+
+\author{Ali Kadhum Idrees$^{a,b}$, Karine Deschinkel$^{a}$, \\
+Michel Salomon$^{a}$ and Rapha\"el Couturier $^{a}$ \\
+ $^{a}${\em{FEMTO-ST Institute, UMR 6174 CNRS, \\
+ University Bourgogne Franche-Comt\'e, Belfort, France}} \\
+ $^{b}${\em{Department of Computer Science, University of Babylon, Babylon, Iraq}}
+}
+
\begin{abstract}
%One of the fundamental challenges in Wireless Sensor Networks (WSNs)
\end{abstract}
\begin{keyword}
-Wireless Sensor Networks, Area Coverage, Network lifetime,
+Wireless Sensor Networks, Area Coverage, Network Lifetime,
Optimization, Scheduling, Distributed Computation.
\end{keyword}
Instead of working with a continuous coverage area, we make it discrete by
considering for each sensor a set of points called primary points. Consequently,
we assume that the sensing disk defined by a sensor is covered if all of its
-primary points are covered. The choice of number and locations of primary points
-is the subject of another study not presented here.
+primary points are covered. The choice of number and locations of primary points is the subject of another study not presented here.
%By knowing the position (point center: ($p_x,p_y$)) of a wireless
%sensor node and its $R_s$, we calculate the primary points directly
\subsection{Decision phase}
-Each WSNL will solve an integer program to select which cover sets will be
+Each WSNL will \textcolor{red}{ execute an optimization algorithm (see section \ref{oa})} to select which cover sets will be
activated in the following sensing phase to cover the subregion to which it
-belongs. The integer program will produce $T$ cover sets, one for each round.
-The WSNL will send an Active-Sleep packet to each sensor in the subregion based
-on the algorithm's results, indicating if the sensor should be active or not in
-each round of the sensing phase. The integer program is based on the model
+belongs. The \textcolor{red}{optimization algorithm} will produce $T$ cover sets, one for each round. The WSNL will send an Active-Sleep packet to each sensor in the subregion based on the algorithm's results, indicating if the sensor should be active or not in
+each round of the sensing phase.
+
+%solve an integer program
+
+\subsection{Sensing phase}
+
+The sensing phase consists of $T$ rounds. Each sensor node in the subregion will
+receive an Active-Sleep packet from WSNL, informing it to stay awake or to go to
+sleep for each round of the sensing phase. Algorithm~\ref{alg:MuDiLCO}, which
+will be executed by each node at the beginning of a period, explains how the
+Active-Sleep packet is obtained.
+
+% In each round during the sensing phase, there is a cover set of sensor nodes, in which the active sensors will execute their sensing task to preserve maximal coverage and lifetime in the subregion and this will continue until finishing the round $T$ and starting new period.
+
+\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{ $RE_j \geq E_{R}$ }{
+ \emph{$s_j.status$ = COMMUNICATION}\;
+ \emph{Send $INFO()$ packet to other nodes in the subregion}\;
+ \emph{Wait $INFO()$ packet from other nodes in the subregion}\;
+ %\emph{UPDATE $RE_j$ for every sent or received INFO Packet}\;
+ %\emph{ Collect information and construct the list L for all nodes in the subregion}\;
+
+ %\If{ the received INFO Packet = No. of nodes in it's subregion -1 }{
+ \emph{LeaderID = Leader election}\;
+ \If{$ s_j.ID = LeaderID $}{
+ \emph{$s_j.status$ = COMPUTATION}\;
+ \emph{$\left\{\left(X_{1,k},\dots,X_{T,k}\right)\right\}_{k \in J}$ =
+ Execute \textcolor{red}{Optimization Algorithm}($T,J$)}\;
+ \emph{$s_j.status$ = COMMUNICATION}\;
+ \emph{Send $ActiveSleep()$ to each node $k$ in subregion a packet \\
+ with vector of activity scheduling $(X_{1,k},\dots,X_{T,k})$}\;
+ \emph{Update $RE_j $}\;
+ }
+ \Else{
+ \emph{$s_j.status$ = LISTENING}\;
+ \emph{Wait $ActiveSleep()$ packet from the Leader}\;
+ % \emph{After receiving Packet, Retrieve the schedule and the $T$ rounds}\;
+ \emph{Update $RE_j $}\;
+ }
+ % }
+ }
+ \Else { Exclude $s_j$ from entering in the current sensing phase}
+
+ % \emph{return X} \;
+\caption{MuDiLCO($s_j$)}
+\label{alg:MuDiLCO}
+
+\end{algorithm}
+
+
+
+
+
+
+\section{\textcolor{red}{ Optimization Algorithm for Multiround Lifetime Coverage Optimization}}
+\label{oa}
+As shown in Algorithm~\ref{alg:MuDiLCO}, the leader will execute an optimization algorithm based on an integer program. The integer program is based on the model
proposed by \cite{pedraza2006} with some modifications, where the objective is
to find a maximum number of disjoint cover sets. To fulfill this goal, the
authors proposed an integer program which forces undercoverage and overcoverage
large compared to $W_{\theta}$.
%The Active-Sleep packet includes the schedule vector with the number of rounds that should be applied by the receiving sensor node during the sensing phase.
-\subsection{Sensing phase}
+This integer program can be solved using two approaches:
-The sensing phase consists of $T$ rounds. Each sensor node in the subregion will
-receive an Active-Sleep packet from WSNL, informing it to stay awake or to go to
-sleep for each round of the sensing phase. Algorithm~\ref{alg:MuDiLCO}, which
-will be executed by each node at the beginning of a period, explains how the
-Active-Sleep packet is obtained.
+\subsection{Optimization solver for Multiround Lifetime Coverage Optimization}
+\label{glpk}
+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. We named the protocol which is based on GLPK solver in the decision phase as MuDiLCO.
-% In each round during the sensing phase, there is a cover set of sensor nodes, in which the active sensors will execute their sensing task to preserve maximal coverage and lifetime in the subregion and this will continue until finishing the round $T$ and starting new period.
+
+%\textcolor{red}{\textbf{\textsc{Answer:} ali }}
+
+
+\subsection{Genetic Algorithm (GA) for Multiround Lifetime Coverage Optimization}
+\label{GA}
+Metaheuristics are a generic search strategies for exploring search spaces for solving the complex problems. These strategies have to dynamically balance between the exploitation of the accumulated search experience and the exploration of the search space. On one hand, this balance can find regions in the search space with high-quality solutions. On the other hand, it prevents waste too much time in regions of the search space which are either already explored or don’t provide high-quality solutions. Therefore, metaheuristic provides an enough good solution to an optimization problem, especially with incomplete information or limited computation capacity \cite{bianchi2009survey}. Genetic Algorithm (GA) is one of the population-based metaheuristic methods that simulates the process of natural selection \cite{hassanien2015applications}. GA starts with a population of random candidate solutions (called individuals or phenotypes) . GA uses genetic operators inspired by natural evolution, such as selection, mutation, evaluation, crossover, and replacement so as to improve the initial population of candidate solutions. This process repeated until a stopping criterion is satisfied. Compared to GLPK optimization solver, GA provides a near optimal solution with acceptible execution time, while GLPK provides optimal solution but it requires high execution time for large problem.
+
+In this section, we present a metaheuristic based GA to solve our multiround lifetime coverage optimization problem. The proposed GA provides a near optimal sechedule for multiround sensing per period. The proposed GA is based on the mathematical model which is presented in Section \ref{oa}. Algorithm \ref{alg:GA} shows the proposed GA to solve the coverage lifetime optimization problem. We named the new protocol which is based on GA in the decision phase as GA-MuDiLCO. The proposed GA can be explained in more details as follow:
\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)}
+ \small
+ \SetKwInput{Input}{Input}
+ \SetKwInput{Output}{Output}
+ \Input{ $ P, J, T, S_{pop}, \alpha_{j,p}^{ind}, X_{t,j}^{ind}, \Theta_{t,p}^{ind}, U_{t,p}^{ind}, Child_{t,j}^{ind}, Ch.\Theta_{t,p}^{ind}, Ch.U_{t,p}^{ind_1}$}
+ \Output{$\left\{\left(X_{1,1},\dots, X_{t,j}, \dots, X_{T,J}\right)\right\}_{t \in T, j \in J}$}
+
\BlankLine
%\emph{Initialize the sensor node and determine it's position and subregion} \;
+ \ForEach {Individual $ind$ $\in$ $S_{pop}$} {
+ \emph{Generate Randomly Chromosome $\left\{\left(X_{1,1},\dots, X_{t,j}, \dots, X_{T,J}\right)\right\}_{t \in T, j \in J}$}\;
+
+ \emph{Update O-U-Coverage $\left\{(P, J, \alpha_{j,p}^{ind}, X_{t,j}^{ind}, \Theta_{t,p}^{ind}, U_{t,p}^{ind})\right\}_{p \in P}$}\;
+
+
+ \emph{Evaluate Individual $(P, J, X_{t,j}^{ind}, \Theta_{t,p}^{ind}, U_{t,p}^{ind})$}\;
+ }
+
+ \While{ Stopping criteria is not satisfied }{
+
+ \emph{Selection $(ind_1, ind_2)$}\;
+ \emph{Crossover $(P_c, X_{t,j}^{ind_1}, X_{t,j}^{ind_2}, Child_{t,j}^{ind_1}, Child_{t,j}^{ind_2})$}\;
+ \emph{Mutation $(P_m, Child_{t,j}^{ind_1}, Child_{t,j}^{ind_2})$}\;
+
+
+ \emph{Update O-U-Coverage $(P, J, \alpha_{j,p}^{ind}, Child_{t,j}^{ind_1}, Ch.\Theta_{t,p}^{ind_1}, Ch.U_{t,p}^{ind_1})$}\;
+ \emph{Update O-U-Coverage $(P, J, \alpha_{j,p}^{ind}, Child_{t,j}^{ind_2}, Ch.\Theta_{t,p}^{ind_2}, Ch.U_{t,p}^{ind_2})$}\;
+
+\emph{Evaluate New Individual$(P, J, Child_{t,j}^{ind_1}, Ch.\Theta_{t,p}^{ind_1}, Ch.U_{t,p}^{ind_1})$}\;
+ \emph{Replacement $(P, J, T, Child_{t,j}^{ind_1}, Ch.\Theta_{t,p}^{ind_1}, Ch.U_{t,p}^{ind_1}, X_{t,j}^{ind}, \Theta_{t,p}^{ind}, U_{t,p}^{ind} )$ }\;
+
+ \emph{Evaluate New Individual$(P, J, Child_{t,j}^{ind_2}, Ch.\Theta_{t,p}^{ind_2}, Ch.U_{t,p}^{ind_2})$}\;
+
+ \emph{Replacement $(P, J, T, Child_{t,j}^{ind_2}, Ch.\Theta_{t,p}^{ind_2}, Ch.U_{t,p}^{ind_2}, X_{t,j}^{ind}, \Theta_{t,p}^{ind}, U_{t,p}^{ind} )$ }\;
- \If{ $RE_j \geq E_{R}$ }{
- \emph{$s_j.status$ = COMMUNICATION}\;
- \emph{Send $INFO()$ packet to other nodes in the subregion}\;
- \emph{Wait $INFO()$ packet from other nodes in the subregion}\;
- %\emph{UPDATE $RE_j$ for every sent or received INFO Packet}\;
- %\emph{ Collect information and construct the list L for all nodes in the subregion}\;
- %\If{ the received INFO Packet = No. of nodes in it's subregion -1 }{
- \emph{LeaderID = Leader election}\;
- \If{$ s_j.ID = LeaderID $}{
- \emph{$s_j.status$ = COMPUTATION}\;
- \emph{$\left\{\left(X_{1,k},\dots,X_{T,k}\right)\right\}_{k \in J}$ =
- Execute Integer Program Algorithm($T,J$)}\;
- \emph{$s_j.status$ = COMMUNICATION}\;
- \emph{Send $ActiveSleep()$ to each node $k$ in subregion a packet \\
- with vector of activity scheduling $(X_{1,k},\dots,X_{T,k})$}\;
- \emph{Update $RE_j $}\;
- }
- \Else{
- \emph{$s_j.status$ = LISTENING}\;
- \emph{Wait $ActiveSleep()$ packet from the Leader}\;
- % \emph{After receiving Packet, Retrieve the schedule and the $T$ rounds}\;
- \emph{Update $RE_j $}\;
- }
- % }
}
- \Else { Exclude $s_j$ from entering in the current sensing phase}
+ \emph{$\left\{\left(X_{1,1},\dots,X_{t,j},\dots,X_{T,J}\right)\right\}$ =
+ Select Best Solution ($S_{pop}$)}\;
+ \emph{return X} \;
+\caption{GA($T, J$)}
+\label{alg:GA}
+
+\end{algorithm}
+
+
+\begin{enumerate} [I)]
+\item \textbf{Representation:} Since the proposed GA's goal is to find the optimal schedule of the sensor nodes which take the responsibility of monitoring the subregion for $T$ rounds in the next phase, the chromosome is defined as a schedule for alive sensors and each chromosome contains $T$ rounds. Each round in the schedule includes J genes, the total alive sensors in the subregion. Therefore, the gene of such a chromosome is a schedule of a sensor. In other words, The genes corresponding to active nodes have the value of one, the others are zero. Figure \ref{chromo} shows solution representation in the proposed GA.
+%[scale=0.3]
+\begin{figure}[h!]
+\centering
+ \includegraphics [scale=0.35] {rep.pdf}
+\caption{Candidate Solution representation by the proposed GA. }
+\label{chromo}
+\end{figure}
+
+
+
+\item \textbf{Initialize Population:} The initial population is randomly generated and each chromosome in the GA population represents a possible sensors schedule solution to cover the entire subregion for $T$ rounds during current period. Each sensor in the chromosome is given a random value (0 or 1) for all rounds. If the random value is 1, the remaining energy of this sensor should be adequate to activate this sensor during current round. Otherwise, the value is set to 0. The energy constraint is applied for each sensor during all rounds.
+
+
+\item \textbf{Update O-U-Coverage:}
+After creating the initial population, The overcoverage $\Theta_{t,p}$ and undercoverage $U_{t,p}$ for each candidate solution are computed (see Algorithm \ref{OU}) so as to use them in the next step.
+
+\begin{algorithm}[h!]
- % \emph{return X} \;
-\caption{MuDiLCO($s_j$)}
-\label{alg:MuDiLCO}
+ \SetKwInput{Input}{Input}
+ \SetKwInput{Output}{Output}
+ \Input{ parameters $P, J, ind, \alpha_{j,p}^{ind}, X_{t,j}^{ind}$}
+ \Output{$U^{ind} = \left\lbrace U_{1,1}^{ind}, \dots, U_{t,p}^{ind}, \dots, U_{T,P}^{ind} \right\rbrace$ and $\Theta^{ind} = \left\lbrace \Theta_{1,1}^{ind}, \dots, \Theta_{t,p}^{ind}, \dots, \Theta_{T,P}^{ind} \right\rbrace$}
+
+ \BlankLine
+
+ \For{$t\leftarrow 1$ \KwTo $T$}{
+ \For{$p\leftarrow 1$ \KwTo $P$}{
+
+ % \For{$i\leftarrow 0$ \KwTo $I_j$}{
+ \emph{$SUM\leftarrow 0$}\;
+ \For{$j\leftarrow 1$ \KwTo $J$}{
+ \emph{$SUM \leftarrow SUM + (\alpha_{j,p}^{ind} \times X_{t,j}^{ind})$ }\;
+ }
+
+ \If { SUM = 0} {
+ \emph{$U_{t,p}^{ind} \leftarrow 0$}\;
+ \emph{$\Theta_{t,p}^{ind} \leftarrow 1$}\;
+ }
+ \Else{
+ \emph{$U_{t,p}^{ind} \leftarrow SUM -1$}\;
+ \emph{$\Theta_{t,p}^{ind} \leftarrow 0$}\;
+ }
+
+ }
+
+ }
+\emph{return $U^{ind}, \Theta^{ind}$ } \;
+\caption{O-U-Coverage}
+\label{OU}
\end{algorithm}
+
+
+\item \textbf{Evaluate Population:}
+After creating the initial population, each individual is evaluated and assigned a fitness value according to the fitness function is illustrated in Eq. \eqref{eqf}. In the proposed GA, the optimal (or near optimal) candidate solution, is the one with the minimum value for the fitness function. The lower the fitness values been assigned to an individual, the better opportunity it get survived. In our works, the function rewards the decrease in the sensor nodes which cover the same primary point and penalizes the decrease to zero in the sensor nodes which cover the primary point.
+
+\begin{equation}
+ F^{ind} \leftarrow \sum_{t=1}^{T} \sum_{p=1}^{P} \left(W_{\theta}* \Theta_{t,p} + W_{U} * U_{t,p} \right) \label{eqf}
+\end{equation}
+
+
+\item \textbf{Selection:} In order to generate a new generation, a portion of the existing population is elected based on a fitness function that ranks the fitness of each candidate solution and preferentially select the best solutions. Two parents should be selected to the mating pool. In the proposed GA-MuDiLCO algorithm, the first parent is selected by using binary tournament selection to select one of the parents \cite{goldberg1991comparative}. In this method, two individuals are chosen at random from population and the better of the two
+individuals is selected. If they have similar fitness values, one of them will be selected randomly. The best individual in the population is selected as a second parent.
+
+
+
+\item \textbf{Crossover:} Crossover is a genetic operator used to take more than one parent solutions and produce a child solution from them. If crossover probability $P_c$ is 100$\%$, then the crossover operation takes place between two individuals. If it is 0$\%$, the two selected individuals in the mating pool will be the new chromosomes without crossover. In the proposed GA, a two-point crossover is used. Figure \ref{cross} gives an example for a two-point crossover for 8 sensors in the subregion and the schedule for 3 rounds.
+
+
+\begin{figure}[h!]
+\centering
+ \includegraphics [scale = 0.3] {crossover.pdf}
+\caption{Two-point crossover. }
+\label{cross}
+\end{figure}
+
+
+\item \textbf{Mutation:}
+Mutation is a divergence operation which introduces random modifications. The purpose of the mutation is to maintain diversity within the population and prevent premature convergence. Mutation is used to add new genetic information (divergence) in order to achieve a global search over the solution search space and avoid to fall in local optima. The mutation oprator in the proposed GA-MuDiLCO works as follow: If mutation probability $P_m$ is 100$\%$, then the mutation operation takes place on the the new individual. The round number is selected randomly within (1..T) in the schedule solution. After that one sensor within this round is selected randomly within (1..J). If the sensor is scheduled as active "1", it should be rescheduled to sleep "0". If the sensor is scheduled as sleep, it rescheduled to active only if it has adequate remaining energy.
+
+
+\item \textbf{Update O-U-Coverage for children:}
+Before evalute each new individual, Algorithm \ref{OU} is called for each new individual to compute the new undercoverage $Ch.U$ and overcoverage $Ch.\Theta$ parameters.
+
+\item \textbf{Evaluate New Individuals:}
+Each new individual is evaluated using Eq. \ref{eqf} but with using the new undercoverage $Ch.U$ and overcoverage $Ch.\Theta$ parameters of the new children.
+
+\item \textbf{Replacement:}
+After evaluatation of new children, Triple Tournament Replacement (TTR) will be applied for each new individual. In TTR strategy, three individuals are selected
+randomly from the population. Find the worst from them and then check its fitness with the new individual fitness. If the fitness of the new individual is better than the fitness of the worst individual, replace the new individual with the worst individual. Otherwise, the replacement is not done.
+
+
+\item \textbf{Stopping criteria:}
+The proposed GA-MuDiLCO stops when the stopping criteria is met. It stops after running for an amount of time in seconds equal to \textbf{Time limit}. The \textbf{Time limit} is the execution time obtained by the optimization solver GLPK for solving the same size of problem divided by two. The best solution will be selected as a schedule of sensors for $T$ rounds during the sensing phase in the current period.
+
+
+
+\end{enumerate}
+
+
+
\section{Experimental study}
\label{exp}
\subsection{Simulation setup}
$E_{R}$ & 36 Joules\\
$R_s$ & 5~m \\
%\hline
-$W_{\Theta}$ & 1 \\
+$W_{\theta}$ & 1 \\
% [1ex] adds vertical space
%\hline
-$W_{U}$ & $|P|^2$
+$W_{U}$ & $|P|^2$ \\
+$P_c$ & 0.95 \\
+$P_m$ & 0.6 \\
+$S_{pop}$ & 50
%inserts single line
\end{tabular}
\label{table3}
% is used to refer this table in the text
\end{table}
-Our protocol is declined into four versions: MuDiLCO-1, MuDiLCO-3, MuDiLCO-5,
+\textcolor{red}{Our first protocol based GLPK optimization solver is declined into four versions: MuDiLCO-1, MuDiLCO-3, MuDiLCO-5,
and MuDiLCO-7, corresponding respectively to $T=1,3,5,7$ ($T$ the number of
-rounds in one sensing period). In the following, we will make comparisons with
+rounds in one sensing period). The second protocol based GA is declined into four versions: GA-MuDiLCO-1, GA-MuDiLCO-3, GA-MuDiLCO-5,
+and GA-MuDiLCO-7 for the same reason of the first protocol. After extensive experiments, we chose the dedicated values for the parameters $P_c$, $P_m$, and $S_{pop}$ because they gave the best results}. In the following, we will make comparisons with
two other methods. The first method, called DESK and proposed by \cite{ChinhVu},
is a full distributed coverage algorithm. The second method, called
GAF~\cite{xu2001geography}, consists in dividing the region into fixed squares.
message is obtained by using the equation in ~\cite{raghunathan2002energy} to
calculate the energy cost for transmitting messages and we propose the same
value for receiving the packets. The energy needed to send or receive a 1-bit
-packet is equal to $0.2575~mW$.
+packet is equal to 0.2575~mW.
The initial energy of each node is randomly set in the interval $[500;700]$. A
sensor node will not participate in the next round if its remaining energy is
% New version with global loops on period
\begin{equation*}
\scriptsize
- \mbox{EC} = \frac{\sum\limits_{m=1}^{M_L} \left[ \left( E^{\mbox{com}}_m+E^{\mbox{list}}_m+E^{\mbox{comp}}_m \right) +\sum\limits_{t=1}^{T_m} \left( E^{a}_t+E^{s}_t \right) \right]}{\sum\limits_{m=1}^{M_L} T_m},
+ \mbox{EC} = \frac{\sum\limits_{m=1}^{M} \left[ \left( E^{\mbox{com}}_m+E^{\mbox{list}}_m+E^{\mbox{comp}}_m \right) +\sum\limits_{t=1}^{T_m} \left( E^{a}_t+E^{s}_t \right) \right]}{\sum\limits_{m=1}^{M} T_m},
\end{equation*}
% Old version -> where $M_L$ and $T_L$ are respectively the number of periods and rounds during
%$Lifetime_{95}$ or $Lifetime_{50}$.
% New version
-where $M_L$ is the number of periods and $T_m$ the number of rounds in a
+where $M$ is the number of periods and $T_m$ the number of rounds in a
period~$m$, both during $Lifetime_{95}$ or $Lifetime_{50}$. The total energy
consumed by the sensors (EC) comes through taking into consideration four main
energy factors. The first one , denoted $E^{\scriptsize \mbox{com}}_m$,
before receiving the decision to go active or sleep in period $m$.
$E^{\scriptsize \mbox{comp}}_m$ refers to the energy needed by all the leader
nodes to solve the integer program during a period. Finally, $E^a_t$ and $E^s_t$
-indicate the energy consummed by the whole network in round $t$.
+indicate the energy consumed by the whole network in round $t$.
%\item {Network Lifetime:} we have defined the network lifetime as the time until all
%nodes have been drained of their energy or each sensor network monitoring an area has become disconnected.
\end{enumerate}
-\section{Results and analysis}
+\subsection{Results and analysis}
-\subsection{Coverage ratio}
+\subsubsection{Coverage ratio}
Figure~\ref{fig3} shows the average coverage ratio for 150 deployed nodes. We
can notice that for the first thirty rounds both DESK and GAF provide a coverage
\begin{figure}[ht!]
\centering
- \includegraphics[scale=0.5] {R1/CR.pdf}
+ \includegraphics[scale=0.5] {R/CR.pdf}
\caption{Average coverage ratio for 150 deployed nodes}
\label{fig3}
\end{figure}
-\subsection{Active sensors ratio}
+\textcolor{red}{ We
+can see that for the first thirty nine rounds GA-MuDiLCO provides a little bit better coverage ratio than MuDiLCO. Both DESK and GAF provide a coverage
+which is a little bit better than the one of MuDiLCO and GA-MuDiLCO for the first thirty rounds because they activate a larger number of nodes during sensing phase. After that GA-MuDiLCO provides a coverage ratio near to the MuDiLCO and better than DESK and GAF. GA-MuDiLCO gives approximate solution with activation a larger number of nodes than MuDiLCO during sensing phase while it activates a less number of nodes in comparison with both DESK and GAF. The results of GA-MuDiLCO seems to be comparable and can maintain the lifetime coverage as long as possible.}
+
+
+
+\subsubsection{Active sensors ratio}
It is crucial to have as few active nodes as possible in each round, in order to
-minimize the communication overhead and maximize the network
-lifetime. Figure~\ref{fig4} presents the active sensor ratio for 150 deployed
+minimize the communication overhead and maximize the network lifetime. Figure~\ref{fig4} presents the active sensor ratio for 150 deployed
nodes all along the network lifetime. It appears that up to round thirteen, DESK
and GAF have respectively 37.6\% and 44.8\% of nodes in ACTIVE status, whereas
-MuDiLCO clearly outperforms them with only 24.8\% of active nodes. After the
-thirty-fifth round, MuDiLCO exhibits larger numbers of active nodes, which agrees
-with the dual observation of higher level of coverage made previously.
-Obviously, in that case DESK and GAF have less active nodes, since they have
-activated many nodes at the beginning. Anyway, MuDiLCO activates the available
-nodes in a more efficient manner.
+MuDiLCO clearly outperforms them with only 24.8\% of active nodes. \textcolor{red}{GA-MuDiLCO activates a number of sensor nodes larger than MuDiLCO but lower than both DESK and GAF. GA-MuDiLCO-1, GA-MuDiLCO-3, and GA-MuDiLCO-5 continue in providing a larger number of active sensors until the forty-sixth round after that it provides less number of active nodes due to the died nodes. GA-MuDiLCO-7 provides a larger number of sensor nodes and maintains a better coverage ratio compared to MuDiLCO-7 until the fifty-seventh round. After the thirty-fifth round, MuDiLCO exhibits larger numbers of active nodes compared with DESK and GAF, which agrees with the dual observation of higher level of coverage made previously}.
+Obviously, in that case DESK and GAF have less active nodes, since they have activated many nodes at the beginning. Anyway, MuDiLCO activates the available nodes in a more efficient manner. \textcolor{red}{GA-MuDiLCO activates near optimal number of sensor nodes also in efficient manner compared with both DESK and GAF}.
\begin{figure}[ht!]
\centering
-\includegraphics[scale=0.5]{R1/ASR.pdf}
+\includegraphics[scale=0.5]{R/ASR.pdf}
\caption{Active sensors ratio for 150 deployed nodes}
\label{fig4}
\end{figure}
-\subsection{Stopped simulation runs}
+%\textcolor{red}{GA-MuDiLCO activates a sensor nodes larger than MuDiLCO but lower than both DESK and GAF }
+
+
+\subsubsection{Stopped simulation runs}
%The results presented in this experiment, is to show the comparison of our MuDiLCO protocol with other two approaches from the point of view the stopped simulation runs per round. Figure~\ref{fig6} illustrates the percentage of stopped simulation
%runs per round for 150 deployed nodes.
Figure~\ref{fig6} reports the cumulative percentage of stopped simulations runs
per round for 150 deployed nodes. This figure gives the breakpoint for each method. DESK stops first, after approximately 45~rounds, because it consumes the
more energy by turning on a large number of redundant nodes during the sensing
-phase. GAF stops secondly for the same reason than DESK. MuDiLCO overcomes
-DESK and GAF because the optimization process distributed on several subregions
-leads to coverage preservation and so extends the network lifetime. Let us
-emphasize that the simulation continues as long as a network in a subregion is
-still connected.
+phase. GAF stops secondly for the same reason than DESK. \textcolor{red}{GA-MuDiLCO stops thirdly for the same reason than DESK and GAF.} \textcolor{red}{MuDiLCO and GA-MuDiLCO overcome}
+DESK and GAF because \textcolor{red}{they activate less number of sensor nodes, as well as }the optimization process distributed on several subregions leads to coverage preservation and so extends the network lifetime.
+Let us emphasize that the simulation continues as long as a network in a subregion is still connected.
%%% The optimization effectively continues as long as a network in a subregion is still connected. A VOIR %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[ht!]
\centering
-\includegraphics[scale=0.5]{R1/SR.pdf}
+\includegraphics[scale=0.5]{R/SR.pdf}
\caption{Cumulative percentage of stopped simulation runs for 150 deployed nodes }
\label{fig6}
\end{figure}
-\subsection{Energy consumption} \label{subsec:EC}
+\subsubsection{Energy consumption} \label{subsec:EC}
We measure the energy consumed by the sensors during the communication,
listening, computation, active, and sleep status for different network densities
\begin{figure}[h!]
\centering
\begin{tabular}{cl}
- \parbox{9.5cm}{\includegraphics[scale=0.5]{R1/EC95.pdf}} & (a) \\
+ \parbox{9.5cm}{\includegraphics[scale=0.5]{R/EC95.pdf}} & (a) \\
\verb+ + \\
- \parbox{9.5cm}{\includegraphics[scale=0.5]{R1/EC50.pdf}} & (b)
+ \parbox{9.5cm}{\includegraphics[scale=0.5]{R/EC50.pdf}} & (b)
\end{tabular}
\caption{Energy consumption for (a) $Lifetime_{95}$ and
(b) $Lifetime_{50}$}
%In fact, a distributed optimization decision, which produces T rounds, on the subregions is greatly reduced the cost of communications and the time of listening as well as the energy needed for sensing phase and computation so thanks to the partitioning of the initial network into several independent subnetworks and producing T rounds for each subregion periodically.
-\subsection{Execution time}
+\subsubsection{Execution time}
We observe the impact of the network size and of the number of rounds on the
computation time. Figure~\ref{fig77} gives the average execution times in
-seconds (needed to solve optimization problem) for different values of $T$. The
+seconds (needed to solve optimization problem) for different values of $T$. The modeling language for Mathematical Programming (AMPL)~\cite{AMPL} is employed to generate the Mixed Integer Linear 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. The
original execution time is computed 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
\begin{figure}[ht!]
\centering
-\includegraphics[scale=0.5]{R1/T.pdf}
+\includegraphics[scale=0.5]{R/T.pdf}
\caption{Execution Time (in seconds)}
\label{fig77}
\end{figure}
%While MuDiLCO-1, 3, and 5 solves the optimization process with suitable execution times to be used on wireless sensor network because it distributed on larger number of small subregions as well as it is used acceptable number of round(s) T. We think that in distributed fashion the solving of the optimization problem to produce T rounds in a subregion can be tackled by sensor nodes. Overall, to be able to deal with very large networks, a distributed method is clearly required.
-\subsection{Network lifetime}
+\subsubsection{Network lifetime}
The next two figures, Figures~\ref{fig8}(a) and \ref{fig8}(b), illustrate the
network lifetime for different network sizes, respectively for $Lifetime_{95}$
\begin{figure}[t!]
\centering
\begin{tabular}{cl}
- \parbox{9.5cm}{\includegraphics[scale=0.5]{R1/LT95.pdf}} & (a) \\
+ \parbox{9.5cm}{\includegraphics[scale=0.5]{R/LT95.pdf}} & (a) \\
\verb+ + \\
- \parbox{9.5cm}{\includegraphics[scale=0.5]{R1/LT50.pdf}} & (b)
+ \parbox{9.5cm}{\includegraphics[scale=0.5]{R/LT50.pdf}} & (b)
\end{tabular}
\caption{Network lifetime for (a) $Lifetime_{95}$ and
(b) $Lifetime_{50}$}
The activity scheduling in each subregion works in periods, where each period
consists of four phases: (i) Information Exchange, (ii) Leader Election, (iii)
Decision Phase to plan the activity of the sensors over $T$ rounds, (iv) Sensing
-Phase itself divided into T rounds.
+Phase itself divided into $T$ rounds.
Simulations results show the relevance of the proposed protocol in terms of
lifetime, coverage ratio, active sensors ratio, energy consumption, execution
