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@@ -73,42 +73,48 @@
 %% \author[label1,label2]{}
 %% \address[label1]{}
 %% \address[label2]{}
 %% \author[label1,label2]{}
 %% \address[label1]{}
 %% \address[label2]{}

-\author{Ali Kadhum Idrees, Karine Deschinkel, \\
-Michel Salomon, and Rapha\"el Couturier}
+%\author{Ali Kadhum Idrees, Karine Deschinkel, \\
+%Michel Salomon, and Rapha\"el Couturier}
+

 %\thanks{are members in the AND team - DISC department - FEMTO-ST Institute, University of Franche-Comt\'e, Belfort, France.
 % e-mail: ali.idness@edu.univ-fcomte.fr, $\lbrace$karine.deschinkel, michel.salomon, raphael.couturier$\rbrace$@univ-fcomte.fr.}% <-this % stops a space
 %\thanks{}% <-this % stops a space
  
 %\thanks{are members in the AND team - DISC department - FEMTO-ST Institute, University of Franche-Comt\'e, Belfort, France.
 % e-mail: ali.idness@edu.univ-fcomte.fr, $\lbrace$karine.deschinkel, michel.salomon, raphael.couturier$\rbrace$@univ-fcomte.fr.}% <-this % stops a space
 %\thanks{}% <-this % stops a space
  

-\address{FEMTO-ST Institute, University of Franche-Comt\'e, Belfort, France. \\ 
-e-mail: ali.idness@edu.univ-fcomte.fr, \\
-$\lbrace$karine.deschinkel, michel.salomon, raphael.couturier$\rbrace$@univ-fcomte.fr.}
+%\address{FEMTO-ST Institute, University of Franche-Comt\'e, Belfort, France. \\ 
+%e-mail: ali.idness@edu.univ-fcomte.fr, \\
+%$\lbrace$karine.deschinkel, michel.salomon, raphael.couturier$\rbrace$@univ-fcomte.fr.}
+
+\author{Ali   Kadhum   Idrees$^{a,b}$,   Karine  Deschinkel$^{a}$,   \\   Michel
+  Salomon$^{a}$,   and  Rapha\"el   Couturier   $^{a}$  \\   $^{a}${\em{FEMTO-ST
+      Institute,  UMR  6174  CNRS,   \\  University  Bourgogne  Franche-Comt\'e,
+      Belfort, France}} \\ $^{b}${\em{Department of Computer Science, University
+      of Babylon, Babylon, Iraq}} }

 
 \begin{abstract}
 
 \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
 Coverage and  lifetime are  two paramount problems  in Wireless  Sensor Networks

-(WSNs). In this paper, a method called Multiround Distributed Lifetime Coverage
+(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
 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 sets of sensor nodes are scheduled to remain active for a number of
-rounds  during the  sensing phase,  to  ensure coverage  so as  to maximize  the
-lifetime of  WSN.  The decision process is  carried out by a  leader node, which
-solves an  integer program to  produce the best  representative sets to  be used
-during the rounds  of the sensing phase. 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.
-
+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, with one set for each round of
+a period, to remain active during the  sensing phase and thus ensure coverage so
+as  to maximize  the  WSN lifetime.   \textcolor{blue}{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.} 
+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.
 Optimization, Scheduling, Distributed Computation.

-

 \end{keyword}
 
 \end{frontmatter}
 \end{keyword}
 
 \end{frontmatter}


@@ -117,51 +123,36 @@ Optimization, Scheduling, Distributed Computation.
  
 \indent  The   fast  developments  of  low-cost  sensor   devices  and  wireless
 communications have allowed the emergence of WSNs. A WSN includes a large number
  
 \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
+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
 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
+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
 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,
+often  costly  or  simply  impossible  to  replace  and/or  recharge  batteries,

 especially in remote and hostile environments. Obviously, to achieve a long life
 especially in remote and hostile environments. Obviously, to achieve a long life

-of the network  it is important to conserve  battery power. Therefore, lifetime
+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
 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
+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,
 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 multirounds 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
-%the area of interest. The limited energy of sensors represents the main challenge in the WSNs
-%design~\cite{Sudip03}, where it is difficult to replace and/or recharge their batteries because the the area of interest nature (such as hostile environments) and the cost. So, it is necessary that a WSN
-%deployed  with high  density because  spatial redundancy  can  then be exploited to increase  the lifetime of the network. However, turn on all the sensor nodes, which monitor the same region at the same time
-%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.
-%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.  
+we concentrate  on the area coverage  problem, with the  objective of maximizing
+the network lifetime by using an optimized multiround scheduling.

 
 The remainder of the paper is organized as follows. The next section
 
 The remainder of the paper is organized as follows. The next section

-% Section~\ref{rw}
-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
+reviews the  related works  in the  field.  Section~\ref{pd}  is devoted  to the
+description of  MuDiLCO protocol. Section~\ref{exp} introduces  the experimental
+framework, it describes  the simulation setup and the different  metrics used to
+assess the  performances.  Section~\ref{analysis}  shows the  simulation results

 obtained using  the discrete event  simulator OMNeT++ \cite{varga}.   They fully
 obtained using  the discrete event  simulator OMNeT++ \cite{varga}.   They fully

-demonstrate  the  usefulness  of   the  proposed  approach.   Finally,  we  give
+demonstrate  the  usefulness  of  the   proposed  approach.   Finally,  we  give

 concluding    remarks   and    some    suggestions   for    future   works    in
 Section~\ref{sec:conclusion}.
 
 concluding    remarks   and    some    suggestions   for    future   works    in
 Section~\ref{sec:conclusion}.
 

-
-%%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 ?
+\section{Related works} 

 \label{rw}
 
 \indent  This section is  dedicated to  the various  approaches proposed  in the
 \label{rw}
 
 \indent  This section is  dedicated to  the various  approaches proposed  in the


@@ -173,214 +164,61 @@ algorithms in WSNs according to several design choices:
 \item  Sensors   scheduling  algorithm  implementation,   i.e.   centralized  or
   distributed/localized algorithms.
 \item The objective of sensor coverage, i.e. to maximize the network lifetime or
 \item  Sensors   scheduling  algorithm  implementation,   i.e.   centralized  or
   distributed/localized algorithms.
 \item The objective of sensor coverage, i.e. to maximize the network lifetime or

-  to minimize the number of sensors during the sensing period.
+  to minimize the number of active sensors during a sensing round.

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

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

 \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.
 \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 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}
+\subsection{Centralized approaches}
+

 The major approach  is to divide/organize the sensors into  a suitable number of
 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
+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
 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.
+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{blue}{Exact or  heuristic
+  approaches  are designed  to provide  cover sets.  Contrary to  exact methods,
+  heuristic  ones can  handle very  large  and centralized  problems.  They  are
+  proposed to reduce  computational overhead such as  energy consumption, delay,
+  and generally allow to increase the network lifetime.}

 
 The first algorithms proposed in the literature consider that the cover sets are
 
 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
+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
 lifetime of the network in comparison  to the disjoint cover set algorithms, but

-designing  algorithms for  non-disjoint cover  sets generally  induces  a higher
+designing  algorithms for  non-disjoint cover  sets generally  induces a  higher

 order  of complexity.   Moreover, in  case of  a sensor's  failure, non-disjoint
 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, the proposed work in ~\cite{cardei2005energy, berman04}    
-
-
-
+scheduling policies  are less  resilient and  reliable because  a sensor  may be
+involved in more than one cover sets.

 
 

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

-Balance (CSB), which choose a set of active nodes using the tuple (data coverage
-range, residual energy).   Then, they have introduced a  new Correlated Node Set
-Computing (CNSC)  algorithm to find  the correlated node  set for a  given node.
-After that,  they proposed  a High Residual  Energy First (HREF)  node selection
-algorithm to  minimize the number of active  nodes so as to  prolong the network
-lifetime. Various centralized methods based on column generation approaches have
-also been proposed~\cite{castano2013column,rossi2012exact,deschinkel2012column}.
-
-
-
-
+Balance  (CSB), which  chooses  a set  of  active nodes  using  the tuple  (data
+coverage range, residual  energy).  Then, they have introduced  a new Correlated
+Node Set Computing (CNSC) algorithm to find  the correlated node set for a given
+node.   After that,  they  proposed a  High Residual  Energy  First (HREF)  node
+selection algorithm to minimize the number of  active nodes so as to prolong the
+network  lifetime.   Various  centralized  methods based  on  column  generation
+approaches                   have                    also                   been
+proposed~\cite{gentili2013,castano2013column,rossi2012exact,deschinkel2012column}.
+\textcolor{blue}{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}
 
 \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.
-
-Some        distributed       algorithms        have        been       developed
-in~\cite{Gallais06,Tian02,Ye03,Zhang05,HeinzelmanCB02, yardibi2010distributed, prasad2007distributed,Misra}
-to perform  the scheduling so  as to preserve coverage.   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  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. 
-
-The  works presented in  \cite{Bang, Zhixin,  Zhang} focuses  on coverage-aware,
-distributed energy-efficient,  and distributed clustering  methods respectively,
-which aims  to extend the network  lifetime, while the coverage  is ensured.  More recently, Shibo et  al. \cite{Shibo} have expressed the coverage
-problem  as  a  minimum weight  submodular  set  cover  problem and  proposed  a
-Distributed Truncated Greedy Algorithm (DTGA)  to solve it.  They take advantage
-from both  temporal and  spatial correlations between  data sensed  by different
-sensors,   and    leverage   prediction,   to   improve    the   lifetime.    In
-\cite{xu2001geography},   Xu  et   al.  have   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.
-
-
-
-
-
-\iffalse
-   
-\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  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 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.
-
-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}

 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
 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


@@ -388,126 +226,66 @@ 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.
 
 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 preserve coverage.   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. 
-
-The  works presented in  \cite{Bang, Zhixin,  Zhang} focuses  on coverage-aware,
+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}.
+
+The  works presented  in  \cite{Bang, Zhixin,  Zhang}  focus on  coverage-aware,

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

-which aims  to extend the network  lifetime, while the coverage  is ensured.  S.
-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.
+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
 
 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.
+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
+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
 in~\cite{idrees2014coverage}.   In~\cite{idrees2014coverage},  the  protocol  is

-deployed over  only two  subregions. Simulation results  have shown that  it was
+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
 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.
-
-
-
+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{blue}{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}.

 
 

-\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}
-
-%Network Lifetime
-
-%\subsection{Network Lifetime}
-%Various   definitions   exist   for   the   lifetime   of   a   sensor
-%network~\cite{die09}.  The main definitions proposed in the literature are
-%related to the  remaining energy of the nodes or  to the coverage percentage. 
-%The lifetime of the  network is mainly defined as the amount
-%of  time during which  the network  can  satisfy its  coverage objective  (the
-%amount of  time that the network  can cover a given  percentage of its
-%area or targets of interest). In this work, we assume that the network
-%is alive  until all  nodes have  been drained of  their energy  or the
-%sensor network becomes disconnected, and we measure the coverage ratio
-%during the WSN lifetime.  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.

 
 \section{MuDiLCO protocol description}
 \label{pd}
 
 
 \section{MuDiLCO protocol description}
 \label{pd}
 

-%Our work will concentrate on the area coverage by design
-%and implementation of a  strategy, which efficiently selects the active
-%nodes   that  must   maintain  both   sensing  coverage   and  network
-%connectivity and at the same time improve the lifetime of the wireless
-%sensor  network. But,  requiring  that  all physical  points  of  the
-%considered region are covered may  be too strict, especially where the
-%sensor network is not dense.   Our approach represents an area covered
-%by a sensor as a set of primary points and tries to maximize the total
-%number  of  primary points  that  are  covered  in each  round,  while
-%minimizing  overcoverage (points  covered by  multiple  active sensors
-%simultaneously).
-
-%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.
-

 \subsection{Assumptions}
 
 We  consider a  randomly and  uniformly  deployed network  consisting of  static
 \subsection{Assumptions}
 
 We  consider a  randomly and  uniformly  deployed network  consisting of  static


@@ -526,58 +304,86 @@ 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
 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

-working nodes in the active mode.
-
-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.
-
-%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
-%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.
-
-%The MuDiLCO protocol works in periods and executed at each sensor node in the network, each sensor node can still sense data while being in
-%LISTENING mode. Thus, by entering the LISTENING mode at the beginning of each round,
-%sensor nodes still executing sensing task while participating in the leader election and decision phases. More specifically, The MuDiLCO protocol algorithm works as follow: 
-%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.
+active nodes.
+
+\indent Instead of working with the coverage area, we consider for each sensor a
+set of  points called  primary points~\cite{idrees2014coverage}. We  assume that
+the sensing  disk defined by a  sensor is covered  if all the primary  points of
+this  sensor are  covered.   By knowing  the position  of  wireless sensor  node
+(centered at  the the  position $\left(p_x,p_y\right)$)  and it's  sensing range
+$R_s$,  we define  up to  25 primary  points $X_1$  to $X_{25}$  as decribed  on
+Figure~\ref{fig1}. The optimal number of primary points is investigated in
+section~\ref{ch4:sec:04:06}.
+
+The coordinates of the primary points are defined as follows:\\
+%$(p_x,p_y)$ = 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 * (\frac{\sqrt{2}}{2})) $\\
+$X_7=( p_x + R_s * (\frac{\sqrt{2}}{2}), p_y + R_s * (\frac{\sqrt{2}}{2})) $\\
+$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 * (0)) $\\
+$X_{11}=( p_x + R_s *  (\frac{\sqrt{2}}{2}), p_y + R_s * (0))$\\
+$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
+  \includegraphics[scale=0.375]{fig26.pdf}
+  \label{fig1}
+  \caption{Wireless sensor node represented by up to 25~primary points}
+\end{figure}

 
 \subsection{Background idea}
 
 \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
-and for each round a set of sensors  (said a cover set) is  responsible for the
-sensing task. A multiround optimization process executed in each period after information exchange and leader election in order to produce a $T$ cover sets of sensors to take the mission of sensing for $T$ rounds.
-\begin{figure}[ht!]
-\centering \includegraphics[width=100mm]{Modelgeneral.pdf} % 70mm
+
+\textcolor{blue}{The WSN  area of  interest is,  at  first,  divided into
+  regular  homogeneous subregions  using  a divide-and-conquer  algorithm. Then, our  protocol  will be  executed  in a  distributed  way in  each
+  subregion  simultaneously  to  schedule  nodes'  activities  for  one  sensing
+  period. Sensor nodes are assumed to be deployed almost uniformly and with high
+  density over the region. The regular  subdivision is made so that the number
+  of hops between any pairs of sensors  inside a subregion is less than or equal
+  to 3.}
+
+As can  be seen  in Figure~\ref{fig2},  our protocol  works in  periods fashion,
+where   each   period   is    divided   into   4~phases:   Information~Exchange,
+Leader~Election,  Decision,  and Sensing.   Each  sensing  phase may  be  itself
+divided into $T$ rounds \textcolor{blue} {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}[t!]
+\centering \includegraphics[width=125mm]{Modelgeneral.pdf} % 70mm

 \caption{The MuDiLCO protocol scheme executed on each node}
 \label{fig2}
 \end{figure} 
 
 \caption{The MuDiLCO protocol scheme executed on each node}
 \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.
-% 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. 
-%%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.
-%%RC so if there are at least one failure per period, the coverage is bad...
+This  protocol minimizes  the  impact of  unexpected node  failure  (not due  to
+batteries running out of energy), 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.   \textcolor{blue}{The   duration   of  the   rounds  is  a   predefined
+   parameter. Round duration  should be long enough to hide  the system control
+   overhead and  short enough to minimize  the negative effects in  case of node
+   failures.}  

 
 The  energy consumption  and some  other constraints  can easily  be  taken into
 account,  since the  sensors  can  update and  then  exchange their  information
 
 The  energy consumption  and some  other constraints  can easily  be  taken into
 account,  since the  sensors  can  update and  then  exchange their  information


@@ -586,8 +392,6 @@ 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.
 
 energy  consuming for some  nodes, even  when they  do not  join the  network to
 monitor the area.
 

-%%%%%%%%%%%%%%%%%parler optimisation%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-

 We define two types of packets that will be used by the proposed protocol:
 \begin{enumerate}[(a)] 
 \item INFO  packet: such a  packet  will be sent by  each sensor node  to all the
 We define two types of packets that will be used by the proposed protocol:
 \begin{enumerate}[(a)] 
 \item INFO  packet: such a  packet  will be sent by  each sensor node  to all the


@@ -602,7 +406,7 @@ There are five status for each sensor node in the network:
 \item LISTENING: sensor node is waiting for a decision (to be active or not);
 \item  COMPUTATION: sensor  node  has been  elected  as leader  and applies  the
   optimization process;
 \item LISTENING: sensor node is waiting for a decision (to be active or not);
 \item  COMPUTATION: sensor  node  has been  elected  as leader  and applies  the
   optimization process;

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

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


@@ -619,48 +423,41 @@ 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.
 
 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{Leader Election phase}
 
 \subsection{Leader Election phase}
 

-This step  consists in  choosing the Wireless  Sensor Node Leader  (WSNL), which
+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
 will be responsible for executing the coverage algorithm.  Each subregion in the
 area of  interest will select its  own WSNL independently for  each period.  All

-the sensor  nodes cooperate to  elect a WSNL.   The nodes in the  same subregion
-will select the  leader based on the received informations  from all other nodes
-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 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.

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

-
-Each  WSNL will solve  an integer  program to  select which  cover sets  will be
-activated in  the following  sensing phase  to cover the  subregion to  which it
-belongs.  The integer  program will produce $T$ cover sets,  one for each round.
-The WSNL will send an Active-Sleep  packet to each sensor in the subregion based
-on the algorithm's results, indicating if  the sensor should be active or not in
-each round  of the  sensing phase.  The  integer program  is based on  the model
-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
+\label{decision}
+
+Each WSNL will  \textcolor{blue}{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.}
+
+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
 of  targets to  become minimal  at  the same  time.  They  use binary  variables
 $x_{jl}$ to indicate if  sensor $j$ belongs to cover set $l$.   In our model, we

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

 
 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:


@@ -694,7 +491,7 @@ We define the Overcoverage variable $\Theta_{t,p}$ as:
 \label{eq13} 
 \end{equation}
 More  precisely, $\Theta_{t,p}$  represents the  number of  active  sensor nodes
 \label{eq13} 
 \end{equation}
 More  precisely, $\Theta_{t,p}$  represents the  number of  active  sensor nodes

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

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


@@ -708,7 +505,7 @@ U_{t,p} = \left \{
 
 Our coverage optimization problem can then be formulated as follows:
 \begin{equation}
 
 Our coverage optimization problem can then be formulated as follows:
 \begin{equation}

- \min \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}
 
 Subject to
 \end{equation}
 
 Subject to


@@ -717,7 +514,7 @@ Subject to
 \end{equation}
 
 \begin{equation}
 \end{equation}
 
 \begin{equation}

-  \sum_{t=1}^{T}  X_{t,j}   \leq  \floor*{RE_{j}/E_{R}} \hspace{6 mm} \forall j \in J, t = 1,\dots,T
+  \sum_{t=1}^{T}  X_{t,j}   \leq  \floor*{RE_{j}/E_{R}} \hspace{10 mm}\forall j \in J\hspace{6 mm} 

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


@@ -733,236 +530,202 @@ U_{t,p} \in \lbrace0,1\rbrace, \hspace{10 mm}\forall p \in P, t = 1,\dots,T  \la
  \Theta_{t,p} \geq 0 \hspace{10 mm}\forall p \in P, t = 1,\dots,T \label{eq178}
 \end{equation}
 
  \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}
-
-%%RC why W_{\theta} is not defined (only one sentence)? How to define in practice Wtheta and Wu?
-
-

 \begin{itemize}
 \item $X_{t,j}$:  indicates whether  or not the  sensor $j$ is  actively sensing
 \begin{itemize}
 \item $X_{t,j}$:  indicates whether  or not the  sensor $j$ is  actively sensing

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

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

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

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

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

   covered).
 \end{itemize}
 
 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
   covered).
 \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
+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.
 
 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. In
-our simulations priority is given  to the coverage by choosing $W_{\theta}$ very
-large compared to $W_U$.
-%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.
+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 $W_{U}$ very
+large compared to $W_{\theta}$.
+
+\textcolor{blue}{The size of the problem depends  on the number of variables and
+  constraints. The number of variables is  linked to the number of alive sensors
+  $A \subseteq J$,  the number of rounds  $T$, and the number  of primary points
+  $P$.  Thus  the integer  program contains $A*T$  variables of  type $X_{t,j}$,
+  $P*T$ overcoverage variables and $P*T$  undercoverage variables. The number of
+  constraints  is equal  to $P*T$  (for constraints  (\ref{eq16})) $+$  $A$ (for
+  constraints (\ref{eq144})).}

 
 \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
 
 \subsection{Sensing phase}
 
 The sensing phase consists of $T$ rounds. Each sensor node in the subregion will
 receive an Active-Sleep packet from WSNL, informing it to stay awake or to go to

-sleep for  each round of the sensing  phase.  Algorithm~\ref{alg:MuDiLCO}, which
-will be  executed by each node  at the beginning  of a period, explains  how the
-Active-Sleep packet is obtained.
-
-% In each round during the sensing phase, there is a cover set of sensor nodes,  in which  the active  sensors will  execute  their sensing  task  to preserve maximal  coverage and lifetime in the subregion and this will continue until finishing the round $T$ and starting new period. 
+sleep for each  round of the sensing  phase.  Algorithm~\ref{alg:MuDiLCO}, which
+will  be executed  by  each sensor  node~$s_j$  at the  beginning  of a  period,
+explains how the Active-Sleep packet is obtained.

 
 \begin{algorithm}[h!]                
 
 \begin{algorithm}[h!]                

- % \KwIn{all the parameters related to information exchange}
-%  \KwOut{$winer-node$ (: the id of the winner sensor node, which is the leader of current round)}
-  \BlankLine
-  %\emph{Initialize the sensor node and determine it's position and subregion} \; 
-  
+  \BlankLine  

   \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}\; 
   \If{ $RE_j \geq E_{R}$ }{
       \emph{$s_j.status$ = COMMUNICATION}\;
       \emph{Send $INFO()$ packet to other nodes in the subregion}\;
       \emph{Wait $INFO()$ packet from other nodes in the subregion}\; 

-      %\emph{UPDATE $RE_j$ for every sent or received INFO Packet}\;
-      %\emph{ Collect information and construct the list L for all nodes in the subregion}\;

       
       

-      %\If{ the received INFO Packet = No. of nodes in it's subregion -1  }{

       \emph{LeaderID = Leader election}\;
       \If{$ s_j.ID = LeaderID $}{
         \emph{$s_j.status$ = COMPUTATION}\;
         \emph{$\left\{\left(X_{1,k},\dots,X_{T,k}\right)\right\}_{k \in J}$ =
           Execute Integer Program Algorithm($T,J$)}\;
         \emph{$s_j.status$ = COMMUNICATION}\;
       \emph{LeaderID = Leader election}\;
       \If{$ s_j.ID = LeaderID $}{
         \emph{$s_j.status$ = COMPUTATION}\;
         \emph{$\left\{\left(X_{1,k},\dots,X_{T,k}\right)\right\}_{k \in J}$ =
           Execute Integer Program Algorithm($T,J$)}\;
         \emph{$s_j.status$ = COMMUNICATION}\;

-        \emph{Send $ActiveSleep()$ to each node $k$ in subregion a packet \\
+        \emph{Send $ActiveSleep()$ packet 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}\;
           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 $}\;
       }  
         \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}
 
 \caption{MuDiLCO($s_j$)}
 \label{alg:MuDiLCO}
 
 \end{algorithm}
 

-\section{Experimental study}
+\section{Experimental framework}

 \label{exp}
 \label{exp}

+

 \subsection{Simulation setup}
 
 \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  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.

 We  performed  simulations for  five  different  densities  varying from  50  to
 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.
-
-%%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 ?
+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.

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

-% title of Table

 \centering
 \centering

-% used for centering table

 \begin{tabular}{c|c}
 \begin{tabular}{c|c}

-% centered columns (4 columns)
-      \hline
-%inserts double horizontal lines
+  \hline

 Parameter & Value  \\ [0.5ex]
 Parameter & Value  \\ [0.5ex]

-   
-%Case & Strategy (with Two Leaders) & Strategy (with One Leader) & Simple Heuristic \\ [0.5ex]
-% inserts table
-%heading

 \hline
 \hline

-% inserts single horizontal line

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

-% inserting body of the table
-%\hline

 Network size &  50, 100, 150, 200 and 250~nodes   \\
 Network size &  50, 100, 150, 200 and 250~nodes   \\

-%\hline

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

-%\hline

 Sensing time for one round & 60 Minutes \\
 $E_{R}$ & 36 Joules\\
 $R_s$ & 5~m   \\     
 Sensing time for one round & 60 Minutes \\
 $E_{R}$ & 36 Joules\\
 $R_s$ & 5~m   \\     

-%\hline
-$w_{\Theta}$ & 1   \\
-% [1ex] adds vertical space
-%\hline
-$w_{U}$ & $|P^2|$
-%inserts single line
+$W_{\theta}$ & 1   \\
+$W_{U}$ & $|P|^2$ \\

 \end{tabular}
 \label{table3}
 \end{tabular}
 \label{table3}

-% is used to refer this table in the text

 \end{table}
 \end{table}

-  
-Our protocol  is declined into  four versions: MuDiLCO-1,  MuDiLCO-3, MuDiLCO-5,
-and  MuDiLCO-7, corresponding  respectively to  $T=1,3,5,7$ ($T$  the  number of
-rounds  in one  sensing period).   In the  following, the  general case  will be
-denoted by  MuDiLCO-T and 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.
+
+\textcolor{blue}{Our  protocol  is  declined   into  four  versions:  MuDiLCO-1,
+  MuDiLCO-3, MuDiLCO-5, and MuDiLCO-7, corresponding respectively to $T=1,3,5,7$
+  ($T$ the  number of rounds in  one sensing period). Since  the time resolution
+  may  be prohibitive  when the  size  of the  problem increases,  a time  limit
+  threshold has  been fixed when  solving large  instances. In these  cases, the
+  solver returns  the best solution  found, which  is not necessary  the optimal
+  one. In practice, we only set time  limit values for $T=5$ and $T=7$. In fact,
+  for $T=5$ we limited the time for  250~nodes, whereas for $T=7$ it was for the
+  three  largest network  sizes.  Therefore  we used  the  following values  (in
+  second): 0.03 for  250~nodes when $T=5$, while for $T=7$  we chose 0.03, 0.06,
+  and 0.08 for respectively 150, 200, and 250~nodes.
+  These time limit thresholds have been  set empirically. The basic idea consists
+  in considering  the average execution  time to  solve the integer  programs to
+  optimality, then in  dividing this average time by three  to set the threshold
+  value.  After that,  this threshold value is increased if  necessary so that
+  the solver is able  to deliver a feasible solution within  the time limit.  In
+  fact, selecting the optimal values for the time limits will be investigated in
+  the future.}
+
+ 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
 
 Some preliminary experiments were performed to study the choice of the number of

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

 sizes. They show that as the number of subregions increases, so does the network
 sizes. They show that as the number of subregions increases, so does the network

-lifetime. Moreover, it  makes the MuDiLCO-T protocol more  robust against random
-network  disconnection due  to  node failures.  However,  too much  subdivisions
-reduces the advantage  of the optimization. In fact, there  is a balance between
+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
 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. 
+it. In the following we have set the number of subregions to 16.

 
 

-\subsection{Energy Model}
+\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  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
-

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

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

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

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

-% title of Table

 \centering
 \centering

-% used for centering table

 \begin{tabular}{|c|c|c|c|c|}
 \begin{tabular}{|c|c|c|c|c|}

-% centered columns (4 columns)
-      \hline
-%inserts double horizontal lines
+  \hline

 Sensor status & MCU & Radio & Sensing & Power (mW) \\ [0.5ex]
 \hline
 Sensor status & MCU & Radio & Sensing & Power (mW) \\ [0.5ex]
 \hline

-% inserts single horizontal line

 LISTENING & on & on & on & 20.05 \\
 LISTENING & on & on & on & 20.05 \\

-% inserting body of the table

 \hline
 ACTIVE & on & off & on & 9.72 \\
 \hline
 SLEEP & off & off & off & 0.02 \\
 \hline
 COMPUTATION & on & on & on & 26.83 \\
 \hline
 ACTIVE & on & off & on & 9.72 \\
 \hline
 SLEEP & off & off & off & 0.02 \\
 \hline
 COMPUTATION & on & on & on & 26.83 \\

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

 \end{tabular}
 
 \label{table4}
 \end{tabular}
 
 \label{table4}

-% is used to refer this table in the text

 \end{table}
 
 \end{table}
 

-For the sake of simplicity we ignore  the energy needed to turn on the radio, to
+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.
 start up the sensor node, to move from one status to another, etc.

-%We also do not consider the need of collecting sensing data. PAS COMPRIS
-Thus, when a sensor becomes active (i.e., it already decides its status), it can
-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
+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
 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.
+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
+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
 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
+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
 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
+(3600 seconds).   According to the interval  of initial energy, a  sensor may be

 alive during at most 20 rounds.
 
 \subsection{Metrics}
 alive during at most 20 rounds.
 
 \subsection{Metrics}


@@ -971,22 +734,18 @@ To evaluate our approach we consider the following performance metrics:
 
 \begin{enumerate}[i]
   
 
 \begin{enumerate}[i]
   

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

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

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

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

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

 of grid points  in the sensing field of  the network. In our simulations $N = 51
 \times 26 = 1326$ grid points.
 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 $\% $.

 
 \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
 
 \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


@@ -998,11 +757,11 @@ of grid points  in the sensing field of  the network. In our simulations $N = 51
 \end{equation*}
 where $A_r^t$ is the number of  active sensors in the subregion $r$ during round
 $t$ in the  current sensing phase, $|J|$  is the total number of  sensors in the
 \end{equation*}
 where $A_r^t$ is the number of  active sensors in the subregion $r$ during round
 $t$ in the  current sensing phase, $|J|$  is the total number of  sensors in the

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

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

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

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


@@ -1014,28 +773,23 @@ network, and $R$ is the total number of the subregions in the network.
   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:
   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:

- \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*}

 
 

-%\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 $M_L$ and  $T_L$ are respectively the number of  periods and rounds during
-$Lifetime_{95}$ or  $Lifetime_{50}$.  The total  energy consumed by  the sensors
-(EC) comes through taking into consideration four main energy factors. The first
-one ,  denoted $E^{\scriptsize \mbox{com}}_m$, represent  the energy consumption
-spent  by  all  the  nodes   for  wireless  communications  during  period  $m$.
-$E^{\scriptsize  \mbox{list}}_m$, the  next  factor, corresponds  to the  energy
-consumed by the sensors in LISTENING  status before receiving the decision to go
-active or  sleep in  period $m$. $E^{\scriptsize  \mbox{comp}}_m$ refers  to the
-energy needed  by all  the leader nodes  to solve  the integer program  during a
-period. Finally, $E^a_t$ and $E^s_t$  indicate the energy consummed by the whole
-network in round $t$.
+  \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*}
+
+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.


@@ -1052,33 +806,83 @@ network in round $t$.
 
 \end{enumerate}
 
 
 \end{enumerate}
 

+\section{Experimental results and analysis}
+\label{analysis}
+
+\subsection{Performance analysis for different number of primary points}
+\label{ch4:sec:04:06}
+
+In this  section, we study the  performance of MuDiLCO-1 approach  for different
+numbers of  primary points. The  objective of this  comparison is to  select the
+suitable number  of primary points  to be used by  a MuDiLCO protocol.   In this
+comparison,  MuDiLCO-1 protocol  is used  with five  primary point  models, each
+model corresponding to a number of  primary points, which are called Model-5 (it
+uses 5 primary points), Model-9, Model-13, Model-17, and Model-21.
+
+\subsubsection{Coverage ratio} 
+
+Figure~\ref{Figures/ch4/R2/CR} shows the average coverage ratio for 150 deployed
+nodes.  As can be seen, 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, the coverage  ratio produced by
+all models  decrease due  to dead nodes.  However, Model-5 is  the one  with the
+slowest decrease due to lower numbers of active sensors in the earlier periods.
+Overall this  model is slightly more  efficient than the other  ones, because it
+offers a good coverage ratio for a larger number of periods.
+\begin{figure}[t!]
+\centering
+ \includegraphics[scale=0.5] {R2/CR.pdf} 
+\caption{Coverage ratio for 150 deployed nodes}
+\label{Figures/ch4/R2/CR}
+\end{figure} 
+
+\subsubsection{Network lifetime}
+
+Finally, we study the effect of increasing the number of primary points on the lifetime of the network. 
+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 which leads to the largest lifetime
+improvement.
+
+\begin{figure}[h!]
+\centering
+\centering
+\includegraphics[scale=0.5]{R2/LT95.pdf}\\~ ~ ~ ~ ~(a) \\
+
+\includegraphics[scale=0.5]{R2/LT50.pdf}\\~ ~ ~ ~ ~(b)

 
 

-\section{Results and analysis}
+\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, as  stated
+before. Therefore, we have chosen the model with five primary points for all the
+experiments presented thereafter.

 
 \subsection{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
 
 \subsection{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-T.  
-%%RC : need to uniformize MuDiLCO or MuDiLCO-T?
-
-%%RC maybe increase the size of the figure for the reviewers, no?
-
-This is due to the fact
-that in comparison with MuDiLCO-T that  uses optimization to put in SLEEP status
+which is a little  bit better than the one of MuDiLCO.  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
 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
+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
 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
+greater than  50\% for far more  rounds.  Overall, the proposed  sensor activity

 scheduling based on optimization in  MuDiLCO maintains higher coverage ratios of
 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-T
-saves more  energy, with less dead nodes,  at most for several  rounds, and thus
-should extend the network lifetime.
+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.  \textcolor{blue}{MuDiLCO-7 seems  to have
+  most of the  time the best coverage  ratio up to round~80,  after MuDiLCO-5 is
+  slightly better.}

 
 

-\begin{figure}[t!]
+\begin{figure}[ht!]

 \centering
 \centering

- \includegraphics[scale=0.5] {R1/CR.pdf} 
+ \includegraphics[scale=0.5] {F/CR.pdf} 

 \caption{Average coverage ratio for 150 deployed nodes}
 \label{fig3}
 \end{figure} 
 \caption{Average coverage ratio for 150 deployed nodes}
 \label{fig3}
 \end{figure} 


@@ -1086,48 +890,40 @@ should extend the network lifetime.
 \subsection{Active sensors ratio} 
 
 It is crucial to have as few active nodes as possible in each round, in order to
 \subsection{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
+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
 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-T clearly outperforms  them with only 24.8\% of  active nodes. After the
-thirty  fifth round,  MuDiLCO-T exhibits  larger number  of active  nodes, which
-agrees with  the dual observation of  higher level of  coverage made previously.
-Obviously, in  that case DESK  and GAF have  less active nodes, since  they have
-activated many nodes at the beginning. Anyway, MuDiLCO-T activates the available
-nodes in a more efficient manner.
+MuDiLCO clearly outperforms  them with only 24.8\% of  active nodes.  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.

 
 

-\begin{figure}[t!]
+\begin{figure}[ht!]

 \centering
 \centering

-\includegraphics[scale=0.5]{R1/ASR.pdf}  
+\includegraphics[scale=0.5]{F/ASR.pdf}  

 \caption{Active sensors ratio for 150 deployed nodes}
 \label{fig4}
 \end{figure} 
 
 \subsection{Stopped simulation runs}
 \caption{Active sensors ratio for 150 deployed nodes}
 \label{fig4}
 \end{figure} 
 
 \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
 
 Figure~\ref{fig6} reports the cumulative  percentage of stopped simulations runs

-per round for  150 deployed nodes. This figure gives the  breakpoint for each of
-the methods.  DESK stops first,  after around 45~rounds, because it consumes the
-more energy by  turning on a large number of redundant  nodes during the sensing
-phase. GAF  stops secondly for the  same reason than  DESK.  MuDiLCO-T overcomes
-DESK and GAF because the  optimization process distributed on several subregions
-leads  to coverage  preservation and  so extends  the network  lifetime.  Let us
-emphasize that the  simulation continues as long as a network  in a subregion is
+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.  Let  us
+emphasize that the simulation  continues as long as a network  in a subregion is

 still connected.
 
 still connected.
 

-%%% The optimization effectively continues as long as a network in a subregion is still connected. A VOIR %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-\begin{figure}[t!]
+\begin{figure}[ht!]

 \centering
 \centering

-\includegraphics[scale=0.5]{R1/SR.pdf} 
+\includegraphics[scale=0.5]{F/SR.pdf} 

 \caption{Cumulative percentage of stopped simulation runs for 150 deployed nodes }
 \label{fig6}
 \end{figure} 
 
 \caption{Cumulative percentage of stopped simulation runs for 150 deployed nodes }
 \label{fig6}
 \end{figure} 
 

-\subsection{Energy Consumption} \label{subsec:EC}
+\subsection{Energy consumption} \label{subsec:EC}

 
 We  measure  the  energy  consumed  by the  sensors  during  the  communication,
 listening, computation, active, and sleep status for different network densities
 
 We  measure  the  energy  consumed  by the  sensors  during  the  communication,
 listening, computation, active, and sleep status for different network densities


@@ -1138,130 +934,130 @@ network sizes, for $Lifetime_{95}$ and $Lifetime_{50}$.
 \begin{figure}[h!]
   \centering
   \begin{tabular}{cl}
 \begin{figure}[h!]
   \centering
   \begin{tabular}{cl}

-    \parbox{9.5cm}{\includegraphics[scale=0.5]{R1/EC95.pdf}} & (a) \\
+    \parbox{9.5cm}{\includegraphics[scale=0.5]{F/EC95.pdf}} & (a) \\

     \verb+ + \\
     \verb+ + \\

-    \parbox{9.5cm}{\includegraphics[scale=0.5]{R1/EC50.pdf}} & (b)
+    \parbox{9.5cm}{\includegraphics[scale=0.5]{F/EC50.pdf}} & (b)

   \end{tabular}
   \caption{Energy consumption for (a) $Lifetime_{95}$ and 
     (b) $Lifetime_{50}$}
   \label{fig7}
 \end{figure} 
 
   \end{tabular}
   \caption{Energy consumption for (a) $Lifetime_{95}$ and 
     (b) $Lifetime_{50}$}
   \label{fig7}
 \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  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.
-
-%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. 
+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.

 
 

+\textcolor{blue}{Energy consumption increases with the  size of the networks and
+  the  number  of  rounds.   The curve  Unlimited-MuDiLCO-7  shows  that  energy
+  consumption due to  the time spent to solve the  integer program to optimality
+  increases drastically with  the size of the network. When  the resolution time
+  is limited for large network sizes, the energy consumption remains of the same
+  order whatever the MuDiLCO version. As can be seen with MuDiLCO-7.}

 
 \subsection{Execution time}
 
 \subsection{Execution time}

-
-We observe  the impact of the  network size and of  the number of  rounds on the
+\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
 computation  time.   Figure~\ref{fig77} gives  the  average  execution times  in

-seconds (needed 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   $\left(
-\frac{35330}{2} \times  \frac{1}{6} \right)$ and  reported on Figure~\ref{fig77}
+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.
 
 for different network sizes.
 

-\begin{figure}[t!]
+\begin{figure}[ht!]

 \centering
 \centering

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

 \caption{Execution Time (in seconds)}
 \label{fig77}
 \end{figure} 
 
 As expected,  the execution time increases  with the number of  rounds $T$ taken
 \caption{Execution Time (in seconds)}
 \label{fig77}
 \end{figure} 
 
 As expected,  the execution time increases  with the number of  rounds $T$ taken

-into account for scheduling of the sensing phase. The times obtained for $T=1,3$
-or $5$ seems bearable, but for $T=7$ they become quickly unsuitable for a sensor
-node, especially when  the sensor network size increases.   Again, we can notice
-that if we want  to schedule the nodes activities for a  large number of rounds,
-we need to choose a relevant number of subregion in order to avoid a complicated
-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.
-
-%While MuDiLCO-1, 3, and 5 solves the optimization process with suitable execution times to be used on wireless sensor network because it distributed on larger number of small subregions as well as it is used acceptable number of round(s) T.  We think that in distributed fashion the solving of the optimization problem to produce T rounds in a subregion can be tackled by sensor nodes. Overall, to be able to deal with very large networks, a distributed method is clearly required.
-
-\subsection{Network Lifetime}
+into  account to  schedule  the sensing  phase. \textcolor{blue}{Obviously,  the
+  number of variables and constraints of the integer program increases with $T$,
+  as  explained  in section~\ref{decision}, the times obtained  for $T=1,3$ or
+  $5$ seem  bearable. But for  $T=7$, without any  limitation of the  time, they
+  become  quickly unsuitable  for  a  sensor node,  especially  when the  sensor
+  network size  increases as  demonstrated by Unlimited-MuDiLCO-7.   Notice that
+  for 250  nodes, we also  limited the execution  time for $T=5$,  otherwise the
+  execution time  would have  been above  MuDiLCO-7.  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. Thus, limiting  the time
+  resolution for large instances allows to reduce the energy consumption without
+  any impact on the coverage quality.}
+
+\subsection{Network lifetime}

 
 The next  two figures,  Figures~\ref{fig8}(a) and \ref{fig8}(b),  illustrate the
 network lifetime  for different network sizes,  respectively for $Lifetime_{95}$
 
 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
+and  $Lifetime_{50}$.  Both  figures show  that the  network lifetime  increases

 together with the  number of sensor nodes, whatever the  protocol, thanks to the
 together with the  number of sensor nodes, whatever the  protocol, thanks to the

-node  density  which  result in  more  and  more  redundant  nodes that  can  be
-deactivated  and  thus save  energy.   Compared  to  the other  approaches,  our
-MuDiLCO-T protocol  maximizes the  lifetime of the  network.  In  particular the
-gain in  lifetime for a coverage over  95\% is greater than  38\% when switching
-from GAF to MuDiLCO-3.  The slight  decrease that can bee observed for MuDiLCO-7
-in case of  $Lifetime_{95}$ with large wireless sensor  networks result 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.
+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\%, and a network of  250~nodes, is greater than
+38\% when switching from GAF to MuDiLCO-5.
+%The lower performance 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{blue}{Overall,  it  clearly appears  that  computing a  scheduling for
+  several rounds is possible and relevant,  providing that the execution time to
+  solve the  optimization problem for  large instances is limited.   Notice that
+  rather than limiting the execution time,  similar results might be obtained by
+  replacing  the  computation of  the  exact  solution  with  the finding  of  a
+  suboptimal  one using  a  heuristic  approach. For  our  simulation setup  and
+  considering  the different  metrics, MuDiLCO-5  seems  to be  the most  suited
+  method in comparison with MuDiLCO-7.}

 
 \begin{figure}[t!]
   \centering
   \begin{tabular}{cl}
 
 \begin{figure}[t!]
   \centering
   \begin{tabular}{cl}

-    \parbox{9.5cm}{\includegraphics[scale=0.5]{R1/LT95.pdf}} & (a) \\
+    \parbox{9.5cm}{\includegraphics[scale=0.5125]{F/LT95.pdf}} & (a) \\

     \verb+ + \\
     \verb+ + \\

-    \parbox{9.5cm}{\includegraphics[scale=0.5]{R1/LT50.pdf}} & (b)
+    \parbox{9.5cm}{\includegraphics[scale=0.5125]{F/LT50.pdf}} & (b)

   \end{tabular}
   \caption{Network lifetime for (a) $Lifetime_{95}$ and 
     (b) $Lifetime_{50}$}
   \label{fig8}
 \end{figure} 
 
   \end{tabular}
   \caption{Network lifetime for (a) $Lifetime_{95}$ and 
     (b) $Lifetime_{50}$}
   \label{fig8}
 \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-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.
-
-
-%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.
-
-
-\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  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. 
-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
+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. 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
 lifetime, coverage  ratio, active  sensors ratio, energy  consumption, execution
 time. Indeed,  when dealing with  large wireless sensor networks,  a distributed

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

 global optimization problem by partitioning it in many smaller problems, one per
 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.
+subregion,  that  can be  solved  more  easily.  \textcolor{blue}{  Furthermore,
+  results  also show  that to  plan the  activity of  sensors for  large network
+  sizes, an  approach to obtain  a near optimal  solution is needed.  Indeed, an
+  exact resolution  of the resulting  optimization problem leads  to prohibitive
+  computation 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.
 
 %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.