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[JournalMultiPeriods.git] / article.tex

diff --git a/article.tex b/article.tex
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--- a/article.tex
+++ b/article.tex
@@ -106,9 +106,14 @@ divided  into subregions and  then the  MuDiLCO protocol  is distributed  on the
 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
 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.
 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.


@@ -183,7 +188,7 @@ algorithms in WSNs according to several design choices:
 \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
 \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.
 \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.


@@ -204,9 +209,11 @@ network. Note that  centralized algorithms have the advantage  of requiring very
 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
 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. \textcolor{red} {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
+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
 the network lifetime. }
 
 The first algorithms proposed in the literature consider that the cover sets are


@@ -232,7 +239,8 @@ node.   After that,  they  proposed a  High  Residual Energy  First (HREF)  node
 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
 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}
 
 \subsection{Distributed approaches}
 %{\bf Distributed approaches}


@@ -297,7 +305,8 @@ computation complexity. Compared to our previous paper, in this one we study the
 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
 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
    
 
 \iffalse
    


@@ -529,10 +538,60 @@ Zhou~\cite{Zhang05} proved that if  the transmission range fulfills the previous
 hypothesis, a complete coverage of  a convex area implies connectivity among the
 active nodes.
 
 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
 
 %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


@@ -549,9 +608,17 @@ primary points are covered. The choice of number and locations of primary points
 
 \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).
 
 \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,
 
 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,


@@ -579,7 +646,7 @@ running out of energy), because it works in periods.
 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
 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. \textcolor{green}{The duration of the period  and 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.}
+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
 
 %%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


@@ -649,70 +716,26 @@ consumption due to the communications.
 
 \subsection{Decision phase}
 
 
 \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
 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.  
+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.  

 
 

-%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}
+%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
 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


@@ -782,7 +805,7 @@ Subject to
 \end{equation}
 
 \begin{equation}
 \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}
 
   \label{eq144} 
 \end{equation}
 


@@ -830,16 +853,70 @@ to guarantee that the maximum number of points are covered during each round.
 %% 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 $W_{\theta}$.
 %% 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.
 
 %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.}
 \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}
 
 \subsection{\textcolor{red}{Genetic Algorithm for Multiround Lifetime Coverage Optimization}}
 \label{GA}


@@ -997,7 +1074,7 @@ The proposed GA-MuDiLCO stops when the stopping criteria is met. It stops after
 
 \end{enumerate} 
 
 
 \end{enumerate} 
 

-
+\fi

 
 \section{Experimental study}
 \label{exp}
 
 \section{Experimental study}
 \label{exp}


@@ -1050,19 +1127,46 @@ $W_{\theta}$ & 1   \\
 % [1ex] adds vertical space
 %\hline
 $W_{U}$ & $|P|^2$ \\
 % [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}
 %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.
 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.


@@ -1248,7 +1352,56 @@ indicate the energy consumed by the whole network in round $t$.
 
 \end{enumerate}
 
 
 \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} 
 
 
 \subsubsection{Coverage ratio} 
 


@@ -1271,16 +1424,17 @@ rounds, and thus should extend the network lifetime.
 
 \begin{figure}[ht!]
 \centering
 
 \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} 
 
 \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.}
 \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} 
 
 
 \subsubsection{Active sensors ratio} 


@@ -1289,12 +1443,14 @@ 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
 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
 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
 
 \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} 
 \caption{Active sensors ratio for 150 deployed nodes}
 \label{fig4}
 \end{figure} 


@@ -1309,15 +1465,16 @@ Obviously, in that case DESK  and GAF have less active nodes, since  they have a
 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
 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
 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} 
 \caption{Cumulative percentage of stopped simulation runs for 150 deployed nodes }
 \label{fig6}
 \end{figure} 


@@ -1333,9 +1490,9 @@ network sizes, for $Lifetime_{95}$ and $Lifetime_{50}$.
 \begin{figure}[h!]
   \centering
   \begin{tabular}{cl}
 \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+ + \\
     \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) $Lifetime_{50}$}
   \end{tabular}
   \caption{Energy consumption for (a) $Lifetime_{95}$ and 
     (b) $Lifetime_{50}$}


@@ -1344,16 +1501,17 @@ network sizes, for $Lifetime_{95}$ and $Lifetime_{50}$.
 
 The  results  show  that  MuDiLCO  is  the  most  competitive  from  the  energy
 consumption point of view.  The  other approaches have a high energy consumption
 
 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}
 %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
 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


@@ -1367,7 +1525,7 @@ for different network sizes.
 
 \begin{figure}[ht!]
 \centering
 
 \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} 
 \caption{Execution Time (in seconds)}
 \label{fig77}
 \end{figure} 


@@ -1400,14 +1558,15 @@ of  $Lifetime_{95}$  with  large  wireless  sensor  networks  results  from  the
 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
 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}
 \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+ + \\
     \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) $Lifetime_{50}$}
   \end{tabular}
   \caption{Network lifetime for (a) $Lifetime_{95}$ and 
     (b) $Lifetime_{50}$}