Update by Ali

[JournalMultiPeriods.git] / article.tex

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--- a/article.tex
+++ b/article.tex
@@ -73,15 +73,25 @@
 %% \author[label1,label2]{}
 %% \address[label1]{}
 %% \address[label2]{}
 %% \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
  
 %\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)
 
 \begin{abstract}
 %One of  the fundamental challenges in Wireless Sensor Networks (WSNs)


@@ -93,7 +103,7 @@ Coverage and  lifetime are  two paramount problems  in Wireless  Sensor Networks
 Optimization  protocol (MuDiLCO)  is proposed  to maintain  the coverage  and to
 improve the lifetime in wireless sensor  networks. The area of interest is first
 divided  into subregions and  then the  MuDiLCO protocol  is distributed  on the
 Optimization  protocol (MuDiLCO)  is proposed  to maintain  the coverage  and to
 improve the lifetime in wireless sensor  networks. The area of interest is first
 divided  into subregions and  then the  MuDiLCO protocol  is distributed  on the

-sensor nodes in each subregion. The proposed MuDiLCO protocol works into periods
+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
 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


@@ -106,7 +116,7 @@ network lifetime and improve the coverage performance.
 \end{abstract}
 
 \begin{keyword}
 \end{abstract}
 
 \begin{keyword}

-Wireless   Sensor   Networks,   Area   Coverage,   Network   lifetime,
+Wireless   Sensor   Networks,   Area   Coverage,   Network   Lifetime,

 Optimization, Scheduling, Distributed Computation.
 
 \end{keyword}
 Optimization, Scheduling, Distributed Computation.
 
 \end{keyword}


@@ -187,13 +197,13 @@ many cover sets) can be added to the above list.
 \subsection{Centralized approaches}
 
 The major approach  is to divide/organize the sensors into  a suitable number of
 \subsection{Centralized approaches}
 
 The major approach  is to divide/organize the sensors into  a suitable number of

-set covers where  each set completely covers an interest  region and to activate
-these set covers successively.  The centralized algorithms always provide nearly
+cover sets where  each set completely covers an interest  region and to activate
+these cover sets successively.  The centralized algorithms always provide nearly

 or close  to optimal solution since the  algorithm has global view  of the whole
 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
 or close  to optimal solution since the  algorithm has global view  of the whole
 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 take the decision needs
+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. Moreover, centralized approaches usually
 suffer from the scalability problem, making them less competitive as the network
 size increases.


@@ -211,16 +221,17 @@ involved in more than one cover sets.
 %For instance, the proposed work in ~\cite{cardei2005energy, berman04}    
 
 In~\cite{yang2014maximum},  the  authors have  considered  a linear  programming
 %For instance, the proposed work in ~\cite{cardei2005energy, berman04}    
 
 In~\cite{yang2014maximum},  the  authors have  considered  a linear  programming

-approach for selecting  the minimum number of working sensor  nodes, in order to
-preserve  a  maximum coverage  and  extend lifetime  of  the  network. Cheng  et
+approach  to select  the minimum  number of  working sensor  nodes, in  order to
+preserve a  maximum coverage  and to  extend lifetime of  the network.  Cheng et

 al.~\cite{cheng2014energy} have defined a  heuristic algorithm called Cover Sets
 al.~\cite{cheng2014energy} have defined a  heuristic algorithm called Cover Sets

-Balance (CSB), which choose a set of active nodes using the tuple (data coverage
-range, residual energy).   Then, they have introduced a  new Correlated Node Set
-Computing (CNSC)  algorithm to find  the correlated node  set for a  given 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 proposed~\cite{castano2013column,rossi2012exact,deschinkel2012column}.
+Balance  (CSB), which  chooses  a set  of  active nodes  using  the tuple  (data
+coverage range, residual  energy).  Then, they have introduced  a new Correlated
+Node Set Computing (CNSC) algorithm to  find the correlated node set for a given
+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
+proposed~\cite{castano2013column,rossi2012exact,deschinkel2012column}.

 
 \subsection{Distributed approaches}
 %{\bf Distributed approaches}
 
 \subsection{Distributed approaches}
 %{\bf Distributed approaches}


@@ -258,18 +269,18 @@ perimeter coverage model from~\cite{Huang:2003:CPW:941350.941367}.
 %heterogeneous energy wireless sensor networks. 
 %In this work, the coverage protocol distributed in each sensor node in the subregion but the optimization take place over the the whole subregion. We consider only distributing the coverage protocol over two subregions. 
 
 %heterogeneous energy wireless sensor networks. 
 %In this work, the coverage protocol distributed in each sensor node in the subregion but the optimization take place over the the whole subregion. We consider only distributing the coverage protocol over two subregions. 
 

-The  works presented in  \cite{Bang, Zhixin,  Zhang} focuses  on coverage-aware,
+The  works presented  in  \cite{Bang, Zhixin,  Zhang}  focus on  coverage-aware,

 distributed energy-efficient,  and distributed clustering  methods respectively,
 distributed energy-efficient,  and distributed clustering  methods respectively,

-which aims to extend the network  lifetime, while the coverage is ensured.  More
-recently, Shibo  et al.  \cite{Shibo} have  expressed the coverage  problem as a
-minimum weight submodular set cover problem and proposed a Distributed Truncated
-Greedy Algorithm (DTGA) to solve it.  They take advantage from both temporal and
-spatial  correlations between  data sensed  by different  sensors,  and leverage
-prediction, to improve the lifetime.  In \cite{xu2001geography}, Xu et al.  have
-described an algorithm, called  Geographical Adaptive Fidelity (GAF), which uses
-geographic location information to divide the area of interest into fixed square
-grids.   Within each grid,  it keeps  only one  node staying  awake to  take the
-responsibility of sensing and communication.
+which  aim at extending  the network  lifetime, while  the coverage  is ensured.
+More recently, Shibo et al.  \cite{Shibo} have expressed the coverage problem as
+a  minimum  weight submodular  set  cover  problem  and proposed  a  Distributed
+Truncated Greedy  Algorithm (DTGA) to solve  it.  They take  advantage from both
+temporal and spatial correlations between  data sensed by different sensors, and
+leverage prediction, to improve  the lifetime.  In \cite{xu2001geography}, Xu et
+al.  have  described an algorithm, called Geographical  Adaptive Fidelity (GAF),
+which uses geographic  location information to divide the  area of interest into
+fixed square grids.   Within each grid, it keeps only one  node staying awake to
+take the responsibility of sensing and communication.

 
 Some  other  approaches (outside  the  scope  of our  work)  do  not consider  a
 synchronized and  predetermined time-slot where  the sensors are active  or not.
 
 Some  other  approaches (outside  the  scope  of our  work)  do  not consider  a
 synchronized and  predetermined time-slot where  the sensors are active  or not.


@@ -335,7 +346,7 @@ sets with a  slight growth rate in execution  time.  When producing non-disjoint
 cover sets,  both Static-CCF  and Dynamic-CCF algorithms,  where CCF  means that
 they  use a cost  function called  Critical Control  Factor, provide  cover sets
 offering longer network lifetime than those produced by \cite{cardei2005energy}.
 cover sets,  both Static-CCF  and Dynamic-CCF algorithms,  where CCF  means that
 they  use a cost  function called  Critical Control  Factor, provide  cover sets
 offering longer network lifetime than those produced by \cite{cardei2005energy}.

-Also, they require  a smaller number of node participations  in order to achieve
+Also, they require  a smaller number of participating nodes  in order to achieve

 these results.
 
 In  the  case  of  non-disjoint algorithms  \cite{pujari2011high},  sensors  may
 these results.
 
 In  the  case  of  non-disjoint algorithms  \cite{pujari2011high},  sensors  may


@@ -419,9 +430,9 @@ proposed in \cite{Huang:2003:CPW:941350.941367}.
 %heterogeneous energy wireless sensor networks. 
 %In this work, the coverage protocol distributed in each sensor node in the subregion but the optimization take place over the the whole subregion. We consider only distributing the coverage protocol over two subregions. 
 
 %heterogeneous energy wireless sensor networks. 
 %In this work, the coverage protocol distributed in each sensor node in the subregion but the optimization take place over the the whole subregion. We consider only distributing the coverage protocol over two subregions. 
 

-The  works presented in  \cite{Bang, Zhixin,  Zhang} focuses  on coverage-aware,
+The  works presented in  \cite{Bang, Zhixin,  Zhang} focus  on coverage-aware,

 distributed energy-efficient,  and distributed clustering  methods respectively,
 distributed energy-efficient,  and distributed clustering  methods respectively,

-which aims  to extend the network  lifetime, while the coverage  is ensured.  S.
+which aim  to extend the network  lifetime, while the coverage  is ensured.  S.

 Misra et al.   \cite{Misra} have proposed a localized  algorithm for coverage in
 sensor networks.  The  algorithm conserve the energy while  ensuring the network
 coverage by activating the subset of  sensors with the minimum overlap area. The
 Misra et al.   \cite{Misra} have proposed a localized  algorithm for coverage in
 sensor networks.  The  algorithm conserve the energy while  ensuring the network
 coverage by activating the subset of  sensors with the minimum overlap area. The


@@ -520,8 +531,7 @@ 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
 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
 
 %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


@@ -596,7 +606,7 @@ There are five status for each sensor node in the network:
 \item LISTENING: sensor node is waiting for a decision (to be active or not);
 \item  COMPUTATION: sensor  node  has been  elected  as leader  and applies  the
   optimization process;
 \item LISTENING: sensor node is waiting for a decision (to be active or not);
 \item  COMPUTATION: sensor  node  has been  elected  as leader  and applies  the
   optimization process;

-\item ACTIVE: sensor node participate to the monitoring of the area;
+\item ACTIVE: sensor node is taking part in the monitoring of the area;

 \item SLEEP: sensor node is turned off to save energy;
 \item COMMUNICATION: sensor node is transmitting or receiving packet.
 \end{enumerate}
 \item SLEEP: sensor node is turned off to save energy;
 \item COMMUNICATION: sensor node is transmitting or receiving packet.
 \end{enumerate}


@@ -623,7 +633,7 @@ This step  consists in  choosing the Wireless  Sensor Node Leader  (WSNL), which
 will be responsible for executing the coverage algorithm.  Each subregion in the
 area of  interest will select its  own WSNL independently for  each period.  All
 the sensor  nodes cooperate to  elect a WSNL.   The nodes in the  same subregion
 will be responsible for executing the coverage algorithm.  Each subregion in the
 area of  interest will select its  own WSNL independently for  each period.  All
 the sensor  nodes cooperate to  elect a WSNL.   The nodes in the  same subregion

-will select the  leader based on the received informations  from all other nodes
+will select the  leader based on the received information  from all other nodes

 in  the same subregion.   The selection  criteria are,  in order  of importance:
 larger  number  of neighbors,  larger  remaining energy,  and  then  in case  of
 equality, larger index. Observations on  previous simulations suggest to use the
 in  the same subregion.   The selection  criteria are,  in order  of importance:
 larger  number  of neighbors,  larger  remaining energy,  and  then  in case  of
 equality, larger index. Observations on  previous simulations suggest to use the


@@ -637,22 +647,81 @@ consumption due to the communications.
 
 \subsection{Decision phase}
 
 
 \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
 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
 of  targets to  become minimal  at  the same  time.  They  use binary  variables
 $x_{jl}$ to indicate if  sensor $j$ belongs to cover set $l$.   In our model, we
 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
 of  targets to  become minimal  at  the same  time.  They  use binary  variables
 $x_{jl}$ to indicate if  sensor $j$ belongs to cover set $l$.   In our model, we

-consider binary  variables $X_{t,j}$ to determine the  possibility of activation
-of sensor $j$ during  the round $t$ of a given sensing  phase.  We also consider
-primary points as targets.  The set of  primary points is denoted by $P$ and the
-set of sensors by  $J$. Only sensors able to be alive  during at least one round
-are involved in the integer program.
+consider binary  variables $X_{t,j}$ to determine the  possibility of activating
+sensor $j$ during round $t$ of  a given sensing phase.  We also consider primary
+points as targets.  The  set of primary points is denoted by  $P$ and the set of
+sensors by  $J$. Only sensors  able to  be alive during  at least one  round are
+involved in the integer program.

 
 %parler de la limite en energie Et pour un round
 
 
 %parler de la limite en energie Et pour un round
 


@@ -688,7 +757,7 @@ We define the Overcoverage variable $\Theta_{t,p}$ as:
 \label{eq13} 
 \end{equation}
 More  precisely, $\Theta_{t,p}$  represents the  number of  active  sensor nodes
 \label{eq13} 
 \end{equation}
 More  precisely, $\Theta_{t,p}$  represents the  number of  active  sensor nodes

-minus  one  that  cover  the  primary  point $p$  during  the  round  $t$.   The
+minus  one  that  cover  the  primary  point $p$  during  round  $t$.   The

 Undercoverage variable  $U_{t,p}$ of the primary  point $p$ during  round $t$ is
 defined by:
 \begin{equation}
 Undercoverage variable  $U_{t,p}$ of the primary  point $p$ during  round $t$ is
 defined by:
 \begin{equation}


@@ -735,11 +804,11 @@ U_{t,p} \in \lbrace0,1\rbrace, \hspace{10 mm}\forall p \in P, t = 1,\dots,T  \la
 
 \begin{itemize}
 \item $X_{t,j}$:  indicates whether  or not the  sensor $j$ is  actively sensing
 
 \begin{itemize}
 \item $X_{t,j}$:  indicates whether  or not the  sensor $j$ is  actively sensing

-  during the round $t$ (1 if yes and 0 if not);
+  during round $t$ (1 if yes and 0 if not);

 \item $\Theta_{t,p}$ - {\it overcoverage}:  the number of sensors minus one that
 \item $\Theta_{t,p}$ - {\it overcoverage}:  the number of sensors minus one that

-  are covering the primary point $p$ during the round $t$;
+  are covering the primary point $p$ during round $t$;

 \item  $U_{t,p}$ -  {\it undercoverage}:  indicates whether  or not  the primary
 \item  $U_{t,p}$ -  {\it undercoverage}:  indicates whether  or not  the primary

-  point $p$  is being covered during  the round $t$ (1  if not covered  and 0 if
+  point $p$  is being covered during round $t$ (1  if not covered  and 0 if

   covered).
 \end{itemize}
 
   covered).
 \end{itemize}
 


@@ -757,60 +826,176 @@ absence  of  monitoring  on  some  parts  of the  subregion  by  minimizing  the
 undercoverage.  The weights  $W_\theta$ and $W_U$ must be  properly chosen so as
 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
 undercoverage.  The weights  $W_\theta$ and $W_U$ must be  properly chosen so as
 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 $W_{\theta}$ very
-large compared to $W_U$.
+In our simulations priority is given  to the coverage by choosing $W_{U}$ very
+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.
 
 %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!]                
 
 \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} \; 
   \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}
 
 
 \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}
 \section{Experimental study}
 \label{exp}
 \subsection{Simulation setup}


@@ -823,8 +1008,7 @@ random topologies and  the results presented hereafter are  the average of these
 25 runs.
 %Based on the results of our proposed work in~\cite{idrees2014coverage}, we found as the region of interest are divided into larger subregions as the network lifetime increased. In this simulation, the network are divided into 16 subregions. 
 We  performed  simulations for  five  different  densities  varying from  50  to
 25 runs.
 %Based on the results of our proposed work in~\cite{idrees2014coverage}, we found as the region of interest are divided into larger subregions as the network lifetime increased. In this simulation, the network are divided into 16 subregions. 
 We  performed  simulations for  five  different  densities  varying from  50  to

-250~nodes. Experimental results are obtained from randomly generated networks in
-which  nodes  are deployed  over  a  $50 \times  25~m^2  $  sensing field.  More
+250~nodes deployed  over  a  $50 \times  25~m^2  $  sensing field.  More

 precisely, the  deployment is controlled  at a coarse  scale in order  to ensure
 that  the deployed  nodes can  cover the  sensing field  with the  given sensing
 range.
 precisely, the  deployment is controlled  at a coarse  scale in order  to ensure
 that  the deployed  nodes can  cover the  sensing field  with the  given sensing
 range.


@@ -859,19 +1043,23 @@ Sensing time for one round & 60 Minutes \\
 $E_{R}$ & 36 Joules\\
 $R_s$ & 5~m   \\     
 %\hline
 $E_{R}$ & 36 Joules\\
 $R_s$ & 5~m   \\     
 %\hline

-$W_{\Theta}$ & 1   \\
+$W_{\theta}$ & 1   \\

 % [1ex] adds vertical space
 %\hline
 % [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}
   
 %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
 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.
 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.


@@ -879,11 +1067,11 @@ During the decision  phase, in each square, one sensor is  then chosen to remain
 active during the sensing phase time.
 
 Some preliminary experiments were performed to study the choice of the number of
 active during the sensing phase time.
 
 Some preliminary experiments were performed to study the choice of the number of

-subregions  which subdivide  the  sensing field,  considering different  network
+subregions  which subdivides  the  sensing field,  considering different  network

 sizes. They show that as the number of subregions increases, so does the network
 lifetime. Moreover,  it makes  the MuDiLCO protocol  more robust  against random
 sizes. They show that as the number of subregions increases, so does the network
 lifetime. Moreover,  it makes  the MuDiLCO protocol  more robust  against random

-network  disconnection due  to node  failures.  However,  too  much subdivisions
-reduces the advantage  of the optimization. In fact, there  is a balance between
+network  disconnection due  to node  failures.  However,  too  many subdivisions
+reduce the advantage  of the optimization. In fact, there  is a balance between

 the  benefit  from the  optimization  and the  execution  time  needed to  solve
 it. Therefore, we have set the number of subregions to 16 rather than 32.
 
 the  benefit  from the  optimization  and the  execution  time  needed to  solve
 it. Therefore, we have set the number of subregions to 16 rather than 32.
 


@@ -902,13 +1090,12 @@ For our  energy consumption model, we  refer to the sensor  node Medusa~II which
 uses an Atmels  AVR ATmega103L microcontroller~\cite{raghunathan2002energy}. The
 typical  architecture  of a  sensor  is composed  of  four  subsystems: the  MCU
 subsystem which is capable of computation, communication subsystem (radio) which
 uses an Atmels  AVR ATmega103L microcontroller~\cite{raghunathan2002energy}. The
 typical  architecture  of a  sensor  is composed  of  four  subsystems: the  MCU
 subsystem which is capable of computation, communication subsystem (radio) which

-is  responsible  for  transmitting/receiving  messages, sensing  subsystem  that
+is responsible  for transmitting/receiving messages, the  sensing subsystem that

 collects  data, and  the  power supply  which  powers the  complete sensor  node
 \cite{raghunathan2002energy}. Each  of the first three subsystems  can be turned
 on or  off depending on  the current status  of the sensor.   Energy consumption
 (expressed in  milliWatt per second) for  the different status of  the sensor is
 collects  data, and  the  power supply  which  powers the  complete sensor  node
 \cite{raghunathan2002energy}. Each  of the first three subsystems  can be turned
 on or  off depending on  the current status  of the sensor.   Energy consumption
 (expressed in  milliWatt per second) for  the different status of  the sensor is

-summarized in Table~\ref{table4}.  The energy  needed to send or receive a 1-bit
-packet is equal to $0.2575~mW$.
+summarized in Table~\ref{table4}.

 
 \begin{table}[ht]
 \caption{The Energy Consumption Model}
 
 \begin{table}[ht]
 \caption{The Energy Consumption Model}


@@ -942,13 +1129,14 @@ COMPUTATION & on & on & on & 26.83 \\
 For the sake of simplicity we ignore  the energy needed to turn on the radio, to
 start up the sensor node, to move from one status to another, etc.
 %We also do not consider the need of collecting sensing data. PAS COMPRIS
 For the sake of simplicity we ignore  the energy needed to turn on the radio, to
 start up the sensor node, to move from one status to another, etc.
 %We also do not consider the need of collecting sensing data. PAS COMPRIS

-Thus, when a sensor becomes active (i.e., it already decides its status), it can
+Thus, when a sensor becomes active (i.e., it has already chosen its status), it can

 turn  its radio  off to  save battery.  MuDiLCO uses  two types  of  packets for
 communication. The size of the  INFO packet and Active-Sleep packet are 112~bits
 and 24~bits  respectively.  The  value of energy  spent to send  a 1-bit-content
 message is  obtained by using  the equation in  ~\cite{raghunathan2002energy} to
 calculate  the energy cost  for transmitting  messages and  we propose  the same
 turn  its radio  off to  save battery.  MuDiLCO uses  two types  of  packets for
 communication. The size of the  INFO packet and Active-Sleep packet are 112~bits
 and 24~bits  respectively.  The  value of energy  spent to send  a 1-bit-content
 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.
+value for receiving the packets. The energy  needed to send or receive a 1-bit
+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
 
 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


@@ -964,16 +1152,16 @@ To evaluate our approach we consider the following performance metrics:
 
 \begin{enumerate}[i]
   
 
 \begin{enumerate}[i]
   

-\item {{\bf Coverage Ratio (CR)}:} the coverage ratio measures how much the area
+\item {{\bf Coverage Ratio (CR)}:} the coverage ratio measures how much of the area

   of a sensor field is covered. In our case, the sensing field is represented as
   of a sensor field is covered. In our case, the sensing field is represented as

-  a connected grid  of points and we use  each grid point as a  sample point for
-  calculating the coverage. The coverage ratio can be calculated by:
+  a connected grid  of points and we use  each grid point as a  sample point to
+  compute the coverage. The coverage ratio can be calculated by:

 \begin{equation*}
 \scriptsize
 \mbox{CR}(\%) = \frac{\mbox{$n^t$}}{\mbox{$N$}} \times 100,
 \end{equation*}
 where $n^t$ is  the number of covered  grid points by the active  sensors of all
 \begin{equation*}
 \scriptsize
 \mbox{CR}(\%) = \frac{\mbox{$n^t$}}{\mbox{$N$}} \times 100,
 \end{equation*}
 where $n^t$ is  the number of covered  grid points by the active  sensors of all

-subregions during round $t$ in the current sensing phase and $N$ is total number
+subregions during round $t$ in the current sensing phase and $N$ is the total number

 of grid points  in the sensing field of  the network. In our simulations $N = 51
 \times 26 = 1326$ grid points.
 %The accuracy of this method depends on the distance between grids. In our
 of grid points  in the sensing field of  the network. In our simulations $N = 51
 \times 26 = 1326$ grid points.
 %The accuracy of this method depends on the distance between grids. In our


@@ -991,11 +1179,11 @@ of grid points  in the sensing field of  the network. In our simulations $N = 51
 \end{equation*}
 where $A_r^t$ is the number of  active sensors in the subregion $r$ during round
 $t$ in the  current sensing phase, $|J|$  is the total number of  sensors in the
 \end{equation*}
 where $A_r^t$ is the number of  active sensors in the subregion $r$ during round
 $t$ in the  current sensing phase, $|J|$  is the total number of  sensors in the

-network, and $R$ is the total number of the subregions in the network.
+network, and $R$ is the total number of subregions in the network.

 
 \item {{\bf Network Lifetime}:} we define the network lifetime as the time until
   the  coverage  ratio  drops  below   a  predefined  threshold.  We  denote  by
 
 \item {{\bf Network Lifetime}:} we define the network lifetime as the time until
   the  coverage  ratio  drops  below   a  predefined  threshold.  We  denote  by

-  $Lifetime_{95}$ (respectively  $Lifetime_{50}$) as  the amount of  time during
+  $Lifetime_{95}$ (respectively  $Lifetime_{50}$) the amount of  time during

   which  the  network   can  satisfy  an  area  coverage   greater  than  $95\%$
   (respectively $50\%$). We assume that the network is alive until all nodes have
   been   drained    of   their   energy   or   the    sensor   network   becomes
   which  the  network   can  satisfy  an  area  coverage   greater  than  $95\%$
   (respectively $50\%$). We assume that the network is alive until all nodes have
   been   drained    of   their   energy   or   the    sensor   network   becomes


@@ -1011,7 +1199,7 @@ network, and $R$ is the total number of the subregions in the network.
   % New version with global loops on period
   \begin{equation*}
     \scriptsize
   % 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*}
 
 
   \end{equation*}
 
 


@@ -1030,17 +1218,17 @@ network, and $R$ is the total number of the subregions in the network.
 % Old version -> where $M_L$ and  $T_L$ are respectively the number of  periods and rounds during
 %$Lifetime_{95}$ or  $Lifetime_{50}$. 
 % New version
 % 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  $E^{\scriptsize  \mbox{com}}_m$,
 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  $E^{\scriptsize  \mbox{com}}_m$,

-represent  the  energy   consumption  spent  by  all  the   nodes  for  wireless
+represents  the  energy   consumption  spent  by  all  the   nodes  for  wireless

 communications  during period  $m$.  $E^{\scriptsize  \mbox{list}}_m$,  the next
 factor, corresponds  to the energy consumed  by the sensors  in LISTENING status
 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 $E^s_t$
 communications  during period  $m$.  $E^{\scriptsize  \mbox{list}}_m$,  the next
 factor, corresponds  to the energy consumed  by the sensors  in LISTENING status
 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.
 
 %\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.


@@ -1057,9 +1245,9 @@ indicate the energy consummed by the whole network in round $t$.
 
 \end{enumerate}
 
 
 \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
 
 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


@@ -1067,7 +1255,7 @@ which is a little bit better than the one of MuDiLCO.
 %%RC : need to uniformize MuDiLCO or MuDiLCO-T? 
 %%MS : MuDiLCO everywhere
 %%RC maybe increase the size of the figure for the reviewers, no?
 %%RC : need to uniformize MuDiLCO or MuDiLCO-T? 
 %%MS : MuDiLCO everywhere
 %%RC maybe increase the size of the figure for the reviewers, no?

-This is due  to the fact that in comparison with  MuDiLCO that uses optimization
+This is due  to the fact that, in comparison with  MuDiLCO which uses optimization

 to put in  SLEEP status redundant sensors, more sensor  nodes remain active with
 DESK and GAF.   As a consequence, when the number of  rounds increases, a larger
 number of node failures  can be observed in DESK and GAF,  resulting in a faster
 to put in  SLEEP status redundant sensors, more sensor  nodes remain active with
 DESK and GAF.   As a consequence, when the number of  rounds increases, a larger
 number of node failures  can be observed in DESK and GAF,  resulting in a faster


@@ -1080,56 +1268,57 @@ rounds, and thus should extend the network lifetime.
 
 \begin{figure}[ht!]
 \centering
 
 \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} 
 
 \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
 
 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
 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 number 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
 
 \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} 
 
 \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
 %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 of
-the methods.  DESK stops first,  after around 45~rounds, because it consumes the
+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
 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
 
 %%% 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} 
 
 \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
 
 We  measure  the  energy  consumed  by the  sensors  during  the  communication,
 listening, computation, active, and sleep status for different network densities


@@ -1140,9 +1329,9 @@ 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]{R1/EC95.pdf}} & (a) \\
+    \parbox{9.5cm}{\includegraphics[scale=0.5]{R/EC95.pdf}} & (a) \\

     \verb+ + \\
     \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) $Lifetime_{50}$}
   \end{tabular}
   \caption{Energy consumption for (a) $Lifetime_{95}$ and 
     (b) $Lifetime_{50}$}


@@ -1163,11 +1352,11 @@ sensors to consider in the integer program.
 %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. 
 
 
 %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
 
 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
 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


@@ -1178,17 +1367,17 @@ for different network sizes.
 
 \begin{figure}[ht!]
 \centering
 
 \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} 
 
 As expected,  the execution time increases  with the number of  rounds $T$ taken
 \caption{Execution Time (in seconds)}
 \label{fig77}
 \end{figure} 
 
 As expected,  the execution time increases  with the number of  rounds $T$ taken

-into account for scheduling of the sensing phase. The times obtained for $T=1,3$
-or $5$ seems bearable, but for $T=7$ they become quickly unsuitable for a sensor
+into account to schedule the sensing phase. The times obtained for $T=1,3$
+or $5$ seem bearable, but for $T=7$ they become quickly unsuitable for a sensor

 node, especially when  the sensor network size increases.   Again, we can notice
 that if we want  to schedule the nodes activities for a  large number of rounds,
 node, especially when  the sensor network size increases.   Again, we can notice
 that if we want  to schedule the nodes activities for a  large number of rounds,

-we need to choose a relevant number of subregion in order to avoid a complicated
+we need to choose a relevant number of subregions in order to avoid a complicated

 and cumbersome optimization.  On the one hand, a large value  for $T$ permits to
 reduce the  energy-overhead due  to the three  pre-sensing phases, on  the other
 hand  a leader  node may  waste a  considerable amount  of energy  to  solve the
 and cumbersome optimization.  On the one hand, a large value  for $T$ permits to
 reduce the  energy-overhead due  to the three  pre-sensing phases, on  the other
 hand  a leader  node may  waste a  considerable amount  of energy  to  solve the


@@ -1196,17 +1385,17 @@ optimization problem.
 
 %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.
 
 
 %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 $Lifetime_{95}$
 and  $Lifetime_{50}$.  Both  figures show  that the  network  lifetime increases
 together with the  number of sensor nodes, whatever the  protocol, thanks to the
 
 The next  two figures,  Figures~\ref{fig8}(a) and \ref{fig8}(b),  illustrate the
 network lifetime  for different network sizes,  respectively for $Lifetime_{95}$
 and  $Lifetime_{50}$.  Both  figures show  that the  network  lifetime increases
 together with the  number of sensor nodes, whatever the  protocol, thanks to the

-node  density  which  result in  more  and  more  redundant  nodes that  can  be
+node  density  which  results in  more  and  more  redundant  nodes that  can  be

 deactivated and thus save energy.  Compared to the other approaches, our MuDiLCO
 protocol  maximizes the  lifetime of  the network.   In particular  the  gain in
 lifetime for a  coverage over 95\% is greater than 38\%  when switching from GAF
 deactivated and thus save energy.  Compared to the other approaches, our MuDiLCO
 protocol  maximizes the  lifetime of  the network.   In particular  the  gain in
 lifetime for a  coverage over 95\% is greater than 38\%  when switching from GAF

-to MuDiLCO-3.  The  slight decrease that can bee observed  for MuDiLCO-7 in case
+to MuDiLCO-3.  The  slight decrease that can be observed  for MuDiLCO-7 in case

 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
 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


@@ -1216,9 +1405,9 @@ linked.
 \begin{figure}[t!]
   \centering
   \begin{tabular}{cl}
 \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+ + \\
     \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) $Lifetime_{50}$}
   \end{tabular}
   \caption{Network lifetime for (a) $Lifetime_{95}$ and 
     (b) $Lifetime_{50}$}


@@ -1236,7 +1425,7 @@ linked.
 \section{Conclusion and future works}
 \label{sec:conclusion}
 
 \section{Conclusion and future works}
 \label{sec:conclusion}
 

-We have addressed  the problem of the coverage and  the lifetime optimization in
+We have addressed  the problem of the coverage and of the lifetime optimization in

 wireless  sensor networks.  This is  a key  issue as  sensor nodes  have limited
 resources in terms of memory, energy, and computational power. To cope with this
 problem,  the field  of sensing  is divided  into smaller  subregions  using the
 wireless  sensor networks.  This is  a key  issue as  sensor nodes  have limited
 resources in terms of memory, energy, and computational power. To cope with this
 problem,  the field  of sensing  is divided  into smaller  subregions  using the


@@ -1252,17 +1441,17 @@ scheduling.
 %subregion using more than one cover set during the sensing phase. 
 The activity  scheduling in each subregion  works in periods,  where each period
 consists of four  phases: (i) Information Exchange, (ii)  Leader Election, (iii)
 %subregion using more than one cover set during the sensing phase. 
 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.
+Decision Phase to plan the activity  of the sensors over $T$ rounds, (iv) Sensing
+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
 time. Indeed,  when dealing with  large wireless sensor networks,  a distributed
 
 Simulations  results show the  relevance of  the proposed  protocol in  terms of
 lifetime, coverage  ratio, active  sensors ratio, energy  consumption, execution
 time. Indeed,  when dealing with  large wireless sensor networks,  a distributed

-approach like  the one we  propose allows to  reduce the difficulty of  a single
+approach, like  the one we  propose, allows to  reduce the difficulty of  a single

 global optimization problem by partitioning it in many smaller problems, one per
 subregion, that can be solved  more easily. Nevertheless, results also show that
 it is not possible to plan the activity of sensors over too many rounds, because
 global optimization problem by partitioning it in many smaller problems, one per
 subregion, that can be solved  more easily. Nevertheless, results also show that
 it is not possible to plan the activity of sensors over too many rounds, because

-the resulting optimization problem leads to too high resolution time and thus to
+the resulting optimization problem leads to too high resolution times and thus to

 an excessive energy consumption.
 
 %In  future work, we plan  to study and propose adjustable sensing range coverage optimization protocol, which computes  all active sensor schedules in one time, by using
 an excessive energy consumption.
 
 %In  future work, we plan  to study and propose adjustable sensing range coverage optimization protocol, which computes  all active sensor schedules in one time, by using