sensor nodes in each subregion. The proposed MuDiLCO protocol works in periods
during which sets of sensor nodes are scheduled to remain active for a number of
rounds during the sensing phase, to ensure coverage so as to maximize the
-lifetime of WSN. The decision process is carried out by a leader node, which
-solves an integer program to produce the best representative sets to be used
-during the rounds of the sensing phase. \textcolor{red}{The integer program is solved by either GLPK solver or Genetic Algorithm (GA)}. Compared with some existing protocols,
+lifetime of WSN. \textcolor{green}{The decision process is carried out by a leader node, which
+solves an optimization problem to produce the best representative sets to be used
+during the rounds of the sensing phase. The optimization problem formulated as an integer program is solved to optimality through a branch-and-Bound method for small instances. For larger instances, the best feasible solution found by the solver after a given time limit threshold is considered. }
+%The decision process is carried out by a leader node, which
+%solves an integer program to produce the best representative sets to be used
+%during the rounds of the sensing phase.
+%\textcolor{red}{The integer program is solved by either GLPK solver or Genetic Algorithm (GA)}.
+Compared with some existing protocols,
simulation results based on multiple criteria (energy consumption, coverage
ratio, and so on) show that the proposed protocol can prolong efficiently the
network lifetime and improve the coverage performance.
\item Sensors scheduling algorithm implementation, i.e. centralized or
distributed/localized algorithms.
\item The objective of sensor coverage, i.e. to maximize the network lifetime or
- to minimize the number of sensors during a sensing round.
+ to minimize the number of active sensors during a sensing round.
\item The homogeneous or heterogeneous nature of the nodes, in terms of sensing
or communication capabilities.
\item The node deployment method, which may be random or deterministic.
low processing power from the sensor nodes, which usually have limited
processing capabilities. The main drawback of this kind of approach is its
higher cost in communications, since the node that will make the decision needs
-information from all the sensor nodes. Moreover, centralized approaches usually
-suffer from the scalability problem, making them less competitive as the network
-size increases.
+information from all the sensor nodes. \textcolor{green} {Exact or heuristics approaches are designed to provide cover sets.
+ %(Moreover, centralized approaches usually
+%suffer from the scalability problem, making them less competitive as the network
+%size increases.)
+Contrary to exact methods, heuristic methods can handle very large and centralized problems. They are proposed to reduce computational overhead such as energy consumption, delay and generally increase in
+the network lifetime. }
The first algorithms proposed in the literature consider that the cover sets are
disjoint: a sensor node appears in exactly one of the generated cover
selection algorithm to minimize the number of active nodes so as to prolong the
network lifetime. Various centralized methods based on column generation
approaches have also been
-proposed~\cite{castano2013column,rossi2012exact,deschinkel2012column}.
+proposed~\cite{gentili2013,castano2013column,rossi2012exact,deschinkel2012column}.
+\textcolor{green}{In~\cite{gentili2013}, authors highlight the trade-off between the network lifetime and the coverage percentage. They show that network lifetime can be hugely improved by decreasing the coverage ratio. }
\subsection{Distributed approaches}
%{\bf Distributed approaches}
possibility of dividing the sensing phase into multiple rounds and we also add
an improved model of energy consumption to assess the efficiency of our
approach. In fact, in this paper we make a multiround optimization, while it was
-a single round optimization in our previous work. \textcolor{red}{In addition, a metaheuristic based GA is proposed to solve our multiround optimization}.
+a single round optimization in our previous work. \textcolor{green}{The idea is to take advantage of the pre-sensing phase
+ to plan the sensor's activity for several rounds instead of one, thus saving energy. In addition, when the optimization problem becomes more complex, its resolution is stopped after a given time threshold}.
\iffalse
hypothesis, a complete coverage of a convex area implies connectivity among the
active nodes.
-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.
+%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.
+
+
+\indent Instead of working with the coverage area, we consider for each sensor a set of points called primary points~\cite{idrees2014coverage}. We assume that the sensing disk defined by a sensor is covered if all the primary points of this sensor are covered. By knowing the position (point center: ($p_x,p_y$)) of a wireless sensor node and it's sensing range $R_s$, we define up to 25 primary points $X_1$ to $X_{25}$ as decribed on Figure~\ref{fig1}. The coordinates of the primary points are the following :\\
+%$(p_x,p_y)$ = point center of wireless sensor node\\
+$X_1=(p_x,p_y)$ \\
+$X_2=( p_x + R_s * (1), p_y + R_s * (0) )$\\
+$X_3=( p_x + R_s * (-1), p_y + R_s * (0)) $\\
+$X_4=( p_x + R_s * (0), p_y + R_s * (1) )$\\
+$X_5=( p_x + R_s * (0), p_y + R_s * (-1 )) $\\
+$X_6= ( p_x + R_s * (\frac{-\sqrt{2}}{2}), p_y + R_s * (0)) $\\
+$X_7=( p_x + R_s * (\frac{\sqrt{2}}{2}), p_y + R_s * (0))$\\
+$X_8=( p_x + R_s * (\frac{-\sqrt{2}}{2}), p_y + R_s * (\frac{-\sqrt{2}}{2})) $\\
+$X_9=( p_x + R_s * (\frac{\sqrt{2}}{2}), p_y + R_s * (\frac{-\sqrt{2}}{2})) $\\
+$X_{10}=( p_x + R_s * (\frac{-\sqrt{2}}{2}), p_y + R_s * (\frac{\sqrt{2}}{2})) $\\
+$X_{11}=( p_x + R_s * (\frac{\sqrt{2}}{2}), p_y + R_s * (\frac{\sqrt{2}}{2})) $\\
+$X_{12}=( p_x + R_s * (0), p_y + R_s * (\frac{\sqrt{2}}{2})) $\\
+$X_{13}=( p_x + R_s * (0), p_y + R_s * (\frac{-\sqrt{2}}{2})) $\\
+$X_{14}=( p_x + R_s * (\frac{\sqrt{3}}{2}), p_y + R_s * (\frac{1}{2})) $\\
+$X_{15}=( p_x + R_s * (\frac{-\sqrt{3}}{2}), p_y + R_s * (\frac{1}{2})) $\\
+$X_{16}=( p_x + R_s * (\frac{\sqrt{3}}{2}), p_y + R_s * (\frac{- 1}{2})) $\\
+$X_{17}=( p_x + R_s * (\frac{-\sqrt{3}}{2}), p_y + R_s * (\frac{- 1}{2})) $\\
+$X_{18}=( p_x + R_s * (\frac{\sqrt{3}}{2}), p_y + R_s * (0)) $\\
+$X_{19}=( p_x + R_s * (\frac{-\sqrt{3}}{2}), p_y + R_s * (0)) $\\
+$X_{20}=( p_x + R_s * (0), p_y + R_s * (\frac{1}{2})) $\\
+$X_{21}=( p_x + R_s * (0), p_y + R_s * (-\frac{1}{2})) $\\
+$X_{22}=( p_x + R_s * (\frac{1}{2}), p_y + R_s * (\frac{\sqrt{3}}{2})) $\\
+$X_{23}=( p_x + R_s * (\frac{- 1}{2}), p_y + R_s * (\frac{\sqrt{3}}{2})) $\\
+$X_{24}=( p_x + R_s * (\frac{- 1}{2}), p_y + R_s * (\frac{-\sqrt{3}}{2})) $\\
+$X_{25}=( p_x + R_s * (\frac{1}{2}), p_y + R_s * (\frac{-\sqrt{3}}{2})) $.
+
+
+
+\begin{figure} %[h!]
+\centering
+ \begin{multicols}{2}
+\centering
+\includegraphics[scale=0.28]{fig21.pdf}\\~ (a)
+\includegraphics[scale=0.28]{principles13.pdf}\\~(c)
+\hfill \hfill
+\includegraphics[scale=0.28]{fig25.pdf}\\~(e)
+\includegraphics[scale=0.28]{fig22.pdf}\\~(b)
+\hfill \hfill
+\includegraphics[scale=0.28]{fig24.pdf}\\~(d)
+\includegraphics[scale=0.28]{fig26.pdf}\\~(f)
+\end{multicols}
+\caption{Wireless Sensor Node represented by (a) 5, (b) 9, (c) 13, (d) 17, (e) 21 and (f) 25 primary points respectively}
+\label{fig1}
+\end{figure}
+
+
+
+
+
%By knowing the position (point center: ($p_x,p_y$)) of a wireless
%sensor node and its $R_s$, we calculate the primary points directly
\subsection{Background idea}
%%RC : we need to clarify the difference between round and period. Currently it seems to be the same (for me at least).
-The area of interest can be divided using the divide-and-conquer strategy into
-smaller areas, called subregions, and then our MuDiLCO protocol will be
-implemented in each subregion in a distributed way.
+%The area of interest can be divided using the divide-and-conquer strategy into
+%smaller areas, called subregions, and then our MuDiLCO protocol will be
+%implemented in each subregion in a distributed way.
+
+\textcolor{green}{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 can be seen in Figure~\ref{fig2}, our protocol works in periods fashion,
where each is divided into 4 phases: Information~Exchange, Leader~Election,
Decision, and Sensing. Each sensing phase may be itself divided into $T$ rounds
-and for each round a set of sensors (a cover set) is responsible for the sensing
+\textcolor{green} {of equal duration} and for each round a set of sensors (a cover set) is responsible for the sensing
task. In this way a multiround optimization process is performed during each
period after Information~Exchange and Leader~Election phases, in order to
produce $T$ cover sets that will take the mission of sensing for $T$ rounds.
decision, the node will not participate to this phase, and, on the other hand,
if the node failure occurs after the decision, the sensing task of the network
will be temporarily affected: only during the period of sensing until a new
-period starts.
+period starts. \textcolor{green}{The duration of the rounds are predefined parameters. Round duration should be long enough to hide the system control overhead and short enough to minimize the negative effects in case of node failure.}
+
%%RC so if there are at least one failure per period, the coverage is bad...
%%MS if we want to be reliable against many node failures we need to have an
%% overcoverage...
\subsection{Decision phase}
-Each WSNL will \textcolor{red}{ execute an optimization algorithm (see section \ref{oa})} to select which cover sets will be
+Each WSNL will \textcolor{green}{ solve an integer program to select which cover sets will be
activated in the following sensing phase to cover the subregion to which it
-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.
+belongs. $T$ cover sets will be produced, 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. }
+%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 \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.
-% 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}
+%solve an integer program
-\end{algorithm}
-\section{\textcolor{red}{ Optimization Algorithm for Multiround Lifetime Coverage Optimization}}
-\label{oa}
+%\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
\end{equation}
\begin{equation}
- \sum_{t=1}^{T} X_{t,j} \leq \floor*{RE_{j}/E_{R}} \hspace{6 mm} \forall j \in J, t = 1,\dots,T
+ \sum_{t=1}^{T} X_{t,j} \leq \floor*{RE_{j}/E_{R}} \hspace{10 mm}\forall j \in J\hspace{6 mm}
\label{eq144}
\end{equation}
%% MS W_theta is smaller than W_u => problem with the following sentence
In our simulations priority is given to the coverage by choosing $W_{U}$ very
large compared to $W_{\theta}$.
+
+\textcolor{green}{The size of the problem depends on the number of variables and constraints. The number of variables is linked to the number of alive sensors $A \subset J$, the number of rounds $T$, and the number of primary points $P$. Thus the integer program contains $A*T$ variables of type $X_{t,j}$, $P*T$ overcoverage variables and $P*T$ undercoverage variables. The number of constraints is equal to $P*T$ (for constraints (\ref{eq16})) $+$ $A$ (for constraints (\ref{eq144})).}
%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}
+
+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}
+
+\iffalse
\textcolor{red}{This integer program can be solved using two approaches:}
\subsection{\textcolor{red}{Optimization solver for Multiround Lifetime Coverage Optimization}}
\label{glpk}
\textcolor{red}{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.}
+\fi
-
-
+\iffalse
\subsection{\textcolor{red}{Genetic Algorithm for Multiround Lifetime Coverage Optimization}}
\label{GA}
\item \textcolor{red}{\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.}
+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. The best solution will be selected as a schedule of sensors for $T$ rounds during the sensing phase in the current period.}
\end{enumerate}
-
+\fi
\section{Experimental study}
\label{exp}
% [1ex] adds vertical space
%\hline
$W_{U}$ & $|P|^2$ \\
-$P_c$ & 0.95 \\
-$P_m$ & 0.6 \\
-$S_{pop}$ & 50
+%$P_c$ & 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}
-
-\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). 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
+
+\textcolor{green}{The MuDilLCO protocol 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). Since the time resolution may be prohibitif when the size of the problem increases, a time limit treshold has been fixed to solve large instances. In these cases, the solver returns the best solution found, which is not necessary the optimal solution.
+ Table \ref{tl} shows time limit values. These time limit treshold have been set empirically. The basic idea consists in considering the average execution time to solve the integer programs to optimality, then by dividing this average time by three to set the threshold value. After that, this treshold value is increased if necessary such that the solver is able to deliver a feasible solution within the time limit. In fact, selecting the optimal values for the time limits will be investigated in future. In Table \ref{tl}, "NO" indicates that the problem has been solved to optimality without time limit. }.
+
+\begin{table}[ht]
+\caption{Time limit values for MuDiLCO protocol versions }
+\centering
+\begin{tabular}{|c|c|c|c|c|}
+ \hline
+ WSN size & MuDiLCO-1 & MuDiLCO-3 & MuDiLCO-5 & MuDiLCO-7 \\ [0.5ex]
+\hline
+ 50 & NO & NO & NO & NO \\
+ \hline
+100 & NO & NO & NO & NO \\
+\hline
+150 & NO & NO & NO & 0.03 \\
+\hline
+200 & NO & NO & NO & 0.06 \\
+ \hline
+ 250 & NO & NO & NO & 0.08 \\
+ \hline
+\end{tabular}
+
+\label{tl}
+
+\end{table}
+
+
+
+
+ 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.
\end{enumerate}
-\subsection{Results and analysis}
+\section{Results and analysis}
+\subsection{Performance Analysis for Different Number of Primary Points}
+\label{ch4:sec:04:06}
+
+In this section, we study the performance of MuDiLCO-1 approach for different numbers of primary points. The objective of this comparison is to select the suitable primary point model to be used by a MuDiLCO protocol. In this comparison, MuDiLCO-1 protocol is used with five models, which are called Model-5 (it uses 5 primary points), Model-9, Model-13, Model-17, and Model-21.
+
+
+%\begin{enumerate}[i)]
+
+%\item {{\bf Coverage Ratio}}
+\subsubsection{Coverage Ratio}
+
+Figure~\ref{Figures/ch4/R2/CR} shows the average coverage ratio for 150 deployed nodes.
+\parskip 0pt
+\begin{figure}[h!]
+\centering
+ \includegraphics[scale=0.5] {R2/CR.pdf}
+\caption{Coverage ratio for 150 deployed nodes}
+\label{Figures/ch4/R2/CR}
+\end{figure}
+As can be seen in Figure~\ref{Figures/ch4/R2/CR}, at the beginning the models which use a larger number of primary points provide slightly better coverage ratios, but latter they are the worst.
+%Moreover, when the number of periods increases, coverage ratio produced by Model-9, Model-13, Model-17, and Model-21 decreases in comparison with Model-5 due to a larger time computation for the decision process for larger number of primary points.
+Moreover, when the number of periods increases, coverage ratio produced by all models decrease, but Model-5 is the one with the slowest decrease due to a smaller time computation of decision process for a smaller number of primary points.
+As shown in Figure ~\ref{Figures/ch4/R2/CR}, coverage ratio decreases when the number of periods increases due to dead nodes. Model-5 is slightly more efficient than other models, because it offers a good coverage ratio for a larger number of periods in comparison with other models.
+
+
+%\item {{\bf Network Lifetime}}
+\subsubsection{Network Lifetime}
+
+Finally, we study the effect of increasing the primary points on the lifetime of the network.
+%In Figure~\ref{Figures/ch4/R2/LT95} and in Figure~\ref{Figures/ch4/R2/LT50}, network lifetime, $Lifetime95$ and $Lifetime50$ respectively, are illustrated for different network sizes.
+As highlighted by Figures~\ref{Figures/ch4/R2/LT}(a) and \ref{Figures/ch4/R2/LT}(b), the network lifetime obviously increases when the size of the network increases, with Model-5 that leads to the larger lifetime improvement.
+
+\begin{figure}[h!]
+\centering
+\centering
+\includegraphics[scale=0.5]{R2/LT95.pdf}\\~ ~ ~ ~ ~(a) \\
+
+\includegraphics[scale=0.5]{R2/LT50.pdf}\\~ ~ ~ ~ ~(b)
+
+\caption{Network lifetime for (a) $Lifetime_{95}$ and (b) $Lifetime_{50}$}
+ \label{Figures/ch4/R2/LT}
+\end{figure}
+
+Comparison shows that Model-5, which uses less number of primary points, is the best one because it is less energy consuming during the network lifetime. It is also the better one from the point of view of coverage ratio. Our proposed Model-5 efficiently prolongs the network lifetime with a good coverage ratio in comparison with other models. Therefore, we have chosen the model with five primary points for all the experiments presented thereafter.
+
+%\end{enumerate}
+
+
+%\subsection{Results and analysis}
\subsubsection{Coverage ratio}
\begin{figure}[ht!]
\centering
- \includegraphics[scale=0.5] {R/CR.pdf}
+ \includegraphics[scale=0.5] {F/CR.pdf}
\caption{Average coverage ratio for 150 deployed nodes}
\label{fig3}
\end{figure}
+\iffalse
\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. MuDiLCO and GA-MuDiLCO clearly outperform DESK and GAF for
a number of periods between 31 and 103. This is because they optimize the coverage and the lifetime in a wireless sensor network by selecting the best representative sensor nodes to take the responsibility of coverage during the sensing phase.}
-
+\fi
\subsubsection{Active sensors ratio}
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. \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}.
+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]{R/ASR.pdf}
+\includegraphics[scale=0.5]{F/ASR.pdf}
\caption{Active sensors ratio for 150 deployed nodes}
\label{fig4}
\end{figure}
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. \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.
+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]{R/SR.pdf}
+\includegraphics[scale=0.5]{F/SR.pdf}
\caption{Cumulative percentage of stopped simulation runs for 150 deployed nodes }
\label{fig6}
\end{figure}
\begin{figure}[h!]
\centering
\begin{tabular}{cl}
- \parbox{9.5cm}{\includegraphics[scale=0.5]{R/EC95.pdf}} & (a) \\
+ \parbox{9.5cm}{\includegraphics[scale=0.5]{F/EC95.pdf}} & (a) \\
\verb+ + \\
- \parbox{9.5cm}{\includegraphics[scale=0.5]{R/EC50.pdf}} & (b)
+ \parbox{9.5cm}{\includegraphics[scale=0.5]{F/EC50.pdf}} & (b)
\end{tabular}
\caption{Energy consumption for (a) $Lifetime_{95}$ and
(b) $Lifetime_{50}$}
The results show that MuDiLCO is the most competitive from the energy
consumption point of view. 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 status of the sensor node. Among the different versions of our protocol, the MuDiLCO-7 one consumes more energy than the other
-versions. This is easy to understand since the bigger the number of rounds and the number of sensors involved in the integer program are, the larger the time computation to solve the optimization problem is. To improve the performances of MuDiLCO-7, we should increase the number of subregions in order to have less sensors to consider in the integer program.
-\textcolor{red}{As shown in Figure~\ref{fig7}, GA-MuDiLCO consumes less energy than both DESK and GAF, but a little bit higher than MuDiLCO because it provides a near optimal solution by activating a larger number of nodes during the sensing phase. GA-MuDiLCO consumes less energy in comparison with MuDiLCO-7 version, especially for the dense networks. However, MuDiLCO protocol and GA-MuDiLCO protocol are the most competitive from the energy
-consumption point of view. The other approaches have a high energy consumption
-due to activating a larger number of redundant nodes.}
+due to activating a larger number of redundant nodes as well as the energy consumed during the different status of the sensor node.
+% Among the different versions of our protocol, the MuDiLCO-7 one consumes more energy than the other
+%versions. This is easy to understand since the bigger the number of rounds and the number of sensors involved in the integer program are, the larger the time computation to solve the optimization problem is. To improve the performances of MuDiLCO-7, we should increase the number of subregions in order to have less sensors to consider in the integer program.
+%\textcolor{red}{As shown in Figure~\ref{fig7}, GA-MuDiLCO consumes less energy than both DESK and GAF, but a little bit higher than MuDiLCO because it provides a near optimal solution by activating a larger number of nodes during the sensing phase. GA-MuDiLCO consumes less energy in comparison with MuDiLCO-7 version, especially for the dense networks. However, MuDiLCO protocol and GA-MuDiLCO protocol are the most competitive from the energy
+%consumption point of view. The other approaches have a high energy consumption
+%due to activating a larger number of redundant nodes.}
%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.
\subsubsection{Execution time}
-
+\label{et}
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 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
\begin{figure}[ht!]
\centering
-\includegraphics[scale=0.5]{R/T.pdf}
+\includegraphics[scale=0.5]{F/T.pdf}
\caption{Execution Time (in seconds)}
\label{fig77}
\end{figure}
difficulty of the optimization problem to be solved by the integer program.
This point was already noticed in subsection \ref{subsec:EC} devoted to the
energy consumption, since network lifetime and energy consumption are directly
-linked. \textcolor{red}{As can be seen in these figures, the lifetime increases with the size of the network, and it is clearly largest for the MuDiLCO
-and the GA-MuDiLCO protocols. GA-MuDiLCO prolongs the network lifetime obviously in comparison with both DESK and GAF, as well as the MuDiLCO-7 version for $lifetime_{95}$. However, comparison shows that MuDiLCO protocol and GA-MuDiLCO protocol, which use distributed optimization over the subregions are the best ones because they are robust to network disconnection during the network lifetime as well as they consume less energy in comparison with other approaches.}
+linked.
+%\textcolor{red}{As can be seen in these figures, the lifetime increases with the size of the network, and it is clearly largest for the MuDiLCO
+%and the GA-MuDiLCO protocols. GA-MuDiLCO prolongs the network lifetime obviously in comparison with both DESK and GAF, as well as the MuDiLCO-7 version for $lifetime_{95}$. However, comparison shows that MuDiLCO protocol and GA-MuDiLCO protocol, which use distributed optimization over the subregions are the best ones because they are robust to network disconnection during the network lifetime as well as they consume less energy in comparison with other approaches.}
\begin{figure}[t!]
\centering
\begin{tabular}{cl}
- \parbox{9.5cm}{\includegraphics[scale=0.5]{R/LT95.pdf}} & (a) \\
+ \parbox{9.5cm}{\includegraphics[scale=0.5]{F/LT95.pdf}} & (a) \\
\verb+ + \\
- \parbox{9.5cm}{\includegraphics[scale=0.5]{R/LT50.pdf}} & (b)
+ \parbox{9.5cm}{\includegraphics[scale=0.5]{F/LT50.pdf}} & (b)
\end{tabular}
\caption{Network lifetime for (a) $Lifetime_{95}$ and
(b) $Lifetime_{50}$}
