Update by Ali

[JournalMultiPeriods.git] / article.tex

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+++ b/article.tex
@@ -1,17 +1,13 @@
 
 

-

 \documentclass[preprint,12pt]{elsarticle}
 
 \documentclass[preprint,12pt]{elsarticle}
 

-

 \usepackage[linesnumbered,ruled,vlined,commentsnumbered]{algorithm2e}
 \usepackage{multicol}
 \usepackage{mathtools}  
 \usepackage[linesnumbered,ruled,vlined,commentsnumbered]{algorithm2e}
 \usepackage{multicol}
 \usepackage{mathtools}  

-
-

 \usepackage{colortbl}
 \usepackage{multirow}
 
 \usepackage{colortbl}
 \usepackage{multirow}
 

-  \SetAlFnt{\small}
+\SetAlFnt{\small}

 \SetAlCapFnt{\large}
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@@ -71,31 +67,61 @@
 %% \address{Address\fnref{label3}}
 %% \fntext[label3]{}
 
 %% \address{Address\fnref{label3}}
 %% \fntext[label3]{}
 

-\title{ Multiperiod Distributed Lifetime Coverage Optimization Protocol in Wireless Sensor Networks}
+\title{Multiround Distributed Lifetime Coverage Optimization Protocol in Wireless Sensor Networks}

 
 %% use optional labels to link authors explicitly to addresses:
 %% \author[label1,label2]{}
 %% \address[label1]{}
 %% \address[label2]{}
 
 %% use optional labels to link authors explicitly to addresses:
 %% \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.}
+
+
+\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}}
+}  

 
 

-\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.}

 
 \begin{abstract}
 %One of  the fundamental challenges in Wireless Sensor Networks (WSNs)
 %is the coverage preservation and the extension of the network lifetime
 %continuously  and  effectively  when  monitoring a  certain  area  (or
 %region) of  interest. 
 
 \begin{abstract}
 %One of  the fundamental challenges in Wireless Sensor Networks (WSNs)
 %is the coverage preservation and the extension of the network lifetime
 %continuously  and  effectively  when  monitoring a  certain  area  (or
 %region) of  interest. 

-Coverage and lifetime are two paramount problems in  Wireless Sensor Networks (WSNs). In this paper, a method called Multiperiod Distributed Lifetime Coverage 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 during which a sets of sensor nodes are scheduled to remaining 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. Compared with some existing protocols, simulation results based on multiple criteria (energy consumption,coverage ratio, ...) show that the proposed protocol can prolong efficiently the network lifetime and improve the coverage performance.
+Coverage and  lifetime are  two paramount problems  in Wireless  Sensor Networks
+(WSNs). In this paper, a method called Multiround Distributed Lifetime Coverage
+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 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. \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.

 
 \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}


@@ -104,18 +130,26 @@ Optimization, Scheduling, Distributed Computation.
 
 \section{Introduction}
  
 
 \section{Introduction}
  

-\indent  The fast developments  in the  low-cost sensor  devices and
-wireless  communications  have allowed  the  emergence  of the WSNs.  WSN
-includes  a large  number of  small, limited-power sensors that can
-sense, process and transmit data over a wireless communication. They
-communicate with each other by using multi-hop wireless communications, cooperate together to monitor the area of interest, 
-and the measured data can be reported to a monitoring center called sink  
-for analysis it~\cite{Sudip03}. There are several applications used the
-WSN including health, home, environmental, military, and industrial
-applications~\cite{Akyildiz02}. Sensor nodes run on batteries with
-limited capacities, and it is often costly or simply impossible to replace and/or recharge batteries, especially in
-remote and hostile environments. To achieve a long life of the network, it is important to conserve battery power.
-Therefore, lifetime optimisation is one of the most critical issues in wireless sensor networks.
+\indent  The   fast  developments  of  low-cost  sensor   devices  and  wireless
+communications have allowed the emergence of WSNs. A WSN includes a large number
+of small, limited-power sensors that  can sense, process, and transmit data over
+a wireless  communication. They communicate  with each other by  using multi-hop
+wireless communications and cooperate together  to monitor the area of interest,
+so that  each measured data can be  reported to a monitoring  center called sink
+for further  analysis~\cite{Sudip03}.  There  are several fields  of application
+covering  a wide  spectrum for  a  WSN, including  health, home,  environmental,
+military, and industrial applications~\cite{Akyildiz02}.
+
+On the one hand sensor nodes run on batteries with limited capacities, and it is
+often  costly  or  simply  impossible  to  replace  and/or  recharge  batteries,
+especially in remote and hostile environments. Obviously, to achieve a long life
+of the  network it is important  to conserve battery  power. Therefore, lifetime
+optimization is one of the most  critical issues in wireless sensor networks. On
+the other hand we must guarantee  coverage over the area of interest. To fulfill
+these two objectives, the main idea  is to take advantage of overlapping sensing
+regions to turn-off redundant sensor nodes  and thus save energy. In this paper,
+we concentrate  on the area coverage  problem, with the  objective of maximizing
+the network lifetime by using an optimized multiround scheduling.

 
 % One of the major scientific research challenges in WSNs, which are addressed by a large number of literature during the last few years is to design energy efficient approaches for coverage and connectivity in WSNs~\cite{conti2014mobile}. The coverage problem is one  of the
 %fundamental challenges in WSNs~\cite{Nayak04} that consists in monitoring efficiently and continuously
 
 % One of the major scientific research challenges in WSNs, which are addressed by a large number of literature during the last few years is to design energy efficient approaches for coverage and connectivity in WSNs~\cite{conti2014mobile}. The coverage problem is one  of the
 %fundamental challenges in WSNs~\cite{Nayak04} that consists in monitoring efficiently and continuously


@@ -125,178 +159,329 @@ Therefore, lifetime optimisation is one of the most critical issues in wireless
 %leads to decrease the lifetime of the network. To extend the lifetime of the network, the main idea is to take advantage of the overlapping sensing regions  of some  sensor nodes to  save energy by  turning off
 %some  of them  during the  sensing phase~\cite{Misra05}. WSNs require energy-efficient solutions to improve the network lifetime that is constrained by the limited power of each sensor node ~\cite{Akyildiz02}. 
 
 %leads to decrease the lifetime of the network. To extend the lifetime of the network, the main idea is to take advantage of the overlapping sensing regions  of some  sensor nodes to  save energy by  turning off
 %some  of them  during the  sensing phase~\cite{Misra05}. WSNs require energy-efficient solutions to improve the network lifetime that is constrained by the limited power of each sensor node ~\cite{Akyildiz02}. 
 

-In this paper, we concentrate on the area coverage problem, with the objective of maximizing the network lifetime by using an optimized multirounds scheduling. 
+%In this paper,  we concentrate on the area coverage  problem, with the objective
+%of maximizing the network lifetime by using an optimized multirounds scheduling.

 %The area of interest is divided into subregions.
 
 % Each period includes four phases starts with a discovery phase to exchange information among the sensors of the subregion, in order  to choose in a  suitable manner a sensor node as leader to carry out a coverage strategy.  This coverage strategy involves the solving of an integer program by the leader,  to optimize the coverage and the lifetime in the subregion by producing a sets of sensor nodes in order to take the mission of coverage preservation during several rounds in the sensing phase. In fact, the nodes in a subregion can be seen as a cluster where each node sends sensing data to the cluster head or the sink node. Furthermore, the activities in a subregion/cluster can continue even if another cluster stops due to too many node failures.  
 
 The remainder of the paper is organized as follows. The next section
 % Section~\ref{rw}
 %The area of interest is divided into subregions.
 
 % Each period includes four phases starts with a discovery phase to exchange information among the sensors of the subregion, in order  to choose in a  suitable manner a sensor node as leader to carry out a coverage strategy.  This coverage strategy involves the solving of an integer program by the leader,  to optimize the coverage and the lifetime in the subregion by producing a sets of sensor nodes in order to take the mission of coverage preservation during several rounds in the sensing phase. In fact, the nodes in a subregion can be seen as a cluster where each node sends sensing data to the cluster head or the sink node. Furthermore, the activities in a subregion/cluster can continue even if another cluster stops due to too many node failures.  
 
 The remainder of the paper is organized as follows. The next section
 % Section~\ref{rw}

-reviews the related work in the field. Section~\ref{pd} is devoted to
-the description of MuDiLCO Protocol. Section~\ref{cp} gives  the coverage model formulation,  which is used
-to schedule the activation of sensors.  Section~\ref{exp}  shows the
-simulation results obtained using the discrete event simulator OMNeT++
-\cite{varga}. They  fully demonstrate  the usefulness of  the proposed
-approach.  Finally,  we give  concluding remarks and  some suggestions
-for future works in Section~\ref{sec:conclusion}.
+reviews  the related works  in the  field.  Section~\ref{pd}  is devoted  to the
+description of MuDiLCO protocol.  Section~\ref{exp} shows the simulation results
+obtained using  the discrete event  simulator OMNeT++ \cite{varga}.   They fully
+demonstrate  the  usefulness  of   the  proposed  approach.   Finally,  we  give
+concluding    remarks   and    some    suggestions   for    future   works    in
+Section~\ref{sec:conclusion}.

 
 
 
 

-\section{Related works}
+%%RC : Related works good for a phd thesis but too long for a paper. Ali you  need to learn to .... summarize :-)
+\section{Related works} % Trop proche de l'etat de l'art de l'article de Zorbas ?

 \label{rw}
 
 \label{rw}
 

-\indent This section is dedicated to the various approaches proposed
-in  the literature  for  the coverage  lifetime maximization  problem,
-where the  objective is to  optimally schedule sensors'  activities in
-order to  extend network lifetime  in WSNs. Cardei and Wu \cite{cardei2006energy} provide a taxonomy for coverage algorithms in WSNs according to several design choices:
+\indent  This section is  dedicated to  the various  approaches proposed  in the
+literature for  the coverage lifetime maximization problem,  where the objective
+is to optimally schedule sensors' activities in order to extend network lifetime
+in WSNs. Cardei  and Wu \cite{cardei2006energy} provide a  taxonomy for coverage
+algorithms in WSNs according to several design choices:

 \begin{itemize}
 \begin{itemize}

-\item Sensors scheduling Algorithms, 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 the sensing period.
-\item The homogeneous or heterogeneous nature of the
-nodes, in terms of sensing or communication capabilities.
+\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 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 node deployment method, which may be random or deterministic.

-\item Additional requirements for energy-efficient
-coverage and connected coverage.
+\item  Additional  requirements  for  energy-efficient and  connected coverage.

 \end{itemize}
 
 \end{itemize}
 

-The choice of non-disjoint or disjoint cover sets (sensors participate or not in many cover sets) can be added to the above list.
+The choice of non-disjoint or disjoint cover sets (sensors participate or not in
+many cover sets) can be added to the above list.

 % The independency in the cover set (i.e. whether the cover sets are disjoint or non-disjoint) \cite{zorbas2010solving} is another design choice that can be added to the above list.
 % The independency in the cover set (i.e. whether the cover sets are disjoint or non-disjoint) \cite{zorbas2010solving} is another design choice that can be added to the above list.

+
+\subsection{Centralized approaches}
+
+The major approach  is to divide/organize the sensors into  a suitable number of
+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
+higher cost in communications, since the  node that will make the decision needs
+information from all the  sensor nodes. \textcolor{green} {Exact or heuristics approaches are designed to provide cover sets.
+ %(Moreover, centralized approaches usually
+%suffer from the scalability problem, making them less competitive as the network
+%size increases.) 
+Contrary to exact methods, heuristic methods can handle very large and centralized problems. They are proposed to reduce computational overhead such as energy consumption, delay and generally increase in
+the network lifetime. }
+
+The first algorithms proposed in the literature consider that the cover sets are
+disjoint:  a  sensor  node  appears  in  exactly  one  of  the  generated  cover
+sets~\cite{abrams2004set,cardei2005improving,Slijepcevic01powerefficient}.     In
+the   case  of  non-disjoint   algorithms  \cite{pujari2011high},   sensors  may
+participate in  more than one  cover set.  In  some cases, this may  prolong the
+lifetime of the network in comparison  to the disjoint cover set algorithms, but
+designing  algorithms for  non-disjoint cover  sets generally  induces  a higher
+order  of complexity.   Moreover, in  case of  a sensor's  failure, non-disjoint
+scheduling  policies are less  resilient and  reliable because  a sensor  may be
+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
+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
+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{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}
+In distributed  and localized coverage  algorithms, the required  computation to
+schedule the  activity of  sensor nodes  will be done  by the  cooperation among
+neighboring nodes. These  algorithms may require more computation  power for the
+processing by the cooperating sensor nodes, but they are more scalable for large
+WSNs.  Localized and distributed algorithms generally result in non-disjoint set
+covers.
+
+Many distributed algorithms have been  developed to perform the scheduling so as
+to          preserve         coverage,          see          for         example
+\cite{Gallais06,Tian02,Ye03,Zhang05,HeinzelmanCB02,       yardibi2010distributed,
+  prasad2007distributed,Misra}.   Distributed  algorithms  typically operate  in
+rounds for  a predetermined duration. At  the beginning of each  round, a sensor
+exchanges information with  its neighbors and makes a  decision to either remain
+turned on or  to go to sleep for  the round. This decision is  basically made on
+simple     greedy     criteria    like     the     largest    uncovered     area
+\cite{Berman05efficientenergy}      or       maximum      uncovered      targets
+\cite{lu2003coverage}.   The  Distributed  Adaptive Sleep  Scheduling  Algorithm
+(DASSA) \cite{yardibi2010distributed}  does not require  location information of
+sensors while  maintaining connectivity and  satisfying a user  defined coverage
+target.  In  DASSA, nodes use the  residual energy levels and  feedback from the
+sink for  scheduling the activity  of their neighbors.  This  feedback mechanism
+reduces  the randomness  in scheduling  that would  otherwise occur  due  to the
+absence of location information.  In  \cite{ChinhVu}, the author have designed a
+novel distributed heuristic,  called Distributed Energy-efficient Scheduling for
+k-coverage (DESK), which  ensures that the energy consumption  among the sensors
+is  balanced  and the  lifetime  maximized  while  the coverage  requirement  is
+maintained.   This heuristic  works in  rounds, requires  only  one-hop neighbor
+information, and each  sensor decides its status (active or  sleep) based on the
+perimeter coverage model from~\cite{Huang:2003:CPW:941350.941367}.
+
+%Our Work, which is presented in~\cite{idrees2014coverage} proposed a coverage optimization protocol to improve the lifetime in
+%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}  focus on  coverage-aware,
+distributed energy-efficient,  and distributed clustering  methods respectively,
+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.
+Indeed, each sensor  maintains its own timer and its  wake-up time is randomized
+\cite{Ye03} or regulated \cite{cardei2005maximum} over time.
+
+The MuDiLCO protocol (for  Multiround Distributed Lifetime Coverage Optimization
+protocol) presented  in this  paper is an  extension of the  approach introduced
+in~\cite{idrees2014coverage}.   In~\cite{idrees2014coverage},  the  protocol  is
+deployed over  only two  subregions. Simulation results  have shown that  it was
+more  interesting  to  divide  the  area  into  several  subregions,  given  the
+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
+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

    
 \subsection{Centralized Approaches}
 %{\bf Centralized approaches}
    
 \subsection{Centralized Approaches}
 %{\bf 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 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 first algorithms  proposed in the  literature consider that  the cover
-sets  are  disjoint: a  sensor  node appears  in  exactly  one of  the
-generated  cover  sets.    For  instance,  Slijepcevic  and  Potkonjak
-\cite{Slijepcevic01powerefficient}   propose    an   algorithm, which
-allocates sensor nodes in mutually independent sets to monitor an area
-divided into  several fields.  Their algorithm builds  a cover  set by
-including in  priority the sensor  nodes, which cover  critical fields,
-that  is to  say fields  that are  covered by  the smallest  number of
-sensors. The time complexity of  their heuristic is $O(n^2)$ where $n$
-is the number of  sensors. Abrams et al.~\cite{abrams2004set}  design  three  approximation
-algorithms  for a  variation of  the  set k-cover  problem, where  the
-objective is to partition the sensors into covers such that the number
-of covers that  includes an area, summed over  all areas, is maximized.
-Their        work        builds        upon       previous        work
-in~\cite{Slijepcevic01powerefficient} and the  generated cover sets do
-not provide complete coverage of the monitoring zone.
-\cite{cardei2005improving} propose a method to efficiently
-compute the maximum  number of disjoint set covers  such that each set
-can  monitor all  targets. They  first  transform the  problem into  a
-maximum  flow   problem, which  is  formulated  as   a  mixed  integer
-programming (MIP). Then their heuristic  uses the output of the MIP to
-compute  disjoint  set  covers.  Results  show  that  this  heuristic
-provides  a   number  of  set  covers  slightly   larger  compared  to
-\cite{Slijepcevic01powerefficient}  but with  a larger  execution time
-due  to the complexity  of the  mixed integer  programming resolution.
-
-Zorbas  et  al.  \cite{zorbas2010solving}  presented a centralised greedy
-algorithm for the efficient production of both node disjoint
-and non-disjoint cover sets. Compared to algorithm's results  of  Slijepcevic and Potkonjak
-\cite{Slijepcevic01powerefficient},  their   heuristic  produces  more
-disjoint cover sets with a slight growth rate in execution time. When producing non-disjoint cover sets, both Static-CCF and Dynamic-CCF 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 these results.
-
-In the  case of non-disjoint algorithms  \cite{pujari2011high}, sensors may
-participate  in more  than  one cover  set.   In some  cases, this  may
-prolong  the lifetime  of the  network in  comparison to  the disjoint
-cover set algorithms, but  designing algorithms for non-disjoint cover
-sets  generally induces a  higher order  of complexity.   Moreover, in
-case of a sensor's  failure, non-disjoint scheduling policies are less
-resilient and less  reliable because a sensor may  be involved in more
-than one  cover sets.  For instance,  Cardei et al.~\cite{cardei2005energy}
-present a  linear programming (LP)  solution and a greedy  approach to
-extend the  sensor network lifetime  by organizing the sensors  into a
-maximal  number of  non-disjoint cover  sets. Simulation  results show
-that by allowing sensors to  participate in multiple sets, the network
-lifetime         increases        compared         with        related
-work~\cite{cardei2005improving}.   In~\cite{berman04}, the
-authors  have formulated  the lifetime  problem and  suggested another
-(LP)  technique to  solve this  problem. A  centralized  solution  based      on      the     Garg-K\"{o}nemann
-algorithm~\cite{garg98}, provably near
-the optimal solution,    is also proposed.
+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
+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
+information from all the  sensor nodes. Moreover, centralized approaches usually
+suffer from the scalability problem, making them less competitive as the network
+size increases.
+
+The first algorithms proposed in the literature consider that the cover sets are
+disjoint: a sensor node appears in exactly one of the generated cover sets.  For
+instance,  Slijepcevic  and  Potkonjak  \cite{Slijepcevic01powerefficient}  have
+proposed an algorithm, which allocates sensor nodes in mutually independent sets
+to monitor an area divided into  several fields.  Their algorithm builds a cover
+set by including in priority the  sensor nodes which cover critical fields, that
+is to say fields  that are covered by the smallest number  of sensors.  The time
+complexity of  their heuristic is $O(n^2)$  where $n$ is the  number of sensors.
+Abrams et al.~\cite{abrams2004set}  have designed three approximation algorithms
+for a variation of the set  k-cover problem, where the objective is to partition
+the sensors  into covers such  that the number  of covers that include  an area,
+summed  over all  areas, is  maximized.  Their  work builds  upon  previous work
+in~\cite{Slijepcevic01powerefficient}  and  the  generated  cover  sets  do  not
+provide complete coverage of the monitoring zone.
+
+In \cite{cardei2005improving}, the authors have proposed a method to efficiently
+compute the maximum number of disjoint set covers such that each set can monitor
+all targets. They first transform the problem into a maximum flow problem, which
+is formulated  as a mixed integer  programming (MIP). Then  their heuristic uses
+the output  of the MIP to compute  disjoint set covers.  Results  show that this
+heuristic  provides  a  number  of   set  covers  slightly  larger  compared  to
+\cite{Slijepcevic01powerefficient}, but with a  larger execution time due to the
+complexity of the mixed integer programming resolution.
+
+Zorbas et al.  \cite{zorbas2010solving} presented a centralized greedy algorithm
+for the efficient production of  both node disjoint and non-disjoint cover sets.
+Compared    to    algorithm's    results    of   Slijepcevic    and    Potkonjak
+\cite{Slijepcevic01powerefficient}, their heuristic produces more disjoint cover
+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}.
+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
+participate in  more than one  cover set.  In  some cases, this may  prolong the
+lifetime of the network in comparison  to the disjoint cover set algorithms, but
+designing  algorithms for  non-disjoint cover  sets generally  induces  a higher
+order  of complexity.   Moreover, in  case of  a sensor's  failure, non-disjoint
+scheduling policies are less resilient and less reliable because a sensor may be
+involved   in   more  than   one   cover   sets.    For  instance,   Cardei   et
+al.~\cite{cardei2005energy}  present a  linear programming  (LP) solution  and a
+greedy approach to extend the  sensor network lifetime by organizing the sensors
+into a maximal number of  non-disjoint cover sets.  Simulation results show that
+by  allowing sensors  to  participate  in multiple  sets,  the network  lifetime
+increases     compared     with     related     work~\cite{cardei2005improving}.
+In~\cite{berman04},  the  authors  have  formulated  the  lifetime  problem  and
+suggested another (LP) technique to  solve this problem.  A centralized solution
+based  on  the  Garg-K\"{o}nemann  algorithm~\cite{garg98},  provably  near  the
+optimal solution, is also proposed.
+
+In~\cite{yang2014maximum},  the  authors  have  proposed  a  linear  programming
+approach for selecting  the minimum number of working sensor  nodes, in order to
+as to preserve  a maximum coverage and extend lifetime of  the network. Cheng et
+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}.

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

-In distributed $\&$ localized coverage algorithms, the required computation to schedule the activity of sensor nodes will be done by the cooperation among the neighbours nodes. These algorithms may require more computation power  for the processing by the cooperated sensor nodes but they are more scaleable for large WSNs.  Localized and distributed algorithms generally result in non-disjoint set covers. 
-
-Some      distributed     algorithms      have      been     developed
-in~\cite{Gallais06,Tian02,Ye03,Zhang05,HeinzelmanCB02, yardibi2010distributed}  to perform the
-scheduling so as to coverage preservation.   Distributed algorithms typically  operate in  rounds for
-a predetermined  duration. At  the  beginning of  each  round, a  sensor
-exchanges information with its neighbors and makes a decision to either
-remain turned  on or to  go to sleep  for the round. This  decision is
-basically made on simple greedy criteria like  the largest uncovered
-area   \cite{Berman05efficientenergy},   maximum   uncovered   targets
-\cite{lu2003coverage}.   In \cite{Tian02}, the  scheduling scheme  is divided
-into rounds, where each round  has a self-scheduling phase followed by
-a sensing phase.  Each sensor  broadcasts a message containing the node ID
-and the node location  to its neighbors at the beginning  of each round. A
-sensor determines  its status by  a rule named off-duty  eligible rule,
-which tells  him to  turn off if  its sensing  area is covered  by its
-neighbors. A  back-off scheme is  introduced to let each  sensor delay
-the decision process  with a random period of time,  in order to avoid
-simultaneous conflicting decisions between nodes and  lack of coverage on any area.
-\cite{prasad2007distributed}  defines a model  for capturing
-the dependencies between different cover sets and proposes localized
-heuristic based on this dependency. The algorithm consists of two
-phases, an initial  setup phase during which each  sensor computes and
-prioritizes the  covers and  a sensing phase  during which  each sensor
-first decides  its on/off status, and  then remains on or off for the
-rest  of the  duration. 
-
-The authors in \cite{yardibi2010distributed}  developed a distributed adaptive sleep scheduling algorithm (DASSA) for WSNs with partial coverage. DASSA does not require location information of sensors while maintaining connectivity and satisfying a user defined coverage target. In DASSA, nodes use the residual energy levels and feedback from the sink for scheduling the activity of their neighbors. This feedback mechanism reduces the randomness in scheduling that would otherwise occur due to the absence of location information. 
-
-In \cite{ChinhVu},  the author proposed a novel  distributed heuristic, called
-Distributed Energy-efficient  Scheduling for k-coverage  (DESK), which
-ensures that the energy consumption  among the sensors is balanced and
-the lifetime  maximized while the coverage  requirement is maintained.
-This  heuristic   works  in  rounds,  requires   only  1-hop  neighbor
-information,  and each  sensor decides  its status  (active  or sleep)
-based    on    the    perimeter    coverage    model    proposed    in
-\cite{Huang:2003:CPW:941350.941367}.
-
+In distributed  and localized coverage  algorithms, the required  computation to
+schedule the  activity of  sensor nodes  will be done  by the  cooperation among
+neighboring nodes. These  algorithms may require more computation  power for the
+processing by the cooperating sensor nodes, but they are more scalable for large
+WSNs.  Localized and distributed algorithms generally result in non-disjoint set
+covers.
+
+Many distributed algorithms have been  developed to perform the scheduling so as
+to          preserve         coverage,          see          for         example
+\cite{Gallais06,Tian02,Ye03,Zhang05,HeinzelmanCB02,yardibi2010distributed}.
+Distributed  algorithms   typically  operate  in  rounds   for  a  predetermined
+duration. At  the beginning of each  round, a sensor  exchanges information with
+its neighbors and makes a decision to  either remain turned on or to go to sleep
+for the  round. This decision is  basically made on simple  greedy criteria like
+the largest  uncovered area \cite{Berman05efficientenergy}  or maximum uncovered
+targets  \cite{lu2003coverage}.   In  \cite{Tian02},  the scheduling  scheme  is
+divided into rounds, where each round  has a self-scheduling phase followed by a
+sensing phase.  Each sensor broadcasts  a message containing the node~ID and the
+node  location to  its  neighbors at  the  beginning of  each  round.  A  sensor
+determines its status by a rule named off-duty eligible rule, which tells him to
+turn off if its  sensing area is covered by its neighbors.  A back-off scheme is
+introduced to let each sensor delay the decision process with a random period of
+time, in  order to  avoid simultaneous conflicting  decisions between  nodes and
+lack  of coverage  on any  area.   In \cite{prasad2007distributed}  a model  for
+capturing  the dependencies  between  different  cover sets  is  defined and  it
+proposes localized heuristic based on this dependency. The algorithm consists of
+two  phases,  an initial  setup  phase during  which  each  sensor computes  and
+prioritizes  the covers  and  a sensing  phase  during which  each sensor  first
+decides  its on/off  status, and  then remains  on or  off for  the rest  of the
+duration. 
+
+The  authors  in  \cite{yardibi2010distributed}  have  developed  a  Distributed
+Adaptive  Sleep Scheduling  Algorithm (DASSA)  for WSNs  with  partial coverage.
+DASSA  does  not  require  location  information of  sensors  while  maintaining
+connectivity and satisfying a user defined coverage target.  In DASSA, nodes use
+the  residual  energy levels  and  feedback from  the  sink  for scheduling  the
+activity of their neighbors.  This  feedback mechanism reduces the randomness in
+scheduling  that  would   otherwise  occur  due  to  the   absence  of  location
+information.  In  \cite{ChinhVu}, the author  have proposed a  novel distributed
+heuristic, called Distributed Energy-efficient Scheduling for k-coverage (DESK),
+which ensures that the energy consumption  among the sensors is balanced and the
+lifetime maximized while the coverage requirement is maintained.  This heuristic
+works in  rounds, requires  only one-hop neighbor  information, and  each sensor
+decides  its status  (active or  sleep) based  on the  perimeter  coverage model
+proposed in \cite{Huang:2003:CPW:941350.941367}.

 
 %Our Work, which is presented in~\cite{idrees2014coverage} proposed a coverage optimization protocol to improve the lifetime in
 %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. 
 
 
 %Our Work, which is presented in~\cite{idrees2014coverage} proposed a coverage optimization protocol to improve the lifetime in
 %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 a Coverage-Aware, Distributed  Energy- Efficient and distributed clustering methods respectively, which aims to extend the network lifetime, while the coverage is ensured.
-S.  Misra  et al.  \cite{Misra}  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 proposed  method  preserves  the
-network connectivity by formation of the network backbone. 
-More    recently,   Shibo   et
-al. \cite{Shibo}  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. proposed 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
-period  of time  where the  sensors are  active or  not.  Indeed, each
-sensor  maintains its  own timer  and its  wake-up time is randomized
-\cite{Ye03} or regulated \cite{cardei2005maximum} over time. 
-
-The MuDiLCO protocol (for Multiperiod Distributed Lifetime Coverage Optimization protocol) presented in this paper is an extension of the approach explained in~\cite{idrees2014coverage}. In~\cite{idrees2014coverage}, the protocol is deployed over only two subregions. Simulation results have shown that it was more interesting to divide the area into several subregions, given the computation complexity. Compared to our previous paper, we study here the possibility of dividing the sensing phase into multiple rounds and we also add a model of energy consumption to assess the efficiency of our approach.
+The  works presented in  \cite{Bang, Zhixin,  Zhang} focus  on coverage-aware,
+distributed energy-efficient,  and distributed clustering  methods respectively,
+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
+proposed method preserves  the network connectivity by formation  of the network
+backbone.  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   proposed  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 period of time  where the sensors  are active or
+not.   Indeed, each  sensor maintains  its  own timer  and its  wake-up time  is
+randomized \cite{Ye03} or regulated \cite{cardei2005maximum} over time.
+
+The MuDiLCO protocol (for Multiround Distributed Lifetime Coverage Optimization
+protocol) presented  in this  paper is an  extension of the  approach introduced
+in~\cite{idrees2014coverage}.   In~\cite{idrees2014coverage},  the  protocol  is
+deployed over  only two  subregions. Simulation results  have shown that  it was
+more  interesting  to  divide  the  area  into  several  subregions,  given  the
+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.
+
+
+
+
+\fi

 %The main contributions of our MuDiLCO Protocol can be summarized as follows:
 %(1) The high coverage ratio, (2) The reduced number of active nodes, (3) The distributed optimization over the subregions in the area of interest, (4) The distributed dynamic leader election at each round based on some priority factors that led to energy consumption balancing among the nodes in the same subregion, (5) The primary point coverage model to represent each sensor node in the network, (6) The activity scheduling based optimization on the subregion, which are based on the primary point coverage model to activate as less number as possible of sensor nodes for a multirounds to take the mission of the coverage in each subregion, (7) The very low energy consumption, (8) The higher network lifetime.
 %\section{Preliminaries}
 %\label{Pr}
 
 %The main contributions of our MuDiLCO Protocol can be summarized as follows:
 %(1) The high coverage ratio, (2) The reduced number of active nodes, (3) The distributed optimization over the subregions in the area of interest, (4) The distributed dynamic leader election at each round based on some priority factors that led to energy consumption balancing among the nodes in the same subregion, (5) The primary point coverage model to represent each sensor node in the network, (6) The activity scheduling based optimization on the subregion, which are based on the primary point coverage model to activate as less number as possible of sensor nodes for a multirounds to take the mission of the coverage in each subregion, (7) The very low energy consumption, (8) The higher network lifetime.
 %\section{Preliminaries}
 %\label{Pr}
 

-
-
-

 %Network Lifetime
 
 %\subsection{Network Lifetime}
 %Network Lifetime
 
 %\subsection{Network Lifetime}


@@ -313,10 +498,7 @@ The MuDiLCO protocol (for Multiperiod Distributed Lifetime Coverage Optimization
 %active sensor node without  connectivity towards a base station cannot
 %transmit information on an event in the area that it monitors.
 
 %active sensor node without  connectivity towards a base station cannot
 %transmit information on an event in the area that it monitors.
 

-
-
-
-\section{ The MuDiLCO Protocol Description}
+\section{MuDiLCO protocol description}

 \label{pd}
 
 %Our work will concentrate on the area coverage by design
 \label{pd}
 
 %Our work will concentrate on the area coverage by design


@@ -331,36 +513,89 @@ The MuDiLCO protocol (for Multiperiod Distributed Lifetime Coverage Optimization
 %minimizing  overcoverage (points  covered by  multiple  active sensors
 %simultaneously).
 
 %minimizing  overcoverage (points  covered by  multiple  active sensors
 %simultaneously).
 

-
-
-%In this section, we introduce a Multiperiod Distributed Lifetime Coverage Optimization protocol, which is called MuDiLCO. It is  distributed on each subregion in the area of interest. It is based on two efficient techniques: network
+%In this section, we introduce a Multiround Distributed Lifetime Coverage Optimization protocol, which is called MuDiLCO. It is  distributed on each subregion in the area of interest. It is based on two efficient techniques: network

 %leader election and sensor activity scheduling for coverage preservation and energy conservation continuously and efficiently to maximize the lifetime in the network.  
 %The main features of our MuDiLCO protocol:
 %i)It divides the area of interest into subregions by using divide-and-conquer concept, ii)It requires only the information of the nodes within the subregion, iii) it divides the network lifetime into periods, which consists in round(s), iv)It based on the autonomous distributed decision by the nodes in the subregion to elect the Leader, v)It apply the activity scheduling based optimization on the subregion, vi)  it achieves an energy consumption balancing among the nodes in the subregion by selecting different nodes as a leader during the network lifetime, vii) It uses the optimization to select the best representative non-disjoint sets of sensors in the subregion by optimize the coverage and the lifetime over the area of interest, viii)It uses our proposed primary point coverage model, which represent the sensing range of the sensor as a set of points, which are used by the our optimization algorithm, ix) It uses a simple energy model that takes communication, sensing and computation energy consumptions into account to evaluate the performance of our Protocol.
 
 %leader election and sensor activity scheduling for coverage preservation and energy conservation continuously and efficiently to maximize the lifetime in the network.  
 %The main features of our MuDiLCO protocol:
 %i)It divides the area of interest into subregions by using divide-and-conquer concept, ii)It requires only the information of the nodes within the subregion, iii) it divides the network lifetime into periods, which consists in round(s), iv)It based on the autonomous distributed decision by the nodes in the subregion to elect the Leader, v)It apply the activity scheduling based optimization on the subregion, vi)  it achieves an energy consumption balancing among the nodes in the subregion by selecting different nodes as a leader during the network lifetime, vii) It uses the optimization to select the best representative non-disjoint sets of sensors in the subregion by optimize the coverage and the lifetime over the area of interest, viii)It uses our proposed primary point coverage model, which represent the sensing range of the sensor as a set of points, which are used by the our optimization algorithm, ix) It uses a simple energy model that takes communication, sensing and computation energy consumptions into account to evaluate the performance of our Protocol.
 

+\subsection{Assumptions}
+
+We  consider a  randomly and  uniformly  deployed network  consisting of  static
+wireless sensors.  The sensors are  deployed in high density to ensure initially
+a high  coverage ratio  of the interested  area.  We  assume that all  nodes are
+homogeneous  in   terms  of  communication  and   processing  capabilities,  and
+heterogeneous  from the  point  of view  of  energy provision.   Each sensor  is
+supposed  to get information  on its  location either  through hardware  such as
+embedded GPS or through location discovery algorithms.
+   
+To model  a sensor node's coverage  area, we consider the  boolean disk coverage
+model   which  is  the   most  widely   used  sensor   coverage  model   in  the
+literature. Thus, each  sensor has a constant sensing range  $R_s$ and all space
+points within  the disk centered  at the sensor  with the radius of  the sensing
+range  is  said  to  be  covered  by  this sensor.   We  also  assume  that  the
+communication   range  satisfies   $R_c  \geq   2R_s$.   In   fact,   Zhang  and
+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.
+
+%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 also 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 calculate the primary  points directly based on the proposed model. We  use these primary points (that can be increased or decreased if necessary)  as references to ensure that the monitored  region  of interest  is  covered by the selected  set  of sensors, instead of using all the points in the area. 
+We can  calculate  the positions of the selected primary
+points in the circle disk of the sensing range of a wireless sensor
+node (see Figure~\ref{fig1}) as follows:\\
+Assuming that the point center of a wireless sensor node is located at $(p_x,p_y)$, we can define up to 25 primary points $X_1$ to $X_{25}$.\\
+%$(p_x,p_y)$ = 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}
+    
+
+
+

 
 

-\subsection{ Assumptions and Models}
-We consider  a randomly and  uniformly deployed network  consisting of
-static  wireless sensors. The  wireless sensors  are deployed  in high
-density to ensure initially a high coverage ratio of the interested area. We
-assume that  all nodes are  homogeneous in terms of  communication and
-processing capabilities and heterogeneous in term of energy provision.
-The  location  information is  available  to  the  sensor node  either
-through hardware such as embedded GPS or through location discovery
-algorithms.   
-\indent We consider a boolean  disk coverage model which is the most
-widely used sensor coverage model in the literature. Each sensor has a
-constant sensing range $R_s$. All  space points within a disk centered
-at  the sensor with  the radius  of the  sensing range  is said  to be
-covered by this sensor. We also assume that the communication range $R_c \geq 2R_s$.
-In  fact,   Zhang  and Zhou~\cite{Zhang05} proved that if the transmission range fulfills the
-previous hypothesis, a complete coverage of a convex area implies
-connectivity among the working nodes in the active mode.
-
-\indent Instead of working with the coverage area, we consider for each
-sensor a set of points called  primary points. We also assume that the
-sensing disk defined  by a sensor is covered if all the primary points of
-this sensor 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


@@ -375,106 +610,144 @@ this sensor are covered. The choice of number and locations of primary points is
 %Initially, the sensor node check it's remaining energy in order to participate in the current round. Each sensor node determines it's position and it's subregion based Embedded GPS  or Location Discovery Algorithm. After that, All the sensors collect position coordinates, current remaining energy, sensor node id, and the number of its one-hop live neighbors during the information exchange. It stores this information into a list $L$.
 %The sensor node enter in listening mode waiting to receive ActiveSleep packet from the leader after the decision to apply multi-round activity scheduling during the sensing phase. Each sensor node will execute the Algorithm~1 to know who is the leader. After that, if the sensor node is leader, It will execute the integer program algorithm ( see section~\ref{cp}) to optimize the coverage and the lifetime in it's subregion. After the decision, the optimization approach will produce the cover sets of sensor nodes to take the mission of coverage during the sensing phase for $T$ rounds. The leader will send ActiveSleep packet to each sensor node in the subregion to inform him to it's schedule for $T$ rounds during the period of sensing, either Active or sleep until the starting of next period. Based on the decision, the leader as other nodes in subregion, either go to be active or go to be sleep based on it's schedule for $T$ rounds during current sensing phase. the other nodes in the same subregion will stay in listening mode waiting the ActiveSleep packet from the leader. After finishing the time period for sensing, which are includes $T$ rounds, all the sensor nodes in the same subregion will start new period by executing the MuDiLCO protocol and the lifetime in the subregion will continue until all the sensor nodes are died or the network becomes disconnected in the subregion.
 
 %Initially, the sensor node check it's remaining energy in order to participate in the current round. Each sensor node determines it's position and it's subregion based Embedded GPS  or Location Discovery Algorithm. After that, All the sensors collect position coordinates, current remaining energy, sensor node id, and the number of its one-hop live neighbors during the information exchange. It stores this information into a list $L$.
 %The sensor node enter in listening mode waiting to receive ActiveSleep packet from the leader after the decision to apply multi-round activity scheduling during the sensing phase. Each sensor node will execute the Algorithm~1 to know who is the leader. After that, if the sensor node is leader, It will execute the integer program algorithm ( see section~\ref{cp}) to optimize the coverage and the lifetime in it's subregion. After the decision, the optimization approach will produce the cover sets of sensor nodes to take the mission of coverage during the sensing phase for $T$ rounds. The leader will send ActiveSleep packet to each sensor node in the subregion to inform him to it's schedule for $T$ rounds during the period of sensing, either Active or sleep until the starting of next period. Based on the decision, the leader as other nodes in subregion, either go to be active or go to be sleep based on it's schedule for $T$ rounds during current sensing phase. the other nodes in the same subregion will stay in listening mode waiting the ActiveSleep packet from the leader. After finishing the time period for sensing, which are includes $T$ rounds, all the sensor nodes in the same subregion will start new period by executing the MuDiLCO protocol and the lifetime in the subregion will continue until all the sensor nodes are died or the network becomes disconnected in the subregion.
 

-\subsection{The Main Idea}
-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
-simultaneously. \\
-
-\noindent Our MuDiLCO protocol works in periods fashion as shown in figure~\ref{fig2}.
+\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.
+
+As  can be seen  in Figure~\ref{fig2},  our protocol  works in  periods fashion,
+where  each is  divided  into 4  phases: Information~Exchange,  Leader~Election,
+Decision, and Sensing.  Each sensing phase may be itself divided into $T$ rounds
+\textcolor{green} {of equal duration} and for each round a set of sensors (a cover set) is responsible for the sensing
+task. In  this way  a multiround optimization  process is performed  during each
+period  after  Information~Exchange  and  Leader~Election phases,  in  order  to
+produce $T$ cover sets that will take the mission of sensing for $T$ rounds.

 \begin{figure}[ht!]
 \begin{figure}[ht!]

-\centering
-\includegraphics[width=95mm]{Modelgeneral.pdf} % 70mm
-\caption{MuDiLCO protocol}
+\centering \includegraphics[width=100mm]{Modelgeneral.pdf} % 70mm
+\caption{The MuDiLCO protocol scheme executed on each node}

 \label{fig2}
 \end{figure} 
 
 \label{fig2}
 \end{figure} 
 

-Each period is divided into 4 phases: Information  Exchange,
-Leader  Election, Decision,  and  Sensing.  Each sensing phase may be itself divided into $T$ rounds.
+%Each period is divided into 4 phases: Information  Exchange,
+%Leader  Election, Decision,  and  Sensing.  Each sensing phase may be itself divided into $T$ rounds.

 % set cover responsible for the sensing task.  
 % set cover responsible for the sensing task.  

-For each round a set of sensors (said a cover set) is responsible for the sensing task.
-This protocol is reliable  against an unexpected node failure  because it works
-in periods. On the one hand,  if a node  failure is  detected before
-making the 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.  
-The energy
-consumption  and  some other  constraints  can  easily  be taken  into
-account  since  the  sensors   can  update  and  then  exchange  their
-information (including their residual energy) at the beginning of each
-period.  However,   the  pre-sensing  phases   (Information Exchange,  Leader
-Election,  Decision) are energy  consuming for  some nodes,  even when
-they do not  join the network to monitor the  area. 
+%For each round a set of sensors (said a cover set) is responsible for the sensing task.
+
+This protocol minimizes the impact of unexpected node failure (not due to batteries
+running out of energy), because it works in periods. 
+%This protocol is reliable against an unexpected node failure, because it works in periods. 
+%%RC : why? I am not convinced
+ On the one hand, if a node failure is detected before  making the
+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 rounds are predefined parameters. Round duration should be long enough to hide the system control overhead and short enough to minimize the negative effects in case of node failure.}
+
+%%RC so if there are at least one failure per period, the coverage is bad...
+%%MS if we want to be reliable against many node failures we need to have an
+%% overcoverage...  
+
+The  energy consumption  and some  other constraints  can easily  be  taken into
+account,  since the  sensors  can  update and  then  exchange their  information
+(including their residual energy) at the beginning of each period.  However, the
+pre-sensing  phases (Information  Exchange, Leader  Election, and  Decision) are
+energy  consuming for some  nodes, even  when they  do not  join the  network to
+monitor the area.

 
 %%%%%%%%%%%%%%%%%parler optimisation%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
 %%%%%%%%%%%%%%%%%parler optimisation%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 

-We define two types of packets used by MuDiLCO protocol :
+We define two types of packets that will be used by the proposed protocol:

 \begin{enumerate}[(a)] 
 \begin{enumerate}[(a)] 

-\item INFO packet: sent by each sensor node to all the nodes inside a subregion for information exchange.
-\item ActiveSleep packet: sent by the leader to all the nodes inside a subregion to inform them to be Active or Sleep during the sensing phase.
+\item INFO  packet: such a  packet  will be sent by  each sensor node  to all the
+  nodes inside a subregion for information exchange.
+\item  Active-Sleep  packet: sent  by  the  leader to  all  the  nodes inside  a
+  subregion to  inform them to remain Active  or to go Sleep  during the sensing
+  phase.

 \end{enumerate}
 
 \end{enumerate}
 

-There are five status for each sensor node in the network :
+There are five status for each sensor node in the network:

 \begin{enumerate}[(a)] 
 \begin{enumerate}[(a)] 

-\item LISTENING: Sensor is waiting for a decision (to be active or not)
-\item COMPUTATION: Sensor applies the optimization process as leader
-\item ACTIVE: Sensor is active
-\item SLEEP: Sensor is turned off
-\item COMMUNICATION: Sensor is transmitting or receiving packet
+\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 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}
 
 Below, we describe each phase in more details.
 
 \subsection{Information Exchange Phase}
 
 \end{enumerate}
 
 Below, we describe each phase in more details.
 
 \subsection{Information Exchange Phase}
 

-Each sensor node $j$ sends its position, remaining energy $RE_j$, and
-the number of neighbours  $NBR_j$ to all wireless sensor nodes in
-its subregion by using an INFO packet (containing information on position coordinates, current remaining energy, sensor node id, number of its one-hop live neighbors)  and then listens to the packets
-sent from  other nodes.  After that, each  node will  have information
-about  all the  sensor  nodes in  the  subregion.  In  our model,  the
-remaining energy corresponds to the time that a sensor can live in the
-active mode.
+Each sensor node $j$ sends its position, remaining energy $RE_j$, and the number
+of neighbors $NBR_j$  to all wireless sensor nodes in its  subregion by using an
+INFO packet  (containing information on position  coordinates, current remaining
+energy, sensor node ID, number of its one-hop live neighbors) and then waits for
+packets sent by other nodes.  After  that, each node will have information about
+all  the sensor  nodes in  the subregion.   In our  model, the  remaining energy
+corresponds to the time that a sensor can live in the active mode.

 
 %\subsection{\textbf Working Phase:}
 
 %The working phase works in rounding fashion. Each round include 3 steps described as follow :
 
 
 %\subsection{\textbf Working Phase:}
 
 %The working phase works in rounding fashion. Each round include 3 steps described as follow :
 

-\subsection{Leader Election Phase}
-This step includes 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
-select WSNL.  The nodes in the  same subregion will  select the leader
-based on  the received  information from all  other nodes in  the same
-subregion.  The selection criteria  in order  of priority  are: larger
-number  of neighbours,  larger remaining  energy, and  then in  case of
-equality, larger index. Observations on previous simulations suggest to use the number of $1-hop$ neighbours as the primary criterion to reduce energy consumption due to the communication.
-
-%the more priority selection factor is the number of $1-hop$ neighbours, $NBR j$, which can  minimize the energy consumption during the communication Significantly.  
+\subsection{Leader Election phase}
+
+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 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
+number  of  one-hop  neighbors  as   the  primary  criterion  to  reduce  energy
+consumption due to the communications.
+
+%the more priority selection factor is the number of $1-hop$ neighbors, $NBR j$, which can  minimize the energy consumption during the communication Significantly.  

 %The pseudo-code for leader election phase is provided in Algorithm~1.
 
 %Where $E_{th}$ is the minimum energy needed to stay active during the sensing phase. As shown in Algorithm~1, the more priority selection factor is the number of $1-hop$ neighbours, $NBR j$, which can  minimize the energy consumption during the communication Significantly.  
 
 %The pseudo-code for leader election phase is provided in Algorithm~1.
 
 %Where $E_{th}$ is the minimum energy needed to stay active during the sensing phase. As shown in Algorithm~1, the more priority selection factor is the number of $1-hop$ neighbours, $NBR j$, which can  minimize the energy consumption during the communication Significantly.  
 

-

 \subsection{Decision phase}
 \subsection{Decision phase}

-The  WSNL will  solve an  integer  program to
-select which cover sets will be activated in the following sensing phase
-to cover  the subregion. The integer program will produce $T$ cover sets (for $T$ rounds). WSNL will send  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.
-\indent  The integer  program   is   based   on  the   model   proposed   by
-\cite{pedraza2006} with some modification, where the objective is  to find a maximum number of
-disjoint  cover sets.   To accomplish  this goal,  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 $X_{t,j}$,  which  determine the possiblity 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.
+
+Each  WSNL will \textcolor{green}{ solve an integer program to  select which  cover sets  will be
+activated in  the following  sensing phase  to cover the  subregion to  which it
+belongs.  $T$ 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
+
+
+
+
+
+
+
+%\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
+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
 

-\noindent  For  a primary  point  $p$,  let  $\alpha_{j,p}$ denote  the
-indicator function of whether the point $p$ is covered, that is:
+For a  primary point  $p$, let $\alpha_{j,p}$  denote the indicator  function of
+whether the point $p$ is covered, that is:

 \begin{equation}
 \alpha_{j,p} = \left \{ 
 \begin{array}{l l}
 \begin{equation}
 \alpha_{j,p} = \left \{ 
 \begin{array}{l l}


@@ -484,8 +757,8 @@ indicator function of whether the point $p$ is covered, that is:
 \end{array} \right.
 %\label{eq12} 
 \end{equation}
 \end{array} \right.
 %\label{eq12} 
 \end{equation}

-The number of active sensors that cover the primary point $p$ during round $t$ is equal
-to $\sum_{j \in J} \alpha_{j,p} * X_{t,j}$ where:
+The number of  active sensors that cover the  primary point $p$ during
+round $t$ is equal to $\sum_{j \in J} \alpha_{j,p} * X_{t,j}$ where:

 \begin{equation}
 X_{t,j} = \left \{ 
 \begin{array}{l l}
 \begin{equation}
 X_{t,j} = \left \{ 
 \begin{array}{l l}


@@ -504,10 +777,10 @@ We define the Overcoverage variable $\Theta_{t,p}$ as:
 \end{array} \right.
 \label{eq13} 
 \end{equation}
 \end{array} \right.
 \label{eq13} 
 \end{equation}

-\noindent 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 Undercoverage variable $U_{t,p}$ of the primary point $p$ during round $t$ is defined
-by:
+More  precisely, $\Theta_{t,p}$  represents the  number of  active  sensor nodes
+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}
 U_{t,p} = \left \{ 
 \begin{array}{l l}
 \begin{equation}
 U_{t,p} = \left \{ 
 \begin{array}{l l}


@@ -517,114 +790,305 @@ U_{t,p} = \left \{
 \label{eq14} 
 \end{equation}
 
 \label{eq14} 
 \end{equation}
 

-\noindent Our coverage optimization problem can then be formulated as follows
-
-
+Our coverage optimization problem can then be formulated as follows:

 \begin{equation}
 \begin{equation}

- Minimize   \sum_{t=1}^{T} \sum_{p=1}^{P} \left(W_{\theta}* \Theta_{t,p} + W_{U} * U_{t,p}  \right)  \label{eq15} 
+ \min \sum_{t=1}^{T} \sum_{p=1}^{P} \left(W_{\theta}* \Theta_{t,p} + W_{U} * U_{t,p}  \right)  \label{eq15} 

 \end{equation}
 
 \end{equation}
 

-\hspace{30 mm} Subject to\\
+Subject to

 \begin{equation}
 \begin{equation}

-  \sum_{j=1}^{J} \alpha_{j,p} * X_{t,j}   = \Theta_{t,p} - U_{t,p} + 1 \label{eq16} \hspace{6 mm} \forall p \in P, t = 1..T
+  \sum_{j=1}^{|J|} \alpha_{j,p} * X_{t,j}   = \Theta_{t,p} - U_{t,p} + 1 \label{eq16} \hspace{6 mm} \forall p \in P, t = 1,\dots,T

 \end{equation}
 
 \begin{equation}
 \end{equation}
 
 \begin{equation}

-  \sum_{t=1}^{T}  X_{t,j}   \leq  \floor*{RE_{j}/E_{th}} \hspace{6 mm} \forall j \in J, t = 1..T
+  \sum_{t=1}^{T}  X_{t,j}   \leq  \floor*{RE_{j}/E_{R}} \hspace{6 mm} \forall j \in J, t = 1,\dots,T

   \label{eq144} 
 \end{equation}
 
 \begin{equation}
   \label{eq144} 
 \end{equation}
 
 \begin{equation}

-X_{t,j} \in \lbrace0,1\rbrace,   \hspace{10 mm} \forall j \in J, t = 1..T \label{eq17} 
+X_{t,j} \in \lbrace0,1\rbrace,   \hspace{10 mm} \forall j \in J, t = 1,\dots,T \label{eq17} 

 \end{equation}
 
 \begin{equation}
 \end{equation}
 
 \begin{equation}

-U_{t,p} \in \lbrace0,1\rbrace, \hspace{10 mm}\forall p \in P, t = 1..T  \label{eq18} 
+U_{t,p} \in \lbrace0,1\rbrace, \hspace{10 mm}\forall p \in P, t = 1,\dots,T  \label{eq18} 

 \end{equation}
 
 \begin{equation}
 \end{equation}
 
 \begin{equation}

- \Theta_{t,p} \geq 0  \hspace{10 mm}\forall p \in P, t = 1..T  \label{eq178} 
+ \Theta_{t,p} \geq 0 \hspace{10 mm}\forall p \in P, t = 1,\dots,T \label{eq178}

 \end{equation}
 
 %\begin{equation}
 %(W_{\theta}+W_{\psi} = P)    \label{eq19} 
 %\end{equation}
 
 \end{equation}
 
 %\begin{equation}
 %(W_{\theta}+W_{\psi} = P)    \label{eq19} 
 %\end{equation}
 

+%%RC why W_{\theta} is not defined (only one sentence)? How to define in practice Wtheta and Wu?

 
 \begin{itemize}
 
 \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);
-\item $\Theta_{t,p}$  : {\it overcoverage}, the number  of sensors minus
-  one that are covering the primary point $p$ during the round $t$;
-\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 covered).
+\item $X_{t,j}$:  indicates whether  or not the  sensor $j$ is  actively sensing
+  during round $t$ (1 if yes and 0 if not);
+\item $\Theta_{t,p}$ - {\it overcoverage}:  the number of sensors minus one that
+  are covering the primary point $p$ during round $t$;
+\item  $U_{t,p}$ -  {\it undercoverage}:  indicates whether  or not  the primary
+  point $p$  is being covered during round $t$ (1  if not covered  and 0 if
+  covered).

 \end{itemize}
 
 \end{itemize}
 

-The first group  of constraints indicates that some  primary point $p$
-should be covered by at least one  sensor and, if it is not always the
-case,  overcoverage  and  undercoverage  variables  help  balancing  the
-restriction  equations by taking  positive values. Constraint \ref{eq144} guarantees that the sensor has enough energy ($RE_j$ its remaining energy) to be alive during the selected rounds knowing that $E_{th}$ is the requiring energy to be alive during one round. 
-There are  two main         
-objectives.  First, we limit the overcoverage of primary points in order to
-activate a minimum number of sensors.  Second we prevent the 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. In our simulations priority is given to the coverage by choosing $W_{\theta}$ very large compared to $W_U$. 
+The first group  of constraints indicates that some primary  point $p$ should be
+covered by at least  one sensor and, if it is not  always the case, overcoverage
+and undercoverage  variables help balancing the restriction  equations by taking
+positive values. The constraint  given by equation~(\ref{eq144}) guarantees that
+the sensor has enough energy ($RE_j$  corresponds to its remaining energy) to be
+alive during  the selected rounds knowing  that $E_{R}$ is the  amount of energy
+required to be alive during one round.
+
+There  are two main  objectives.  First,  we limit  the overcoverage  of primary
+points in order to activate a  minimum number of sensors.  Second we prevent the
+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
+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.
 
 
 \subsection{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. 
-% 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. 

 
 

+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}                
+\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
  % \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 it's subregion} \; 
+  %\emph{Initialize the sensor node and determine it's position and subregion} \; 

   
   

-  \If{ $RE_j \geq E_{th}$ }{
-      \emph{ $s_j.status$ = LISTENING}\;
-      \emph{ Send and Receive INFO Packet to and from other nodes in the 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{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{Selection of LeaderID}\;
-           \If{ $ s_j.ID = LeaderID $}{ 
-               \emph{Execute Integer Program Algorithm $(Schedule_{T,J})$ }\;
-               \emph{ Send $ActiveSleep()$ Packet with $Schedule_{1..T,k}$  }\;
-                 \emph{UPDATE $RE_j $}\;
-                         }       
-           \Else{ 
-                 \emph{Wait $ActiveSleep()$ Packet from the Leader}\;
-                % \emph{After receiving Packet, Retrieve the schedule and the $T$ rounds}\;
-                 \emph{UPDATE $RE_j $}\;
-           }  
-    %  }
-
+      \emph{LeaderID = Leader election}\;
+      \If{$ 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}
   }
   \Else { Exclude $s_j$ from entering in the current sensing phase}

-   
+  

  %   \emph{return X} \;
 \caption{MuDiLCO($s_j$)}
 \label{alg:MuDiLCO}
 
 \end{algorithm}
 
  %   \emph{return X} \;
 \caption{MuDiLCO($s_j$)}
 \label{alg:MuDiLCO}
 
 \end{algorithm}
 

+\iffalse
+\textcolor{red}{This integer program can be solved using two approaches:}
+
+\subsection{\textcolor{red}{Optimization solver for Multiround Lifetime Coverage Optimization}}
+\label{glpk}
+\textcolor{red}{The modeling language for Mathematical Programming (AMPL)~\cite{AMPL} is  employed to generate the integer program instance  in a  standard format, which  is then read  and solved  by the optimization solver  GLPK (GNU  linear Programming Kit  available in  the public domain) \cite{glpk} through a Branch-and-Bound method. We named the protocol which is based on GLPK solver in the decision phase as MuDiLCO.}
+\fi
+
+\iffalse
+
+\subsection{\textcolor{red}{Genetic Algorithm for Multiround Lifetime Coverage Optimization}}
+\label{GA}
+\textcolor{red}{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. In comparison with GLPK optimization solver, GA provides a near optimal solution with acceptable execution time, as well as it requires a less amount of memory especially for large size problems. GLPK provides optimal solution, but it requires higher execution time and amount of memory for large problem.}
+
+\textcolor{red}{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!]    
+       
+ \small
+ \SetKwInput{Input}{\textcolor{red}{Input}}
+ \SetKwInput{Output}{\textcolor{red}{Output}}
+ \Input{ \textcolor{red}{$ 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}$}}
+ \Output{\textcolor{red}{$\left\{\left(X_{1,1},\dots, X_{t,j}, \dots, X_{T,J}\right)\right\}_{t \in T, j \in J}$}}
+
+  \BlankLine
+  %\emph{Initialize the sensor node and determine it's position and subregion} \; 
+  \ForEach {\textcolor{red}{Individual $ind$ $\in$ $S_{pop}$}} {
+     \emph{\textcolor{red}{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{\textcolor{red}{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{\textcolor{red}{Evaluate Individual $(P, J, X_{t,j}^{ind}, \Theta_{t,p}^{ind}, U_{t,p}^{ind})$}}\;  
+  }
+  
+  \While{\textcolor{red}{ Stopping criteria is not satisfied} }{
+  
+  \emph{\textcolor{red}{Selection $(ind_1, ind_2)$}}\;
+    \emph{\textcolor{red}{Crossover $(P_c, X_{t,j}^{ind_1}, X_{t,j}^{ind_2}, Child_{t,j}^{ind_1}, Child_{t,j}^{ind_2})$}}\;
+    \emph{\textcolor{red}{Mutation $(P_m, Child_{t,j}^{ind_1}, Child_{t,j}^{ind_2})$}}\;
+   
+   
+   \emph{\textcolor{red}{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{\textcolor{red}{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{\textcolor{red}{Evaluate New Individual$(P, J, Child_{t,j}^{ind_1}, Ch.\Theta_{t,p}^{ind_1}, Ch.U_{t,p}^{ind_1})$}}\;  
+ \emph{\textcolor{red}{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{\textcolor{red}{Evaluate New Individual$(P, J, Child_{t,j}^{ind_2}, Ch.\Theta_{t,p}^{ind_2}, Ch.U_{t,p}^{ind_2})$}}\;  
+  
+ \emph{\textcolor{red}{Replacement $(P, J, T, Child_{t,j}^{ind_2}, Ch.\Theta_{t,p}^{ind_2}, Ch.U_{t,p}^{ind_2}, X_{t,j}^{ind}, \Theta_{t,p}^{ind}, U_{t,p}^{ind}  )$ }}\;
+  
+      
+  }
+  \emph{\textcolor{red}{$\left\{\left(X_{1,1},\dots,X_{t,j},\dots,X_{T,J}\right)\right\}$ =
+            Select Best Solution ($S_{pop}$)}}\;
+ \emph{\textcolor{red}{return X}} \;
+\caption{\textcolor{red}{GA($T, J$)}}
+\label{alg:GA}
+
+\end{algorithm}
+
+
+\begin{enumerate} [I)]
+
+\item \textcolor{red}{\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 sensing phase, the chromosome is defined as a schedule for alive  sensors and each chromosome contains $T$ rounds. The proposed GA uses binary representation, where 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 \textcolor{red}{\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 the current round. Otherwise, the value is set to 0. The energy constraint is applied for each sensor during all rounds. }
+
+
+\item \textcolor{red}{\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!]                
+  
+ \SetKwInput{Input}{\textcolor{red}{Input}}
+ \SetKwInput{Output}{\textcolor{red}{Output}}
+ \Input{ \textcolor{red}{parameters $P, J, ind, \alpha_{j,p}^{ind}, X_{t,j}^{ind}$}}
+ \Output{\textcolor{red}{$U^{ind} = \left\lbrace U_{1,1}^{ind}, \dots, U_{t,p}^{ind}, \dots, U_{T,P}^{ind} \right\rbrace$ and $\Theta^{ind} = \left\lbrace \Theta_{1,1}^{ind}, \dots, \Theta_{t,p}^{ind}, \dots, \Theta_{T,P}^{ind} \right\rbrace$}}
+
+  \BlankLine
+
+  \For{\textcolor{red}{$t\leftarrow 1$ \KwTo $T$}}{
+  \For{\textcolor{red}{$p\leftarrow 1$ \KwTo $P$}}{
+     
+ %    \For{$i\leftarrow 0$ \KwTo $I_j$}{
+       \emph{\textcolor{red}{$SUM\leftarrow 0$}}\;
+         \For{\textcolor{red}{$j\leftarrow 1$ \KwTo $J$}}{
+              \emph{\textcolor{red}{$SUM \leftarrow SUM + (\alpha_{j,p}^{ind} \times X_{t,j}^{ind})$ }}\;
+         }
+         
+         \If { \textcolor{red}{SUM = 0}} {
+         \emph{\textcolor{red}{$U_{t,p}^{ind} \leftarrow 0$}}\;
+         \emph{\textcolor{red}{$\Theta_{t,p}^{ind} \leftarrow 1$}}\;
+         }
+         \Else{
+         \emph{\textcolor{red}{$U_{t,p}^{ind} \leftarrow SUM -1$}}\;
+         \emph{\textcolor{red}{$\Theta_{t,p}^{ind} \leftarrow 0$}}\;
+         }
+     
+     }
+     
+  }
+\emph{\textcolor{red}{return $U^{ind}, \Theta^{ind}$ }} \;
+\caption{O-U-Coverage}
+\label{OU}
+
+\end{algorithm}
+
+
+
+\item \textcolor{red}{\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 gets 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 \textcolor{red}{\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 the 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.}

 
 
 
 

-\section{Simulations}
+
+\item \textcolor{red}{\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 \textcolor{red}{\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 operator in the proposed GA-MuDiLCO works as follow: If mutation probability $P_m$ is 100$\%$, then the mutation operation takes place on 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 \textcolor{red}{\textbf{Update O-U-Coverage for children:}
+Before evaluating 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 \textcolor{red}{\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 \textcolor{red}{\textbf{Replacement:}
+After evaluation 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 \textcolor{red}{\textbf{Stopping criteria:}
+The proposed GA-MuDiLCO stops when the stopping criteria is met. It stops after running for an amount of time in seconds equal to \textbf{Time limit}. The \textbf{Time limit} is the execution time obtained by the optimization solver GLPK for solving the same size of problem. The best solution will be selected as a schedule of sensors for $T$ rounds during the sensing phase in the current period.}
+
+
+
+\end{enumerate} 
+
+\fi
+
+\section{Experimental study}

 \label{exp}
 \label{exp}

-\subsection{Simulation Framework}
-We conducted  a series of simulations to evaluate the
-efficiency and the relevance of our approach,  using the  discrete event
-simulator  OMNeT++  \cite{varga}. The simulation parameters are summarized in
-Table~\ref{table3}. \\
+\subsection{Simulation setup}
+
+We  conducted  a  series of  simulations  to  evaluate  the efficiency  and  the
+relevance  of   our  approach,  using  the  discrete   event  simulator  OMNeT++
+\cite{varga}.     The     simulation     parameters    are     summarized     in
+Table~\ref{table3}.  Each experiment  for  a network  is  run over  25~different
+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
+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.
+
+%%RC these parameters are realistic?
+%% maybe we can increase the field and sensing range. 5mfor Rs it seems very small... what do the other good papers consider ?

 
 \begin{table}[ht]
 \caption{Relevant parameters for network initializing.}
 
 \begin{table}[ht]
 \caption{Relevant parameters for network initializing.}


@@ -642,48 +1106,72 @@ Parameter & Value  \\ [0.5ex]
 %heading
 \hline
 % inserts single horizontal line
 %heading
 \hline
 % inserts single horizontal line

-Sensing  Field  & $(50 \times 25)~m^2 $   \\
+Sensing field size & $(50 \times 25)~m^2 $   \\

 % inserting body of the table
 %\hline
 % inserting body of the table
 %\hline

-Nodes Number &  50, 100, 150, 200 and 250~nodes   \\
+Network size &  50, 100, 150, 200 and 250~nodes   \\

 %\hline
 %\hline

-Initial Energy  & 500-700~joules  \\  
+Initial energy  & 500-700~joules  \\  

 %\hline
 %\hline

-Sensing Time for One Round & 60 Minutes \\
-$E_{th}$ & 36 Joules\\
+Sensing time for one round & 60 Minutes \\
+$E_{R}$ & 36 Joules\\

 $R_s$ & 5~m   \\     
 %\hline
 $R_s$ & 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}

-
-25 simulation runs are performed with different network topologies. 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 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 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.\\
-

   
   

-Our MuDiLCO protocol is declined into four versions: MuDiLCO-1, MuDiLCO-3, MuDiLCO-5, and MuDiLCO-7, corresponding  to $T=1$, $T=3$, $T=5$ or $T=7$ ($T$ the number of rounds in one sensing period).  We call the method MuDiLCO-T for the general case. We compare MuDiLCO-T 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.   During the  decision phase,  in  each square,  one sensor  is chosen to remain on during the sensing phase time.\\
-
-
+\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
+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.
+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
+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
+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.
+
+\subsection{Energy model}
+
+We  use an  energy consumption  model  proposed by~\cite{ChinhVu}  and based  on
+\cite{raghunathan2002energy} with slight  modifications.  The energy consumption
+for  sending/receiving the packets  is added,  whereas the  part related  to the
+sensing range is removed because we consider a fixed sensing range.

 
 

-
-\subsection{Energy Model}
-
-We use an energy consumption model proposed by~\cite{ChinhVu} and based on ~\cite{raghunathan2002energy} with slight modifications.
-The energy consumption for sending/receiving the packets is added whereas the part related to the sensing range is removed because we consider a fixed sensing range.

 % We are took into account the energy consumption needed for the high computation during executing the algorithm on the sensor node. 
 %The new energy consumption model will take into account the energy consumption for communication (packet transmission/reception), the radio of the sensor node, data sensing, computational energy of Micro-Controller Unit (MCU) and high computation energy of MCU. 
 %revoir la phrase
 
 % We are took into account the energy consumption needed for the high computation during executing the algorithm on the sensor node. 
 %The new energy consumption model will take into account the energy consumption for communication (packet transmission/reception), the radio of the sensor node, data sensing, computational energy of Micro-Controller Unit (MCU) and high computation energy of MCU. 
 %revoir la phrase
 

-For our energy consumption model, we refer to the sensor node (Medusa II) which uses 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 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 is equal to $0.2575 mW$.
+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
+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
+summarized in Table~\ref{table4}.

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


@@ -694,17 +1182,17 @@ transmitting/receiving messages, sensing subsystem that collects data, and the p
 % centered columns (4 columns)
       \hline
 %inserts double horizontal lines
 % centered columns (4 columns)
       \hline
 %inserts double horizontal lines

-Sensor mode & MCU   & Radio & Sensing & Power (mWs) \\ [0.5ex]
+Sensor status & MCU & Radio & Sensing & Power (mW) \\ [0.5ex]

 \hline
 % inserts single horizontal line
 \hline
 % inserts single horizontal line

-Listening & ON & ON & ON & 20.05 \\
+LISTENING & on & on & on & 20.05 \\

 % inserting body of the table
 \hline
 % inserting body of the table
 \hline

-Active & ON & OFF & ON & 9.72 \\
+ACTIVE & on & off & on & 9.72 \\

 \hline
 \hline

-Sleep & OFF & OFF & OFF & 0.02 \\
+SLEEP & off & off & off & 0.02 \\

 \hline
 \hline

-Computation & ON & ON & ON & 26.83 \\
+COMPUTATION & on & on & on & 26.83 \\

 %\hline
 %\multicolumn{4}{|c|}{Energy needed to send/receive a 1-bit} & 0.2575\\
  \hline
 %\hline
 %\multicolumn{4}{|c|}{Energy needed to send/receive a 1-bit} & 0.2575\\
  \hline


@@ -714,235 +1202,397 @@ Computation & ON & ON & ON & 26.83 \\
 % is used to refer this table in the text
 \end{table}
 
 % is used to refer this table in the text
 \end{table}
 

-For sake of  simplicity we ignore the energy needed to turn on the
-radio, to start up the sensor node, the transition from mode to another, etc. 
+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
 %We also do not consider the need of collecting sensing data. PAS COMPRIS

-Thus, when a sensor becomes active (i.e., it already decides it's 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 Status-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.
-
-
-The initial energy of  each node is  randomly set in the interval $[500-700]$.  Each  sensor  node will  not participate in the next round if its remaining energy is less than $E_{th}=36 Joules$, the minimum energy needed for the node to stay alive during one round. This value has been computed by multiplying the energy consumed in active state (9.72 mWs) by the time in second for one round (3600 seconds). According to the interval of initial energy, a sensor may be alive during at most 20 rounds.\\
-
-
-
+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
+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
+less than  $E_{R}=36~\mbox{Joules}$, the minimum  energy needed for the  node to
+stay alive  during one round.  This value has  been computed by  multiplying the
+energy consumed in  active state (9.72 mW)  by the time in second  for one round
+(3600 seconds).  According to the  interval of initial  energy, a sensor  may be
+alive during at most 20 rounds.

 
 

- 

 \subsection{Metrics}
 
 \subsection{Metrics}
 

-We introduce the following performance metrics for evaluating our approach: 
+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 of a sensor field is  covered. In our case, we treated the sensing fields as a grid, and used each grid point as a sample point
-for calculating the coverage. The coverage ratio can be calculated by:
+\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
+  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
 \begin{equation*}
 \scriptsize

-\mbox{CR}(\%) = \frac{\mbox{$n^t$}}{\mbox{$N$}} \times 100.
+\mbox{CR}(\%) = \frac{\mbox{$n^t$}}{\mbox{$N$}} \times 100,

 \end{equation*}
 \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 of grid points in the sensing field of the network.
+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 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
 %simulations, the sensing field has been divided into 50 by 25 grid points, which means
 %there are $51 \times 26~ = ~ 1326$ points in total.
 % Therefore, for our simulations, the error in the coverage calculation is less than ~ 1 $\% $.
 
 %The accuracy of this method depends on the distance between grids. In our
 %simulations, the sensing field has been divided into 50 by 25 grid points, which means
 %there are $51 \times 26~ = ~ 1326$ points in total.
 % Therefore, for our simulations, the error in the coverage calculation is less than ~ 1 $\% $.
 

-\item{{\bf Number of Active Sensors Ratio(ASR)}:} It is important to have as few active nodes as possible in each round,
-in  order to  minimize  the communication  overhead  and maximize  the
-network lifetime. The Active Sensors Ratio is defined as follows:
+\item{{\bf Number  of Active Sensors Ratio  (ASR)}:} it is important  to have as
+  few  active  nodes  as  possible  in  each  round, in  order  to  minimize  the
+  communication overhead  and maximize the network lifetime.  The Active Sensors
+  Ratio is defined as follows:

 \begin{equation*}
 \begin{equation*}

-\scriptsize
-\mbox{ASR}(\%) =  \frac{\sum\limits_{r=1}^R \mbox{$A_r^t$}}{\mbox{$S$}} \times 100 .
-\end{equation*}
-Where: $A_r^t$ is the number of active sensors in the subregion $r$ during round $t$ in the current sensing phase, $S$ is the total number of sensors in the network, and $R$ is the total number of the 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 denoted by $Lifetime95$ (respectively  $Lifetime50$) as the amount of  time during which  the network  can  satisfy an area  coverage greater than $95\%$ (repectively $50\%$). We assume that the network
-is alive  until all  nodes have  been drained of  their energy  or the
-sensor network becomes disconnected. Network connectivity is important because an
-active sensor node without  connectivity towards a base station cannot
-transmit information on an event in the area that it monitors.
- 
-
-\item {{\bf Energy Consumption}:}
-
- Energy Consumption (EC) can be seen as the total energy consumed by the sensors during the $Lifetime95$ or $Lifetime50$ divided by the number of rounds. The EC can be computed as follow: \\
- \begin{equation*}
-\scriptsize
-\mbox{EC} =  \frac{\mbox{$\sum\limits_{d=1}^D \left( E^c_d + E^l_d + E^a_d + E^s_d + E^p_d \right)$ }}{\mbox{$D$}}   .
+\scriptsize  \mbox{ASR}(\%) = \frac{\sum\limits_{r=1}^R
+  \mbox{$A_r^t$}}{\mbox{$|J|$}} \times 100,

 \end{equation*}
 \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 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
+  $Lifetime_{95}$ (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
+  disconnected. Network connectivity is  important because an active sensor node
+  without connectivity towards a base  station cannot transmit information on an
+  event in the area that it monitors.
+
+\item {{\bf  Energy Consumption  (EC)}:} the average  energy consumption  can be
+  seen as the total energy consumed by the sensors during the $Lifetime_{95}$ or
+  $Lifetime_{50}$  divided  by the  number  of rounds.  EC  can  be computed  as
+  follows:
+
+  % New version with global loops on period
+  \begin{equation*}
+    \scriptsize
+    \mbox{EC} = \frac{\sum\limits_{m=1}^{M} \left[ \left( E^{\mbox{com}}_m+E^{\mbox{list}}_m+E^{\mbox{comp}}_m \right) +\sum\limits_{t=1}^{T_m} \left( E^{a}_t+E^{s}_t \right) \right]}{\sum\limits_{m=1}^{M} T_m},
+  \end{equation*}
+
+
+% Old version with loop on round outside the loop on period
+%  \begin{equation*}
+%    \scriptsize
+%    \mbox{EC} = \frac{\sum\limits_{m=1}^{M_L} \left( E^{\mbox{com}}_m+E^{\mbox{list}}_m+E^{\mbox{comp}}_m \right) +\sum\limits_{t=1}^{T_L} \left( E^{a}_t+E^{s}_t \right)}{T_L},
+%  \end{equation*}
+
+% Ali version 

 %\begin{equation*}
 %\scriptsize
 %\mbox{EC} =  \frac{\mbox{$\sum\limits_{d=1}^D E^c_d$}}{\mbox{$D$}} + \frac{\mbox{$\sum\limits_{d=1}^D %E^l_d$}}{\mbox{$D$}} + \frac{\mbox{$\sum\limits_{d=1}^D E^a_d$}}{\mbox{$D$}} + %\frac{\mbox{$\sum\limits_{d=1}^D E^s_d$}}{\mbox{$D$}}.
 %\end{equation*}
 
 %\begin{equation*}
 %\scriptsize
 %\mbox{EC} =  \frac{\mbox{$\sum\limits_{d=1}^D E^c_d$}}{\mbox{$D$}} + \frac{\mbox{$\sum\limits_{d=1}^D %E^l_d$}}{\mbox{$D$}} + \frac{\mbox{$\sum\limits_{d=1}^D E^a_d$}}{\mbox{$D$}} + %\frac{\mbox{$\sum\limits_{d=1}^D E^s_d$}}{\mbox{$D$}}.
 %\end{equation*}
 

-Where: D is the number of rounds during $Lifetime95$ or $Lifetime50$. 
-The total energy consumed by the sensors (EC) comes through taking into consideration four main energy factors, which are $E^c_d$, $E^l_d$, $E^a_d$, $E^s_d$ and $E^p_d$.
-The energy consumption $E^c_d$ for wireless  communications  is  calculated by taking into account the  energy spent by  all the nodes while  transmitting and
-receiving  packets during round $d$. The $E^l_d$ represents the energy consumed by all the sensors during the listening mode before taking the decision to go Active or Sleep in round $d$. $E^a_d$ and $E^s_d$  refer to energy consumed in the active mode or in the sleeping mode. The $E^p_d$ refers to energy consumed by the computation (processing) to solve the integer program.
+% 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  $M$ 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  $E^{\scriptsize  \mbox{com}}_m$,
+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 $E^s_t$
+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.
 

-
-
-\item {{\bf Execution Time}:} a  sensor  node has  limited  energy  resources  and computing  power,
-therefore it is important that the proposed algorithm has the shortest
-possible execution  time. The energy of  a sensor node  must be mainly
-used   for  the  sensing   phase,  not   for  the   pre-sensing  ones.   
+\item {{\bf  Execution Time}:}  a sensor node  has limited energy  resources and
+  computing power, therefore it is important that the proposed algorithm has the
+  shortest possible execution  time. The energy of a sensor  node must be mainly
+  used for the sensing phase, not for the pre-sensing ones.

   
   

-\item {{\bf Stopped simulation runs}:} A simulation
-ends  when the  sensor network  becomes
-disconnected (some nodes are dead and are not able to send information to the base station). We report the number of simulations that are stopped due to network disconnections and for which round it occurs.
+\item {{\bf Stopped simulation runs}:} a simulation ends when the sensor network
+  becomes disconnected (some nodes are dead and are not able to send information
+  to the base station). We report the number of simulations that are stopped due
+  to network disconnections and for which round it occurs.

 
 \end{enumerate}
 
 
 \end{enumerate}
 

+\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. 

 
 

-%%%%%%%%%%%%%%%%%%%%%%%%VU JUSQU ICI**************************************************

 
 

+%\begin{enumerate}[i)]

 
 

+%\item {{\bf Coverage Ratio}}
+\subsubsection{Coverage Ratio} 

 
 

-\section{Results and analysis}
-\subsection{Coverage ratio} 
-Figure~\ref{fig3} shows the average coverage ratio for 150 deployed nodes.  
+Figure~\ref{Figures/ch4/R2/CR} shows the average coverage ratio for 150 deployed nodes.  

 \parskip 0pt    
 \begin{figure}[h!]
 \centering
 \parskip 0pt    
 \begin{figure}[h!]
 \centering

- \includegraphics[scale=0.5] {R1/CR.pdf} 
-\caption{The coverage ratio for 150 deployed nodes}
-\label{fig3}
+ \includegraphics[scale=0.5] {R2/CR.pdf} 
+\caption{Coverage ratio for 150 deployed nodes}
+\label{Figures/ch4/R2/CR}

 \end{figure} 
 \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.

 
 

-DESK and GAF provide a very little better coverage ratio than MuDiLCO-T (in the first thirty rounds. This is due to the fact that MuDiLCO  put in sleep mode redundant sensors using optimization (which slightly decreases the coverage ratio) while there are more active nodes in the case of DESK and GAF. When the number of rounds increases, coverage ratio produced by DESK and GAF decreases. This is due to dead nodes. However,  MuDiLCO-T maintains the coverage ratio greater than 50$\%$ for a larger number of rounds in comparison with DESK and GAF. Although some nodes are dead, sensor activity scheduling based on optimization in MuDiLCO allows to prolong the coverage of the area of interest. The simulation results shows the superiority of our method, that  keeps high coverage for a larger number of rounds, so the network lifetime is extended.

 
 

+%\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. 

 
 

-\subsection{Active sensors ratio} 
- It is important 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. 

 \begin{figure}[h!]
 \centering
 \begin{figure}[h!]
 \centering

-\includegraphics[scale=0.5]{R1/ASR.pdf}  
-\caption{The active sensors ratio for 150 deployed nodes }
-\label{fig4}
-\end{figure} 
+\centering
+\includegraphics[scale=0.5]{R2/LT95.pdf}\\~ ~ ~ ~ ~(a) \\

 
 

+\includegraphics[scale=0.5]{R2/LT50.pdf}\\~ ~ ~ ~ ~(b)

 
 

-We can observe that DESK and GAF have 37.6 $\%$ and 44.8 $\%$ of active nodes and MuDiLCO-T competes perfectly with only 24.8$\%$  of active nodes for the first thirteen rounds.
-From the thirty fifth round, MuDiLCO-T has a larger number of active nodes in comparison with DESK and GAF but it maintains a higher level of coverage compared to the two other methods. DESK and GAF have less number of active nodes because many nodes are died.
+\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 Model-5 for all the experiments presented thereafter. 

 
 

-\subsection{Stopped simulation runs}
-%The results presented in this experiment, is to show the comparison of our MuDiLCO protocol with other two approaches from the point of view the stopped simulation runs per round. Figure~\ref{fig6} illustrates the percentage of stopped simulation
-%runs per round for 150 deployed nodes. 
-Figure~\ref{fig6} reports the cumulative percentage of stopped simulations runs per round for 150 deployed nodes. This figure gives the breakpoint for each of the methods.
-\begin{figure}[h!]
+%\end{enumerate}
+
+
+\subsection{Results and analysis}
+
+\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
+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?
+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
+decrease of the coverage ratio.   Furthermore, our protocol allows to maintain a
+coverage ratio  greater than  50\% for far  more rounds.  Overall,  the proposed
+sensor  activity scheduling based  on optimization  in MuDiLCO  maintains higher
+coverage ratios of the  area of interest for a larger number  of rounds. It also
+means that MuDiLCO saves more energy,  with less dead nodes, at most for several
+rounds, and thus should extend the network lifetime.
+
+\begin{figure}[ht!]

 \centering
 \centering

-\includegraphics[scale=0.5]{R1/SR.pdf} 
-\caption{Cumulative percentage of stopped simulation runs for 150 deployed nodes }
-\label{fig6}
+ \includegraphics[scale=0.5] {R/CR.pdf} 
+\caption{Average coverage ratio for 150 deployed nodes}
+\label{fig3}

 \end{figure} 
 \end{figure} 

-DESK stops first (around 45 rounds) because it consumes more energy for turning on a large number of redundant nodes during the sensing phase. GAF stops secondly for the same reason of DESK. MuDiLCO-T overcomes DESK and GAF because the optimization process distributed on several subregions  leads to coverage preservation and so extends the network lifetime.
-%%% The optimization effectively continues as long as a network in a subregion is still connected. A VOIR %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 
 

+\iffalse
+\textcolor{red}{ We
+can see that for the first thirty nine rounds GA-MuDiLCO provides a little bit better coverage ratio  than MuDiLCO. Both DESK and GAF provide a coverage
+which is a little bit better than the one of MuDiLCO and GA-MuDiLCO for the first thirty rounds because they activate a larger number of nodes during sensing phase. After that GA-MuDiLCO provides a coverage ratio near to the  MuDiLCO and better than DESK and GAF. GA-MuDiLCO gives approximate solution with activation a larger number of nodes than MuDiLCO during sensing phase while it activates a less number of nodes in comparison with both DESK and GAF. MuDiLCO and GA-MuDiLCO clearly outperform DESK and GAF for
+a number of periods between 31 and 103. This is because they optimize the coverage and the lifetime in a wireless sensor network by selecting the best representative sensor nodes to take the responsibility of coverage during the sensing phase.}
+\fi

 
 

-\subsection{Energy Consumption}
-We measure the energy consumed by the sensors during the communication, listening, computation, active, and sleep modes for different network densities and compare it with the two other methods. Figures~\ref{fig95} and ~\ref{fig7} illustrate the energy consumption for different network sizes for $Lifetime95$ and $Lifetime50$. 

 
 

-\begin{figure}[h!]
+\subsubsection{Active sensors ratio} 
+
+It is crucial to have as few active nodes as possible in each round, in order to
+minimize the communication overhead and maximize    the network lifetime. Figure~\ref{fig4}  presents the active  sensor ratio for  150 deployed
+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}.
+
+\begin{figure}[ht!]

 \centering
 \centering

-\includegraphics[scale=0.5]{R1/EC95.pdf} 
-\caption{The Energy Consumption with $95\%-Lifetime$}
-\label{fig95}
+\includegraphics[scale=0.5]{R/ASR.pdf}  
+\caption{Active sensors ratio for 150 deployed nodes}
+\label{fig4}

 \end{figure} 
 
 \end{figure} 
 

-The results show that MuDiLCO-T 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 modes of the sensor node.\\
- 
-As shown in Figures~\ref{fig95}\ref{fig7} and \ref{fig7}, MuDiLCO-7 consumes more energy than the other versions of MuDiLCO, especially for large sizes of network. This is easy to understand since the bigger the number of rounds and the number of sensors involved in the integer program, the larger the time computation to solve the optimization problem.  
-\begin{figure}[h!]
+%\textcolor{red}{GA-MuDiLCO activates a sensor nodes larger than MuDiLCO but lower than both DESK and GAF }
+
+
+\subsubsection{Stopped simulation runs}
+%The results presented in this experiment, is to show the comparison of our MuDiLCO protocol with other two approaches from the point of view the stopped simulation runs per round. Figure~\ref{fig6} illustrates the percentage of stopped simulation
+%runs per round for 150 deployed nodes. 
+
+Figure~\ref{fig6} reports the cumulative  percentage of stopped simulations runs
+per round for  150 deployed nodes. This figure gives the  breakpoint for each method.  DESK stops first,  after approximately 45~rounds, because it consumes the
+more energy by  turning on a large number of redundant  nodes during the sensing
+phase. GAF  stops secondly for the  same reason than  DESK. 
+%\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
 \centering

-\includegraphics[scale=0.5]{R1/EC50.pdf} 
-\caption{The Energy Consumption with $Lifetime50$}
-\label{fig7}
+\includegraphics[scale=0.5]{R/SR.pdf} 
+\caption{Cumulative percentage of stopped simulation runs for 150 deployed nodes }
+\label{fig6}

 \end{figure} 
 
 \end{figure} 
 

+\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
+and   compare   it   with   the  two   other   methods.    Figures~\ref{fig7}(a)
+and~\ref{fig7}(b)  illustrate  the  energy  consumption,  considering  different
+network sizes, for $Lifetime_{95}$ and $Lifetime_{50}$.

 
 

+\begin{figure}[h!]
+  \centering
+  \begin{tabular}{cl}
+    \parbox{9.5cm}{\includegraphics[scale=0.5]{R/EC95.pdf}} & (a) \\
+    \verb+ + \\
+    \parbox{9.5cm}{\includegraphics[scale=0.5]{R/EC50.pdf}} & (b)
+  \end{tabular}
+  \caption{Energy consumption for (a) $Lifetime_{95}$ and 
+    (b) $Lifetime_{50}$}
+  \label{fig7}
+\end{figure} 
+
+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.}

 %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}
-We observe the impact of the network size and of the number of rounds $T$ on the computation time. Figure~\ref{fig77} gives the average execution times in seconds  (times to solve optimization problem) for different values of $T$. 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 AVR ATmega103L microcontroller (6 MHz) and a MIPS rate equal to 6 to run the optimization resolution, this time is multiplied by 2944.2 $\left( \frac{35330}{2} \times 6\right)$ and reported on Figure~\ref{fig77} for different network sizes.
+\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
+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
+AVR ATmega103L  microcontroller (6 MHz) and  a MIPS rate  equal to 6 to  run the
+optimization   resolution,   this  time   is   multiplied   by  2944.2   $\left(
+\frac{35330}{2} \times  \frac{1}{6} \right)$ and  reported on Figure~\ref{fig77}
+for different network sizes.

 
 

-Figure~\ref{fig77} shows that the execution time increases with the number of rounds $T$ taken into account for the scheduling of the sensing phase. MuDiLCO-7 results in execution time that quickly becomes unsuitable for a sensor network, especially when the sensor network size increases. 
-%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.
-
-\begin{figure}[h!]
+\begin{figure}[ht!]

 \centering
 \centering

-\includegraphics[scale=0.5]{R1/T.pdf}  
+\includegraphics[scale=0.5]{R/T.pdf}  

 \caption{Execution Time (in seconds)}
 \label{fig77}
 \end{figure} 
 
 \caption{Execution Time (in seconds)}
 \label{fig77}
 \end{figure} 
 

+As expected,  the execution time increases  with the number of  rounds $T$ taken
+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,
+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
+optimization problem.

 
 

-\subsection{Network Lifetime}
-In Figure~\ref{fig9} and in Figure~\ref{fig8}, network lifetime, $Lifetime95$ and $Lifetime50$ respectively, are illustrated for different  network sizes. 
+%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.

 
 

-\begin{figure}[h!]
-\centering
-\includegraphics[scale=0.5]{R1/LT95.pdf}  
-\caption{The Network Lifetime for $Lifetime95$}
-\label{fig9}
+\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
+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
+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
+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.}
+\begin{figure}[t!]
+  \centering
+  \begin{tabular}{cl}
+    \parbox{9.5cm}{\includegraphics[scale=0.5]{R/LT95.pdf}} & (a) \\
+    \verb+ + \\
+    \parbox{9.5cm}{\includegraphics[scale=0.5]{R/LT50.pdf}} & (b)
+  \end{tabular}
+  \caption{Network lifetime for (a) $Lifetime_{95}$ and 
+    (b) $Lifetime_{50}$}
+  \label{fig8}

 \end{figure} 
 
 \end{figure} 
 

+% By choosing the best suited nodes, for each round, by optimizing the coverage and lifetime of the network to cover the area of interest with a maximum number rounds and by letting the other nodes sleep in order to be used later in next rounds, our MuDiLCO protocol efficiently prolonges the network lifetime. 

 
 

-As highlighted by Figure~\ref{fig9}, network lifetime obviously
-increases when the size of the network increases. MuDiLCO-T (whatever values of $T$) maximizes the lifetime of the network compared with other approaches. The gain in lifetime for a coverage over $95\%$ is greater than $38\%$ between GAF and MuDiLCO-3.
-
-% By choosing the best suited nodes, for each round, by optimizing the coverage and lifetime of the network to cover the area of interest with a maximum number rounds and by letting the other nodes sleep in order to be used later in next rounds, our MuDiLCO-T protocol efficiently prolonges the network lifetime. 
-
-%In Figure~\ref{fig8}, Comparison shows that our MuDiLCO-T protocol, which are used distributed optimization on the subregions with the ability of producing T rounds, is the best one because it is robust to network disconnection during the network lifetime as well as it consume less energy in comparison with other approaches. It also means that distributing the protocol in each sensor node and subdividing the sensing field into many subregions, which are managed independently and simultaneously, is the most relevant way to maximize the lifetime of a network.
+%In Figure~\ref{fig8}, Comparison shows that our MuDiLCO protocol, which are used distributed optimization on the subregions with the ability of producing T rounds, is the best one because it is robust to network disconnection during the network lifetime as well as it consume less energy in comparison with other approaches. It also means that distributing the protocol in each sensor node and subdividing the sensing field into many subregions, which are managed independently and simultaneously, is the most relevant way to maximize the lifetime of a network.

 
 
 %We see that our MuDiLCO-7 protocol results in execution times that quickly become unsuitable for a sensor network as well as the energy consumption seems to be huge because it used a larger number of rounds T during performing the optimization decision in the subregions, which is led to decrease the network lifetime. On the other side, our MuDiLCO-1, 3, and 5 protocol seems to be more efficient in comparison with other approaches because they are prolonged the lifetime of the network more than DESK and GAF.
 
 
 %We see that our MuDiLCO-7 protocol results in execution times that quickly become unsuitable for a sensor network as well as the energy consumption seems to be huge because it used a larger number of rounds T during performing the optimization decision in the subregions, which is led to decrease the network lifetime. On the other side, our MuDiLCO-1, 3, and 5 protocol seems to be more efficient in comparison with other approaches because they are prolonged the lifetime of the network more than DESK and GAF.

- 
-

 
 

-\begin{figure}[h!]
-\centering
-\includegraphics[scale=0.5]{R1/LT50.pdf}  
-\caption{The Network Lifetime for $Lifetime50$}
-\label{fig8}
-\end{figure} 

 
 

-\section{Conclusion and Future Works}
+\section{Conclusion and future works}

 \label{sec:conclusion}
 
 \label{sec:conclusion}
 

-In this paper, we have  addressed the problem of the coverage and 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 concept of divide-and-conquer method, and then a MuDiLCO protocol optimizes  coverage and  lifetime performances in each subregion.
-The  proposed  protocol  combines  two efficient  techniques:  network
-leader election  and sensor activity scheduling.
+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
+concept  of divide-and-conquer  method, and  then  we propose  a protocol  which
+optimizes coverage  and lifetime performances in each  subregion.  Our protocol,
+called MuDiLCO (Multiround  Distributed Lifetime Coverage Optimization) combines
+two  efficient   techniques:  network   leader  election  and   sensor  activity
+scheduling.

 %,  where the challenges
 %include how to select the  most efficient leader in each subregion and
 %the best cover sets %of active nodes that will optimize the network lifetime
 %while taking the responsibility of covering the corresponding
 %subregion using more than one cover set during the sensing phase. 
 %,  where the challenges
 %include how to select the  most efficient leader in each subregion and
 %the best cover sets %of active nodes that will optimize the network lifetime
 %while taking the responsibility of covering the corresponding
 %subregion using more than one cover set during the sensing phase. 

-The activity scheduling in each subregion works in periods, 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. 
-
-Simulations results show the relevance  of the proposed MuDiLCO
-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 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 an excessive energy consumption.
+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.
+
+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
+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 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
 %optimization  methods. This protocol can prolong the network lifetime by minimizing the number of the active sensor nodes near the borders by optimizing the sensing range of sensor nodes.
 % use section* for acknowledgement
 
 %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
 %optimization  methods. This protocol can prolong the network lifetime by minimizing the number of the active sensor nodes near the borders by optimizing the sensing range of sensor nodes.
 % use section* for acknowledgement

-%\section*{Acknowledgment}
+
+\section*{Acknowledgment}
+This work is  partially funded by the Labex ACTION program (contract ANR-11-LABX-01-01).
+As a Ph.D.  student, Ali Kadhum IDREES would like to gratefully acknowledge the
+University  of Babylon  - Iraq  for the  financial support,  Campus  France (The
+French  national agency  for the  promotion of  higher  education, international
+student   services,  and   international  mobility).%,   and  the   University  ofFranche-Comt\'e - France for all the support in France. 
+
+
+

 
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