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 %% \address{Address\fnref{label3}}
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-\title{ Multiperiod Distributed Lifetime Coverage Optimization Protocol in Wireless Sensor Networks}
+\title{Multiround Distributed Lifetime Coverage Optimization Protocol in Wireless Sensor Networks}

 
 %% use optional labels to link authors explicitly to addresses:
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-\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 (WSNs). In this paper, a method called Multiperiod Distributed Lifetime Coverage Optimization protocol (MuDiLCO), is proposed to maintain the coverage and to improve the lifetime in wireless sensor networks. The area  of interest is first divided into subregions and then the MuDiLCO protocol is distributed on the sensor nodes in each subregion. The proposed MuDiLCO protocol works into periods during which a sets of sensor nodes are scheduled to remaining active for a number of rounds during the sensing phase, to ensure coverage so as to maximize the lifetime of WSN. The  decision process is carried  out  by a leader node,  which  solves an integer program to produce the best representative sets to be used during the rounds of the sensing phase. Compared with some existing protocols, simulation results based on multiple criteria (energy consumption,coverage ratio, ...) show that the proposed protocol can prolong efficiently the network lifetime and improve the coverage performance.
-
+Coverage and  lifetime are  two paramount problems  in Wireless  Sensor Networks
+(WSNs). In this paper, a  method called Multiround Distributed Lifetime Coverage
+Optimization  protocol (MuDiLCO)  is proposed  to maintain  the coverage  and to
+improve the lifetime in wireless sensor  networks. The area of interest is first
+divided into  subregions and  then the  MuDiLCO protocol  is distributed  on the
+sensor nodes in  each subregion. The proposed MuDiLCO protocol  works in periods
+during which sets of sensor nodes are  scheduled, 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}
 
 \section{Introduction}
  
 \end{keyword}
 
 \end{frontmatter}
 
 \section{Introduction}
  

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

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

 \label{rw}
 
 \label{rw}
 

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

 \begin{itemize}
 \begin{itemize}

-\item Sensors scheduling Algorithms, i.e. centralized or distributed/localized algorithms.
-\item The objective of sensor coverage, i.e. to maximize the network lifetime
-or to minimize the number of sensors during the sensing period.
-\item The homogeneous or heterogeneous nature of the
-nodes, in terms of sensing or communication capabilities.
+\item  Sensors   scheduling  algorithm  implementation,   i.e.   centralized  or
+  distributed/localized algorithms.
+\item The objective of sensor coverage, i.e. to maximize the network lifetime or
+  to minimize the number of active sensors during a sensing round.
+\item The homogeneous or heterogeneous nature  of the nodes, in terms of sensing
+  or communication capabilities.

 \item The node deployment method, which may be random or deterministic.
 \item The node deployment method, which may be random or deterministic.

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

 \end{itemize}
 
 \end{itemize}
 

-The choice of non-disjoint or disjoint cover sets (sensors participate or not in many cover sets) can be added to the above list.
-% The 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}
-%{\bf Centralized approaches}
-The major approach is
-to divide/organize  the sensors into  a suitable number of  set covers
-where each  set completely covers  an interest region and  to activate
-these set covers successively. The centralized algorithms always provide nearly or close to optimal solution since the algorithm has global view of the whole network. Note that centralized algorithms have the advantage of requiring very low processing power from the sensor nodes, which usually have
-limited processing capabilities. 
-
-The first algorithms  proposed in the  literature consider that  the cover
-sets  are  disjoint: a  sensor  node appears  in  exactly  one of  the
-generated  cover  sets.    For  instance,  Slijepcevic  and  Potkonjak
-\cite{Slijepcevic01powerefficient}   propose    an   algorithm, which
-allocates sensor nodes in mutually independent sets to monitor an area
-divided into  several fields.  Their algorithm builds  a cover  set by
-including in  priority the sensor  nodes, which cover  critical fields,
-that  is to  say fields  that are  covered by  the smallest  number of
-sensors. The time complexity of  their heuristic is $O(n^2)$ where $n$
-is the number of  sensors. Abrams et al.~\cite{abrams2004set}  design  three  approximation
-algorithms  for a  variation of  the  set k-cover  problem, where  the
-objective is to partition the sensors into covers such that the number
-of covers that  includes an area, summed over  all areas, is maximized.
-Their        work        builds        upon       previous        work
-in~\cite{Slijepcevic01powerefficient} and the  generated cover sets do
-not provide complete coverage of the monitoring zone.
-\cite{cardei2005improving} propose a method to efficiently
-compute the maximum  number of disjoint set covers  such that each set
-can  monitor all  targets. They  first  transform the  problem into  a
-maximum  flow   problem, which  is  formulated  as   a  mixed  integer
-programming (MIP). Then their heuristic  uses the output of the MIP to
-compute  disjoint  set  covers.  Results  show  that  this  heuristic
-provides  a   number  of  set  covers  slightly   larger  compared  to
-\cite{Slijepcevic01powerefficient}  but with  a larger  execution time
-due  to the complexity  of the  mixed integer  programming resolution.
-
-Zorbas  et  al.  \cite{zorbas2010solving}  presented a centralised greedy
-algorithm for the efficient production of both node disjoint
-and non-disjoint cover sets. Compared to algorithm's results  of  Slijepcevic and Potkonjak
-\cite{Slijepcevic01powerefficient},  their   heuristic  produces  more
-disjoint cover sets with a slight growth rate in execution time. When producing non-disjoint cover sets, both Static-CCF and Dynamic-CCF provide cover sets offering longer network lifetime than those produced by
-\cite{cardei2005energy}. Also, they require a smaller number of node participations in order to
-achieve these results.
-
-In the  case of non-disjoint algorithms  \cite{pujari2011high}, sensors may
-participate  in more  than  one cover  set.   In some  cases, this  may
-prolong  the lifetime  of the  network in  comparison to  the disjoint
-cover set algorithms, but  designing algorithms for non-disjoint cover
-sets  generally induces a  higher order  of complexity.   Moreover, in
-case of a sensor's  failure, non-disjoint scheduling policies are less
-resilient and less  reliable because a sensor may  be involved in more
-than one  cover sets.  For instance,  Cardei et al.~\cite{cardei2005energy}
-present a  linear programming (LP)  solution and a greedy  approach to
-extend the  sensor network lifetime  by organizing the sensors  into a
-maximal  number of  non-disjoint cover  sets. Simulation  results show
-that by allowing sensors to  participate in multiple sets, the network
-lifetime         increases        compared         with        related
-work~\cite{cardei2005improving}.   In~\cite{berman04}, the
-authors  have formulated  the lifetime  problem and  suggested another
-(LP)  technique to  solve this  problem. A  centralized  solution  based      on      the     Garg-K\"{o}nemann
-algorithm~\cite{garg98}, provably near
-the optimal solution,    is also proposed.
+The choice of non-disjoint or disjoint cover sets (sensors participate or not in
+many cover sets) can be added to the above list.
+
+\subsection{Centralized approaches}
+
+The major approach  is to divide/organize the sensors into  a suitable number of
+cover sets where  each set completely covers an interest  region and to activate
+these cover sets successively.  The centralized algorithms always provide nearly
+or close to  optimal solution since the  algorithm has global view  of the whole
+network. Note that  centralized algorithms have the advantage  of requiring very
+low  processing  power  from  the  sensor  nodes,  which  usually  have  limited
+processing  capabilities. The  main drawback  of this  kind of  approach is  its
+higher cost in communications, since the  node that will make the decision needs
+information  from all  the  sensor nodes.   \textcolor{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
+disjoint:  a  sensor  node  appears  in  exactly  one  of  the  generated  cover
+sets~\cite{abrams2004set,cardei2005improving,Slijepcevic01powerefficient}.    In
+the  case   of  non-disjoint   algorithms  \cite{pujari2011high},   sensors  may
+participate in  more than one  cover set.  In some  cases, this may  prolong the
+lifetime of the network in comparison  to the disjoint cover set algorithms, but
+designing  algorithms for  non-disjoint cover  sets generally  induces a  higher
+order  of complexity.   Moreover, in  case of  a sensor's  failure, non-disjoint
+scheduling policies  are less  resilient and  reliable because  a sensor  may be
+involved in more than one cover sets.
+
+In~\cite{yang2014maximum},  the authors  have  considered  a linear  programming
+approach  to select  the minimum  number of  working sensor  nodes, in  order to
+preserve a  maximum coverage and  to extend lifetime  of the network.   Cheng et
+al.~\cite{cheng2014energy} have defined a  heuristic algorithm called Cover Sets
+Balance  (CSB), which  chooses  a set  of  active nodes  using  the tuple  (data
+coverage range, residual  energy).  Then, they have introduced  a new Correlated
+Node Set Computing (CNSC) algorithm to find  the correlated node set for a given
+node.   After that,  they  proposed a  High Residual  Energy  First (HREF)  node
+selection algorithm to minimize the number of  active nodes so as to prolong the
+network  lifetime.   Various  centralized  methods based  on  column  generation
+approaches                   have                    also                   been
+proposed~\cite{gentili2013,castano2013column,rossi2012exact,deschinkel2012column}.
+\textcolor{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 $\&$ localized coverage algorithms, the required computation to schedule the activity of sensor nodes will be done by the cooperation among the neighbours nodes. These algorithms may require more computation power  for the processing by the cooperated sensor nodes but they are more scaleable for large WSNs.  Localized and distributed algorithms generally result in non-disjoint set covers. 
-
-Some      distributed     algorithms      have      been     developed
-in~\cite{Gallais06,Tian02,Ye03,Zhang05,HeinzelmanCB02, yardibi2010distributed}  to perform the
-scheduling so as to coverage preservation.   Distributed algorithms typically  operate in  rounds for
-a predetermined  duration. At  the  beginning of  each  round, a  sensor
-exchanges information with its neighbors and makes a decision to either
-remain turned  on or to  go to sleep  for the round. This  decision is
-basically made on simple greedy criteria like  the largest uncovered
-area   \cite{Berman05efficientenergy},   maximum   uncovered   targets
-\cite{lu2003coverage}.   In \cite{Tian02}, the  scheduling scheme  is divided
-into rounds, where each round  has a self-scheduling phase followed by
-a sensing phase.  Each sensor  broadcasts a message containing the node ID
-and the node location  to its neighbors at the beginning  of each round. A
-sensor determines  its status by  a rule named off-duty  eligible rule,
-which tells  him to  turn off if  its sensing  area is covered  by its
-neighbors. A  back-off scheme is  introduced to let each  sensor delay
-the decision process  with a random period of time,  in order to avoid
-simultaneous conflicting decisions between nodes and  lack of coverage on any area.
-\cite{prasad2007distributed}  defines a model  for capturing
-the dependencies between different cover sets and proposes localized
-heuristic based on this dependency. The algorithm consists of two
-phases, an initial  setup phase during which each  sensor computes and
-prioritizes the  covers and  a sensing phase  during which  each sensor
-first decides  its on/off status, and  then remains on or off for the
-rest  of the  duration. 
-
-The authors in \cite{yardibi2010distributed}  developed a distributed adaptive sleep scheduling algorithm (DASSA) for WSNs with partial coverage. DASSA does not require location information of sensors while maintaining connectivity and satisfying a user defined coverage target. In DASSA, nodes use the residual energy levels and feedback from the sink for scheduling the activity of their neighbors. This feedback mechanism reduces the randomness in scheduling that would otherwise occur due to the absence of location information. 
-
-In \cite{ChinhVu},  the author proposed a novel  distributed heuristic, called
-Distributed Energy-efficient  Scheduling for k-coverage  (DESK), which
-ensures that the energy consumption  among the sensors is balanced and
-the lifetime  maximized while the coverage  requirement is maintained.
-This  heuristic   works  in  rounds,  requires   only  1-hop  neighbor
-information,  and each  sensor decides  its status  (active  or sleep)
-based    on    the    perimeter    coverage    model    proposed    in
-\cite{Huang:2003:CPW:941350.941367}.
-
-
-%Our Work, which is presented in~\cite{idrees2014coverage} proposed a coverage optimization protocol to improve the lifetime in
-%heterogeneous energy wireless sensor networks. 
-%In this work, the coverage protocol distributed in each sensor node in the subregion but the optimization take place over the the whole subregion. We consider only distributing the coverage protocol over two subregions. 
-
-The works presented in \cite{Bang, Zhixin, Zhang}  focuses on a Coverage-Aware, Distributed  Energy- Efficient and distributed clustering methods respectively, which aims to extend the network lifetime, while the coverage is ensured.
-S.  Misra  et al.  \cite{Misra}  proposed  a  localized algorithm  for
-coverage in  sensor networks. The algorithm conserve  the energy while
-ensuring  the network coverage  by activating  the subset  of sensors,
-with  the  minimum  overlap  area. The proposed  method  preserves  the
-network connectivity by formation of the network backbone. 
-More    recently,   Shibo   et
-al. \cite{Shibo}  expressed the coverage  problem as a  minimum weight
-submodular  set cover  problem  and proposed  a Distributed  Truncated
-Greedy Algorithm  (DTGA) to  solve it. They  take advantage  from both
-temporal  and spatial  correlations between  data sensed  by different
-sensors,  and   leverage  prediction,  to  improve   the  lifetime. 
-
-In \cite{xu2001geography}, Xu et al. proposed an algorithm, called Geographical Adaptive Fidelity (GAF), which uses geographic location information to divide the area of interest into fixed square grids. Within each grid, it keeps only one node staying awake to take the responsibility of sensing and communication.
-
-Some other approaches (outside the scope of our work) do  not consider a synchronized and predetermined
-period  of time  where the  sensors are  active or  not.  Indeed, each
-sensor  maintains its  own timer  and its  wake-up time is randomized
-\cite{Ye03} or regulated \cite{cardei2005maximum} over time. 
-
-The MuDiLCO protocol (for Multiperiod Distributed Lifetime Coverage Optimization protocol) presented in this paper is an extension of the approach explained in~\cite{idrees2014coverage}. In~\cite{idrees2014coverage}, the protocol is deployed over only two subregions. Simulation results have shown that it was more interesting to divide the area into several subregions, given the computation complexity. Compared to our previous paper, we study here the possibility of dividing the sensing phase into multiple rounds and we also add a model of energy consumption to assess the efficiency of our approach.
-%The 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{ The MuDiLCO Protocol Description}
+
+In distributed  and localized coverage  algorithms, the required  computation to
+schedule the  activity of  sensor nodes  will be done  by the  cooperation among
+neighboring nodes. These  algorithms may require more computation  power for the
+processing by the cooperating sensor nodes, but they are more scalable for large
+WSNs.  Localized and distributed algorithms generally result in non-disjoint set
+covers.
+
+Many distributed algorithms have been  developed to perform the scheduling so as
+to          preserve         coverage,          see          for         example
+\cite{Gallais06,Tian02,Ye03,Zhang05,HeinzelmanCB02,       yardibi2010distributed,
+  prasad2007distributed,Misra}.   Distributed  algorithms  typically operate  in
+rounds for  a predetermined duration. At  the beginning of each  round, a sensor
+exchanges information with  its neighbors and makes a  decision to either remain
+turned on or  to go to sleep for  the round. This decision is  basically made on
+simple     greedy     criteria    like     the     largest    uncovered     area
+\cite{Berman05efficientenergy}      or       maximum      uncovered      targets
+\cite{lu2003coverage}.   The  Distributed  Adaptive Sleep  Scheduling  Algorithm
+(DASSA) \cite{yardibi2010distributed}  does not require  location information of
+sensors while  maintaining connectivity and  satisfying a user  defined coverage
+target.  In  DASSA, nodes use the  residual energy levels and  feedback from the
+sink for  scheduling the activity  of their neighbors.  This  feedback mechanism
+reduces  the randomness  in scheduling  that would  otherwise occur  due  to the
+absence of location information.  In  \cite{ChinhVu}, the author have designed a
+novel distributed heuristic,  called Distributed Energy-efficient Scheduling for
+k-coverage (DESK), which  ensures that the energy consumption  among the sensors
+is  balanced  and the  lifetime  maximized  while  the coverage  requirement  is
+maintained.   This heuristic  works in  rounds, requires  only  one-hop neighbor
+information, and each  sensor decides its status (active or  sleep) based on the
+perimeter coverage model from~\cite{Huang:2003:CPW:941350.941367}.
+
+The  works presented  in  \cite{Bang, Zhixin,  Zhang}  focus on  coverage-aware,
+distributed energy-efficient,  and distributed clustering  methods respectively,
+which  aim at extending  the network  lifetime, while  the coverage  is ensured.
+More recently, Shibo et al.  \cite{Shibo} have expressed the coverage problem as
+a  minimum  weight submodular  set  cover  problem  and proposed  a  Distributed
+Truncated Greedy  Algorithm (DTGA) to solve  it.  They take  advantage from both
+temporal and spatial correlations between  data sensed by different sensors, and
+leverage prediction, to improve  the lifetime.  In \cite{xu2001geography}, Xu et
+al.  have  described an algorithm, called Geographical  Adaptive Fidelity (GAF),
+which uses geographic  location information to divide the  area of interest into
+fixed square grids.   Within each grid, it keeps only one  node staying awake to
+take the responsibility of sensing and communication.
+
+Some  other  approaches (outside  the  scope  of our  work)  do  not consider  a
+synchronized and  predetermined time-slot where  the sensors are active  or not.
+Indeed, each sensor  maintains its own timer and its  wake-up time is randomized
+\cite{Ye03} or regulated \cite{cardei2005maximum} over time.
+
+The MuDiLCO protocol (for  Multiround Distributed Lifetime Coverage Optimization
+protocol) presented  in this paper  is an  extension of the  approach introduced
+in~\cite{idrees2014coverage}.   In~\cite{idrees2014coverage},  the  protocol  is
+deployed over  only two subregions.  Simulation results  have shown that  it was
+more  interesting  to  divide  the  area  into  several  subregions,  given  the
+computation complexity. Compared to our previous paper, in this one we study the
+possibility of dividing  the sensing phase into multiple rounds  and we also add
+an  improved  model of  energy  consumption  to  assess  the efficiency  of  our
+approach. In fact, in this paper we make a multiround optimization, while it was
+a single round  optimization in our previous work.  \textcolor{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}.
+
+
+\section{MuDiLCO protocol description}

 \label{pd}
 
 \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 Multiperiod Distributed Lifetime Coverage Optimization protocol, which is called MuDiLCO. It is  distributed on each subregion in the area of interest. It is based on two efficient techniques: network
-%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 and Models}
-We consider  a randomly and  uniformly deployed network  consisting of
-static  wireless sensors. The  wireless sensors  are deployed  in high
-density to ensure initially a high coverage ratio of the interested area. We
-assume that  all nodes are  homogeneous in terms of  communication and
-processing capabilities and heterogeneous in term of energy provision.
-The  location  information is  available  to  the  sensor node  either
-through hardware such as embedded GPS or through location discovery
-algorithms.   
-\indent We consider a boolean  disk coverage model which is the most
-widely used sensor coverage model in the literature. Each sensor has a
-constant sensing range $R_s$. All  space points within a disk centered
-at  the sensor with  the radius  of the  sensing range  is said  to be
-covered by this sensor. We also assume that the communication range $R_c \geq 2R_s$.
-In  fact,   Zhang  and Zhou~\cite{Zhang05} proved that if the transmission range fulfills the
-previous hypothesis, a complete coverage of a convex area implies
-connectivity among the working nodes in the active mode.
-
-\indent Instead of working with the coverage area, we consider for each
-sensor a set of points called  primary points. We also assume that the
-sensing disk defined  by a sensor is covered if all the primary points of
-this sensor are covered. The choice of number and locations of primary points is the subject of another study not presented here.
-
-%By  knowing the  position (point  center: ($p_x,p_y$))  of  a wireless
-%sensor node  and its $R_s$,  we calculate the primary  points directly
-%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.
-
-\subsection{The Main Idea}
-The   area  of  interest   can  be  divided using the
-divide-and-conquer strategy into smaller areas,  called subregions and
-then  our MuDiLCO protocol  will be implemented  in each  subregion
-simultaneously. \\
-
-\noindent Our MuDiLCO protocol works in periods fashion as shown in figure~\ref{fig2}.
-\begin{figure}[ht!]
-\centering
-\includegraphics[width=95mm]{Modelgeneral.pdf} % 70mm
-\caption{MuDiLCO protocol}
+\subsection{Assumptions}
+
+We  consider a  randomly and  uniformly  deployed network  consisting of  static
+wireless sensors.  The sensors are  deployed in high density to ensure initially
+a high  coverage ratio  of the interested  area.  We  assume that all  nodes are
+homogeneous  in   terms  of  communication  and   processing  capabilities,  and
+heterogeneous  from the  point  of view  of  energy provision.   Each sensor  is
+supposed  to get information  on its  location either  through hardware  such as
+embedded GPS or through location discovery algorithms.
+   
+To model  a sensor node's coverage  area, we consider the  boolean disk coverage
+model   which  is  the   most  widely   used  sensor   coverage  model   in  the
+literature. Thus, each  sensor has a constant sensing range  $R_s$ and all space
+points within  the disk centered  at the sensor  with the radius of  the sensing
+range  is  said  to  be  covered  by  this sensor.   We  also  assume  that  the
+communication   range  satisfies   $R_c  \geq   2R_s$.   In   fact,   Zhang  and
+Zhou~\cite{Zhang05} proved that if  the transmission range fulfills the previous
+hypothesis, a complete coverage of  a convex area implies connectivity among the
+active nodes.
+
+\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 its  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}
+
+\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} 
 
 \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. On the one hand,  if a node  failure is  detected before
-making the decision, the node will not participate to this phase, and,
-on the other hand, if the  node failure occurs after the decision, the
-sensing task of the network  will be temporarily affected: only during
-the period of sensing until a new period starts.  
-The energy
-consumption  and  some other  constraints  can  easily  be taken  into
-account  since  the  sensors   can  update  and  then  exchange  their
-information (including their residual energy) at the beginning of each
-period.  However,   the  pre-sensing  phases   (Information Exchange,  Leader
-Election,  Decision) are energy  consuming for  some nodes,  even when
-they do not  join the network to monitor the  area. 
-
-%%%%%%%%%%%%%%%%%parler optimisation%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-We define two types of packets used by MuDiLCO protocol :
+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
+(including their residual energy) at the beginning of each period.  However, the
+pre-sensing  phases (Information  Exchange, Leader  Election, and  Decision) are
+energy  consuming for some  nodes, even  when they  do not  join the  network to
+monitor the area.
+
+We define two types of packets that will be used by the proposed protocol:

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

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

 \end{enumerate}
 
 \end{enumerate}
 

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

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

-\item LISTENING: Sensor is waiting for a decision (to be active or not)
-\item COMPUTATION: Sensor applies the optimization process as leader
-\item ACTIVE: Sensor is active
-\item SLEEP: Sensor is turned off
-\item COMMUNICATION: Sensor is transmitting or receiving packet
+\item LISTENING: sensor node is waiting for a decision (to be active or not);
+\item  COMPUTATION: sensor  node  has been  elected  as leader  and applies  the
+  optimization process;
+\item ACTIVE: sensor node is taking part in the monitoring of the area;
+\item SLEEP: sensor node is turned off to save energy;
+\item COMMUNICATION: sensor node is transmitting or receiving packet.

 \end{enumerate}
 
 Below, we describe each phase in more details.
 
 \subsection{Information Exchange Phase}
 
 \end{enumerate}
 
 Below, we describe each phase in more details.
 
 \subsection{Information Exchange Phase}
 

-Each sensor node $j$ sends its position, remaining energy $RE_j$, and
-the number of neighbours  $NBR_j$ to all wireless sensor nodes in
-its subregion by using an INFO packet (containing information on position coordinates, current remaining energy, sensor node id, number of its one-hop live neighbors)  and then listens to the packets
-sent from  other nodes.  After that, each  node will  have information
-about  all the  sensor  nodes in  the  subregion.  In  our model,  the
-remaining energy corresponds to the time that a sensor can live in the
-active mode.
-
-%\subsection{\textbf Working Phase:}
-
-%The working phase works in rounding fashion. Each round include 3 steps described as follow :
-
-\subsection{Leader Election Phase}
-This step includes choosing the Wireless  Sensor Node  Leader (WSNL),
-which  will  be  responsible  for executing  the coverage  algorithm.  Each
-subregion  in  the   area  of  interest  will  select   its  own  WSNL
-independently  for each  period.  All the  sensor  nodes cooperate  to
-select WSNL.  The nodes in the  same subregion will  select the leader
-based on  the received  information from all  other nodes in  the same
-subregion.  The selection criteria  in order  of priority  are: larger
-number  of neighbours,  larger remaining  energy, and  then in  case of
-equality, larger index. Observations on previous simulations suggest to use the number of $1-hop$ neighbours as the primary criterion to reduce energy consumption due to the communication.
-
-%the more priority selection factor is the number of $1-hop$ neighbours, $NBR j$, which can  minimize the energy consumption during the communication Significantly.  
-%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.  
-
+Each sensor node $j$ sends its position, remaining energy $RE_j$, and the number
+of neighbors $NBR_j$  to all wireless sensor nodes in its  subregion by using an
+INFO packet  (containing information on position  coordinates, current remaining
+energy, sensor node ID, number of its one-hop live neighbors) and then waits for
+packets sent by other nodes.  After  that, each node will have information about
+all  the sensor  nodes in  the subregion.   In our  model, the  remaining energy
+corresponds to the time that a sensor can live in the active mode.
+
+\subsection{Leader Election phase}
+
+This step  consists in choosing  the Wireless  Sensor Node Leader  (WSNL), which
+will be responsible for executing the coverage algorithm.  Each subregion in the
+area of  interest will select its  own WSNL independently for  each period.  All
+the sensor  nodes cooperate to  elect a WSNL.  The  nodes in the  same subregion
+will select the leader based on the received information from all other nodes in
+the same subregion.  The selection criteria  are, in order of importance: larger
+number of  neighbors, larger  remaining energy,  and then  in case  of equality,
+larger index. Observations on previous simulations  suggest to use the number of
+one-hop neighbors as  the primary criterion to reduce energy  consumption due to
+the communications.

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

-The  WSNL will  solve an  integer  program to
-select which cover sets will be activated in the following sensing phase
-to cover  the subregion. The integer program will produce $T$ cover sets (for $T$ rounds). WSNL will send  Active-Sleep packet to each
-sensor in the subregion based on the algorithm's results, indicating if the sensor should be active or not in each round of the sensing phase.
-\indent  The integer  program   is   based   on  the   model   proposed   by
-\cite{pedraza2006} with some modification, where the objective is  to find a maximum number of
-disjoint  cover sets.   To accomplish  this goal,  authors  proposed an
-integer program, which forces undercoverage and overcoverage of targets
-to  become  minimal at  the  same  time.   They use  binary  variables
-$x_{jl}$ to indicate  if sensor $j$ belongs to cover  set $l$.  In our
-model, we  consider  binary  variables $X_{t,j}$,  which  determine the possiblity of activation of sensor $j$ during the round $t$ of a given sensing phase. We also consider primary points  as targets.   The set  of primary  points is
-denoted by $P$ and the set of sensors by $J$. Only sensors able to be alive during at least one round are involved in the integer program.
-
-%parler de la limite en energie Et pour un round
-
-\noindent  For  a primary  point  $p$,  let  $\alpha_{j,p}$ denote  the
-indicator function of whether the point $p$ is covered, that is:
+\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
+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:

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


@@ -484,8 +470,8 @@ indicator function of whether the point $p$ is covered, that is:
 \end{array} \right.
 %\label{eq12} 
 \end{equation}
 \end{array} \right.
 %\label{eq12} 
 \end{equation}

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

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


@@ -504,10 +490,10 @@ We define the Overcoverage variable $\Theta_{t,p}$ as:
 \end{array} \right.
 \label{eq13} 
 \end{equation}
 \end{array} \right.
 \label{eq13} 
 \end{equation}

-\noindent More precisely, $\Theta_{t,p}$ represents the number of active
-sensor  nodes  minus  one  that  cover the  primary  point  $p$ during the round $t$.\\
-The Undercoverage variable $U_{t,p}$ of the primary point $p$ during round $t$ is defined
-by:
+More  precisely, $\Theta_{t,p}$  represents the  number of  active  sensor nodes
+minus  one  that  cover  the  primary  point $p$  during  round  $t$.   The
+Undercoverage variable  $U_{t,p}$ of the primary  point $p$ during  round $t$ is
+defined by:

 \begin{equation}
 U_{t,p} = \left \{ 
 \begin{array}{l l}
 \begin{equation}
 U_{t,p} = \left \{ 
 \begin{array}{l l}


@@ -517,432 +503,575 @@ U_{t,p} = \left \{
 \label{eq14} 
 \end{equation}
 
 \label{eq14} 
 \end{equation}
 

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

 \begin{equation}
 \begin{equation}

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

 \end{equation}
 
 \end{equation}
 

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

 \begin{equation}
 \begin{equation}

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

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

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

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

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

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

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

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

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

 \end{equation}
 
 \end{equation}
 

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

 \begin{itemize}
 \begin{itemize}

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

 \end{itemize}
 
 \end{itemize}
 

-The first group  of constraints indicates that some  primary point $p$
-should be covered by at least one  sensor and, if it is not always the
-case,  overcoverage  and  undercoverage  variables  help  balancing  the
-restriction  equations by taking  positive values. Constraint \ref{eq144} guarantees that the sensor has enough energy ($RE_j$ its remaining energy) to be alive during the selected rounds knowing that $E_{th}$ is the requiring energy to be alive during one round. 
-There are  two main         
-objectives.  First, we limit the overcoverage of primary points in order to
-activate a minimum number of sensors.  Second we prevent the absence of monitoring on
- some parts of the subregion by  minimizing the undercoverage.   The
-weights  $w_\theta$  and  $w_U$  must  be properly  chosen  so  as  to
-guarantee that  the maximum number  of points are covered  during each
-round. In our simulations priority is given to the coverage by choosing $W_{\theta}$ very large compared to $W_U$. 
-%The Active-Sleep packet includes the schedule vector with the number of rounds that should be applied by the receiving sensor node during the sensing phase.
-
+The first group  of constraints indicates that some primary  point $p$ should be
+covered by at least  one sensor and, if it is not  always the case, overcoverage
+and undercoverage variables  help balancing the restriction  equations by taking
+positive values. The constraint  given by equation~(\ref{eq144}) guarantees that
+the sensor has enough energy ($RE_j$  corresponds to its remaining energy) to be
+alive during  the selected rounds knowing  that $E_{R}$ is the  amount of energy
+required to be alive during one round.
+
+There are  two main  objectives.  First,  we limit  the overcoverage  of primary
+points in order to activate a minimum  number of sensors.  Second we prevent the
+absence  of  monitoring  on  some  parts of  the  subregion  by  minimizing  the
+undercoverage.  The weights  $W_\theta$ and $W_U$ must be properly  chosen so as
+to guarantee that the maximum number of points are covered during each round.
+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}
 
 \subsection{Sensing phase}

-The sensing phase consists of $T$ rounds. Each sensor node in the subregion will receive  an Active-Sleep  packet from WSNL  informing it to stay awake or to go to sleep  for each round of the sensing phase. 
-% In each round during the sensing phase, there is a cover set of sensor nodes,  in which  the active  sensors will  execute  their sensing  task  to preserve maximal  coverage and lifetime in the subregion and this will continue until finishing the round $T$ and starting new period. 

 
 

-
-
-\begin{algorithm}                
- % \KwIn{all the parameters related to information exchange}
-%  \KwOut{$winer-node$ (: the id of the winner sensor node, which is the leader of current round)}
-  \BlankLine
-  \emph{Initialize the sensor node and determine it's position and it's subregion} \; 
-  
-  \If{ $RE_j \geq E_{th}$ }{
-      \emph{ $s_j.status$ = LISTENING}\;
-      \emph{ Send and Receive INFO Packet to and from other nodes in the subregion}\;
-      %\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}\;
+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 sensor  node~$s_j$  at the  beginning  of a  period,
+explains how the Active-Sleep packet is obtained.
+
+\begin{algorithm}[h!]                
+  \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{ the received INFO Packet = No. of nodes in it's subregion -1  }{
-           \emph{Selection of LeaderID}\;
-           \If{ $ s_j.ID = LeaderID $}{ 
-               \emph{Execute Integer Program Algorithm $(Schedule_{T,J})$ }\;
-               \emph{ Send $ActiveSleep()$ Packet with $Schedule_{1..T,k}$  }\;
-                 \emph{UPDATE $RE_j $}\;
-                         }       
-           \Else{ 
-                 \emph{Wait $ActiveSleep()$ Packet from the Leader}\;
-                % \emph{After receiving Packet, Retrieve the schedule and the $T$ rounds}\;
-                 \emph{UPDATE $RE_j $}\;
-           }  
-    %  }
-
+      \emph{LeaderID = Leader election}\;
+      \If{$ s_j.ID = LeaderID $}{
+        \emph{$s_j.status$ = COMPUTATION}\;
+        \emph{$\left\{\left(X_{1,k},\dots,X_{T,k}\right)\right\}_{k \in J}$ =
+          Execute Integer Program Algorithm($T,J$)}\;
+        \emph{$s_j.status$ = COMMUNICATION}\;
+        \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}\;
+        \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 framework}
+\label{exp}

 
 

+\subsection{Simulation setup}

 
 

-
-\section{Simulations}
-\label{exp}
-\subsection{Simulation Framework}
-We conducted  a series of simulations to evaluate the
-efficiency and the relevance of our approach,  using the  discrete event
-simulator  OMNeT++  \cite{varga}. The simulation parameters are summarized in
-Table~\ref{table3}. \\
+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
+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  & $(50 \times 25)~m^2 $   \\
-% inserting body of the table
-%\hline
-Nodes Number &  50, 100, 150, 200 and 250~nodes   \\
-%\hline
-Initial Energy  & 500-700~joules  \\  
-%\hline
-Sensing Time for One Round & 60 Minutes \\
-$E_{th}$ & 36 Joules\\
+Sensing field size & $(50 \times 25)~m^2 $   \\
+Network size &  50, 100, 150, 200 and 250~nodes   \\
+Initial energy  & 500-700~joules  \\  
+Sensing time for one round & 60 Minutes \\
+$E_{R}$ & 36 Joules\\

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

-25 simulation runs are performed with different network topologies. The results presented hereafter are the average of these 25 runs. 
-%Based on the results of our proposed work in~\cite{idrees2014coverage}, we found as the region of interest are divided into larger subregions as the network lifetime increased. In this simulation, the network are divided into 16 subregions. 
-We  performed  simulations for five different densities varying from 50 to 250~nodes. Experimental results are obtained from  randomly generated  networks in  which  nodes are
-deployed over a  $(50 \times 25)~m^2 $ sensing  field. More precisely, the deployment is controlled at a coarse scale in   order to ensure that the  deployed nodes can cover the sensing field with the given sensing range.\\
-
-  
-Our MuDiLCO protocol is declined into four versions: MuDiLCO-1, MuDiLCO-3, MuDiLCO-5, and MuDiLCO-7, corresponding  to $T=1$, $T=3$, $T=5$ or $T=7$ ($T$ the number of rounds in one sensing period).  We call the method MuDiLCO-T for the general case. We compare MuDiLCO-T with two other methods. The first method, called DESK and proposed by ~\cite{ChinhVu} is a full distributed coverage algorithm. The second method, called GAF ~\cite{xu2001geography}, consists in dividing the region into fixed squares.   During the  decision phase,  in  each square,  one sensor  is chosen to remain on during the sensing phase time.\\
-
-
-
-
-\subsection{Energy Model}
-
-We use an energy consumption model proposed by~\cite{ChinhVu} and based on ~\cite{raghunathan2002energy} with slight modifications.
-The energy consumption for sending/receiving the packets is added whereas the part related to the sensing range is removed because we consider a fixed sensing range.
-% We are took into account the energy consumption needed for the high computation during executing the algorithm on the sensor node. 
-%The new energy consumption model will take into account the energy consumption for communication (packet transmission/reception), the radio of the sensor node, data sensing, computational energy of Micro-Controller Unit (MCU) and high computation energy of MCU. 
-%revoir la phrase
-
-For our energy consumption model, we refer to the sensor node (Medusa II) which uses Atmels AVR ATmega103L microcontroller~\cite{raghunathan2002energy}. The typical architecture of a sensor is composed of four subsystems : the MCU subsystem which is capable of computation, communication subsystem (radio) which is responsible for
-transmitting/receiving messages, sensing subsystem that collects data, and the power supply which  powers the complete sensor node ~\cite{raghunathan2002energy}. Each of the first three subsystems can be turned on or off depending on the current status of the sensor. Energy consumption (expressed in milliWatt per second)  for the different status of the sensor is summarized in Table~\ref{table4}. The energy needed to send or receive a 1-bit is equal to $0.2575 mW$.
+\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  is  to consider  the
+  average execution  time to solve  the integer  programs to optimality  for 100
+  nodes and then to adjust the time linearly according to the increasing network
+  size. 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
+subregions  which subdivides  the sensing  field, considering  different network
+sizes. They show that as the number of subregions increases, so does the network
+lifetime. Moreover,  it makes  the MuDiLCO protocol  more robust  against random
+network  disconnection due  to node  failures.  However,  too many  subdivisions
+reduce the  advantage of the optimization.  In fact, there is  a balance between
+the benefit from the optimization and the  execution time needed to solve it. In
+the following we have set the number of subregions to 16.
+
+\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.
+
+For our  energy consumption model, we  refer to the sensor  node Medusa~II which
+uses an Atmels  AVR ATmega103L microcontroller~\cite{raghunathan2002energy}. The
+typical  architecture  of a  sensor  is composed  of  four  subsystems: the  MCU
+subsystem which is capable of computation, communication subsystem (radio) which
+is responsible  for transmitting/receiving messages, the  sensing subsystem that
+collects  data, and  the  power supply  which  powers the  complete sensor  node
+\cite{raghunathan2002energy}. Each  of the first three subsystems  can be turned
+on or  off depending on  the current status  of the sensor.   Energy consumption
+(expressed in  milliWatt per second) for  the different status of  the sensor is
+summarized in Table~\ref{table4}.

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

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

 \hline
 \hline

-% inserts single horizontal line
-Listening & ON & ON & ON & 20.05 \\
-% inserting body of the table
+LISTENING & on & on & on & 20.05 \\

 \hline
 \hline

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

 \hline
 \hline

-Sleep & OFF & OFF & OFF & 0.02 \\
+SLEEP & off & off & off & 0.02 \\
+\hline
+COMPUTATION & on & on & on & 26.83 \\

 \hline
 \hline

-Computation & ON & ON & ON & 26.83 \\
-%\hline
-%\multicolumn{4}{|c|}{Energy needed to send/receive a 1-bit} & 0.2575\\
- \hline

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

-% is used to refer this table in the text

 \end{table}
 
 \end{table}
 

-For sake of  simplicity we ignore the energy needed to turn on the
-radio, to start up the sensor node, the transition from mode to another, etc. 
-%We also do not consider the need of collecting sensing data. PAS COMPRIS
-Thus, when a sensor becomes active (i.e., it already decides it's status), it can turn its radio off to save battery. MuDiLCO uses two types of packets for communication. The size of the INFO-Packet and Status-Packet are 112 bits and 24 bits respectively. 
-The value of energy spent to send a 1-bit-content message is obtained by using the equation in ~\cite{raghunathan2002energy} to calculate the energy cost for transmitting messages and we propose the same value for receiving the packets.
-
-
-The initial energy of  each node is  randomly set in the interval $[500-700]$.  Each  sensor  node will  not participate in the next round if its remaining energy is less than $E_{th}=36 Joules$, the minimum energy needed for the node to stay alive during one round. This value has been computed by multiplying the energy consumed in active state (9.72 mWs) by the time in second for one round (3600 seconds). According to the interval of initial energy, a sensor may be alive during at most 20 rounds.\\
-
+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.
+Thus, when a sensor becomes active (i.e.,  it has already chosen its status), it
+can turn its radio  off to save battery.  MuDiLCO uses two  types of packets for
+communication. The size of the INFO  packet and Active-Sleep packet are 112~bits
+and 24~bits  respectively.  The value  of energy  spent to send  a 1-bit-content
+message is  obtained by using  the equation in  ~\cite{raghunathan2002energy} to
+calculate the  energy cost  for transmitting  messages and  we propose  the same
+value for receiving  the packets. The energy  needed to send or  receive a 1-bit
+packet is equal to 0.2575~mW.
+
+The initial energy of each node is  randomly set in the interval $[500;700]$.  A
+sensor node will  not participate in the  next round if its  remaining energy is
+less than  $E_{R}=36~\mbox{Joules}$, the minimum  energy needed for the  node to
+stay alive  during one round.  This  value has been computed  by multiplying the
+energy consumed in  active state (9.72 mW)  by the time in second  for one round
+(3600 seconds).   According to the interval  of initial energy, a  sensor may be
+alive during at most 20 rounds.

 
 

-
-
- 

 \subsection{Metrics}
 
 \subsection{Metrics}
 

-We introduce the following performance metrics for evaluating our approach: 
+To evaluate our approach we consider the following performance metrics:

 
 

-\begin{enumerate}[i)]
+\begin{enumerate}[i]

   
   

-\item {{\bf Coverage Ratio (CR)}:} the coverage ratio measures how much the area of a sensor field is  covered. In our case, we treated the sensing fields as a grid, and used each grid point as a sample point
-for calculating the coverage. The coverage ratio can be calculated by:
+\item {{\bf Coverage Ratio (CR)}:} the coverage ratio measures how much of the area
+  of a sensor field is covered. In our case, the sensing field is represented as
+  a connected grid  of points and we use  each grid point as a  sample point to
+  compute the coverage. The coverage ratio can be calculated by:

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

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

 \end{equation*}
 \end{equation*}

-Where: $n^t$ is the number of covered grid points by the active sensors of all subregions during round $t$ in the current sensing phase and $N$ is total number of grid points in the sensing field of the network.
-%The accuracy of this method depends on the distance between grids. In our
-%simulations, the sensing field has been divided into 50 by 25 grid points, which means
-%there are $51 \times 26~ = ~ 1326$ points in total.
-% Therefore, for our simulations, the error in the coverage calculation is less than ~ 1 $\% $.
-
-\item{{\bf Number of Active Sensors Ratio(ASR)}:} It is important to have as few active nodes as possible in each round,
-in  order to  minimize  the communication  overhead  and maximize  the
-network lifetime. The Active Sensors Ratio is defined as follows:
+where $n^t$ is  the number of covered  grid points by the active  sensors of all
+subregions during round $t$ in the current sensing phase and $N$ is the total number
+of grid points  in the sensing field of  the network. In our simulations $N = 51
+\times 26 = 1326$ grid points.
+
+\item{{\bf Number  of Active Sensors Ratio  (ASR)}:} it is important  to have as
+  few  active  nodes  as  possible  in  each  round, in  order  to  minimize  the
+  communication overhead  and maximize the network lifetime.  The Active Sensors
+  Ratio is defined as follows:

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

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

 \end{equation*}
 \end{equation*}

-
-%\begin{equation*}
-%\scriptsize
-%\mbox{EC} =  \frac{\mbox{$\sum\limits_{d=1}^D E^c_d$}}{\mbox{$D$}} + \frac{\mbox{$\sum\limits_{d=1}^D %E^l_d$}}{\mbox{$D$}} + \frac{\mbox{$\sum\limits_{d=1}^D E^a_d$}}{\mbox{$D$}} + %\frac{\mbox{$\sum\limits_{d=1}^D E^s_d$}}{\mbox{$D$}}.
-%\end{equation*}
-
-Where: D is the number of rounds during $Lifetime95$ or $Lifetime50$. 
-The total energy consumed by the sensors (EC) comes through taking into consideration four main energy factors, which are $E^c_d$, $E^l_d$, $E^a_d$, $E^s_d$ and $E^p_d$.
-The energy consumption $E^c_d$ for wireless  communications  is  calculated by taking into account the  energy spent by  all the nodes while  transmitting and
-receiving  packets during round $d$. The $E^l_d$ represents the energy consumed by all the sensors during the listening mode before taking the decision to go Active or Sleep in round $d$. $E^a_d$ and $E^s_d$  refer to energy consumed in the active mode or in the sleeping mode. The $E^p_d$ refers to energy consumed by the computation (processing) to solve the integer program.
+where $A_r^t$ is the number of  active sensors in the subregion $r$ during round
+$t$ in the  current sensing phase, $|J|$  is the total number of  sensors in the
+network, and $R$ is the total number of subregions in the network.
+
+\item {{\bf Network Lifetime}:} we define the network lifetime as the time until
+  the  coverage  ratio  drops  below   a  predefined  threshold.  We  denote  by
+  $Lifetime_{95}$ (respectively  $Lifetime_{50}$) the amount of  time during
+  which  the  network   can  satisfy  an  area  coverage   greater  than  $95\%$
+  (respectively $50\%$). We assume that the network is alive until all nodes have
+  been   drained    of   their   energy   or   the    sensor   network   becomes
+  disconnected. Network connectivity is  important because an active sensor node
+  without connectivity towards a base  station cannot transmit information on an
+  event in the area that it monitors.
+
+\item {{\bf  Energy Consumption  (EC)}:} the average  energy consumption  can be
+  seen as the total energy consumed by the sensors during the $Lifetime_{95}$ or
+  $Lifetime_{50}$  divided  by the  number  of rounds.  EC  can  be computed  as
+  follows:
+
+  \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.
 

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

   
   

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

 
 \end{enumerate}
 
 
 \end{enumerate}
 

+\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} 

 
 

-%%%%%%%%%%%%%%%%%%%%%%%%VU JUSQU ICI**************************************************
+\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)
+
+\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.

 
 

-\section{Results and analysis}

 \subsection{Coverage ratio} 
 \subsection{Coverage ratio} 

-Figure~\ref{fig3} shows the average coverage ratio for 150 deployed nodes.  
-\parskip 0pt    
-\begin{figure}[h!]
+
+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.  This  is due to the fact
+that, in comparison with MuDiLCO which  uses optimization to put in SLEEP status
+redundant sensors,  more sensor  nodes remain  active with DESK  and GAF.   As a
+consequence,  when the  number  of rounds  increases, a  larger  number of  node
+failures can be observed in DESK and  GAF, resulting in a faster decrease of the
+coverage ratio.  Furthermore,  our protocol allows to maintain  a coverage ratio
+greater than  50\% for far more  rounds.  Overall, the proposed  sensor activity
+scheduling based on optimization in  MuDiLCO maintains higher coverage ratios of
+the area of interest  for a larger number of rounds. It  also means that MuDiLCO
+saves more energy,  with less dead nodes,  at most for several  rounds, and thus
+should  extend the  network lifetime.  \textcolor{blue}{MuDiLCO-7 seems  to have
+  most of the  time the best coverage  ratio up to round~80,  after that MuDiLCO-5 is
+  slightly better.}
+
+\begin{figure}[ht!]

 \centering
 \centering

- \includegraphics[scale=0.5] {R1/CR.pdf} 
-\caption{The coverage ratio for 150 deployed nodes}
+ \includegraphics[scale=0.5] {F/CR.pdf} 
+\caption{Average coverage ratio for 150 deployed nodes}

 \label{fig3}
 \end{figure} 
 
 \label{fig3}
 \end{figure} 
 

-DESK and GAF provide a very little better coverage ratio than MuDiLCO-T (in the first thirty rounds. This is due to the fact that MuDiLCO  put in sleep mode redundant sensors using optimization (which slightly decreases the coverage ratio) while there are more active nodes in the case of DESK and GAF. When the number of rounds increases, coverage ratio produced by DESK and GAF decreases. This is due to dead nodes. However,  MuDiLCO-T maintains the coverage ratio greater than 50$\%$ for a larger number of rounds in comparison with DESK and GAF. Although some nodes are dead, sensor activity scheduling based on optimization in MuDiLCO allows to prolong the coverage of the area of interest. The simulation results shows the superiority of our method, that  keeps high coverage for a larger number of rounds, so the network lifetime is extended.
+\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
+lifetime. Figure~\ref{fig4}  presents the active  sensor ratio for  150 deployed
+nodes all along the network lifetime. It appears that up to round thirteen, DESK
+and GAF have  respectively 37.6\% and 44.8\% of nodes  in ACTIVE status, whereas
+MuDiLCO clearly outperforms  them with only 24.8\% of  active nodes.  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.

 
 

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

 \centering
 \centering

-\includegraphics[scale=0.5]{R1/ASR.pdf}  
-\caption{The active sensors ratio for 150 deployed nodes }
+\includegraphics[scale=0.5]{F/ASR.pdf}  
+\caption{Active sensors ratio for 150 deployed nodes}

 \label{fig4}
 \end{figure} 
 
 \label{fig4}
 \end{figure} 
 

+\subsection{Stopped simulation runs}

 
 

-We can observe that DESK and GAF have 37.6 $\%$ and 44.8 $\%$ of active nodes and MuDiLCO-T competes perfectly with only 24.8$\%$  of active nodes for the first thirteen rounds.
-From the thirty fifth round, MuDiLCO-T has a larger number of active nodes in comparison with DESK and GAF but it maintains a higher level of coverage compared to the two other methods. DESK and GAF have less number of active nodes because many nodes are died.
-
+Figure~\ref{fig6} reports the cumulative  percentage of stopped simulations runs
+per round  for 150  deployed nodes.  This figure gives  the breakpoint  for each
+method.  DESK  stops first, after  approximately 45~rounds, because  it consumes
+the more  energy by  turning on  a large  number of  redundant nodes  during the
+sensing  phase. GAF  stops  secondly for  the  same reason  than  DESK.  Let  us
+emphasize that the simulation  continues as long as a network  in a subregion is
+still connected.

 
 

-\subsection{Stopped simulation runs}
-%The results presented in this experiment, is to show the comparison of our MuDiLCO protocol with other two approaches from the point of view the stopped simulation runs per round. Figure~\ref{fig6} illustrates the percentage of stopped simulation
-%runs per round for 150 deployed nodes. 
-Figure~\ref{fig6} reports the cumulative percentage of stopped simulations runs per round for 150 deployed nodes. This figure gives the breakpoint for each of the methods.
-\begin{figure}[h!]
+\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} 

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

 
 

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

 
 

-\subsection{Energy Consumption}
-We measure the energy consumed by the sensors during the communication, listening, computation, active, and sleep modes for different network densities and compare it with the two other methods. Figures~\ref{fig95} and ~\ref{fig7} illustrate the energy consumption for different network sizes for $Lifetime95$ and $Lifetime50$. 
+We  measure  the  energy  consumed  by the  sensors  during  the  communication,
+listening, computation, active, and sleep status for different network densities
+and   compare   it   with   the  two   other   methods.    Figures~\ref{fig7}(a)
+and~\ref{fig7}(b)  illustrate  the  energy  consumption,  considering  different
+network sizes, for $Lifetime_{95}$ and $Lifetime_{50}$.

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

-\centering
-\includegraphics[scale=0.5]{R1/EC95.pdf} 
-\caption{The Energy Consumption with $95\%-Lifetime$}
-\label{fig95}
+  \centering
+  \begin{tabular}{cl}
+    \parbox{9.5cm}{\includegraphics[scale=0.5]{F/EC95.pdf}} & (a) \\
+    \verb+ + \\
+    \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{figure} 
 

-The results show that MuDiLCO-T is the most competitive from the energy consumption point of view.  The other approaches have a  high energy  consumption  due  to  activating a larger number of redundant nodes as well as the energy consumed during the different modes of the sensor node.\\
- 
-As shown in Figures~\ref{fig95}\ref{fig7} and \ref{fig7}, MuDiLCO-7 consumes more energy than the other versions of MuDiLCO, especially for large sizes of network. This is easy to understand since the bigger the number of rounds and the number of sensors involved in the integer program, the larger the time computation to solve the optimization problem.  
-\begin{figure}[h!]
-\centering
-\includegraphics[scale=0.5]{R1/EC50.pdf} 
-\caption{The Energy Consumption with $Lifetime50$}
-\label{fig7}
-\end{figure} 
-
-
-
-%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  optimally solve the  integer program
+  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 $T$ on the computation time. Figure~\ref{fig77} gives the average execution times in seconds  (times to solve optimization problem) for different values of $T$. The original execution time is computed on a laptop DELL with intel Core i3 2370 M (2.4 GHz) processor (2 cores) and the MIPS (Million Instructions Per Second) rate equal to 35330. To be consistent with the use of a sensor node with Atmels AVR ATmega103L microcontroller (6 MHz) and a MIPS rate equal to 6 to run the optimization resolution, this time is multiplied by 2944.2 $\left( \frac{35330}{2} \times 6\right)$ and reported on Figure~\ref{fig77} for different network sizes.
+\label{et}
+We observe  the impact of the  network size and of  the number of rounds  on the
+computation  time.   Figure~\ref{fig77} gives  the  average  execution times  in
+seconds (needed to solve optimization problem)  for different values of $T$. The
+modeling language for Mathematical Programming (AMPL)~\cite{AMPL} is employed to
+generate the Mixed  Integer Linear Program instance in a  standard format, which
+is then read and solved by  the optimization solver GLPK (GNU linear Programming
+Kit  available in  the  public domain)  \cite{glpk}  through a  Branch-and-Bound
+method. The  original execution  time is  computed on a  laptop DELL  with Intel
+Core~i3~2370~M (2.4 GHz) processor (2  cores) and the MIPS (Million Instructions
+Per Second) rate equal to 35330. To be  consistent with the use of a sensor node
+with Atmels AVR ATmega103L microcontroller (6 MHz) and a MIPS rate equal to 6 to
+run  the optimization  resolution, this  time  is multiplied  by 2944.2  $\left(
+\frac{35330}{2} \times  \frac{1}{6} \right)$ and reported  on Figure~\ref{fig77}
+for different network sizes.

 
 

-Figure~\ref{fig77} shows that the execution time increases with the number of rounds $T$ taken into account for the scheduling of the sensing phase. MuDiLCO-7 results in execution time that quickly becomes unsuitable for a sensor network, especially when the sensor network size increases. 
-%While MuDiLCO-1, 3, and 5 solves the optimization process with suitable execution times to be used on wireless sensor network because it distributed on larger number of small subregions as well as it is used acceptable number of round(s) T.  We think that in distributed fashion the solving of the optimization problem to produce T rounds in a subregion can be tackled by sensor nodes. Overall, to be able to deal with very large networks, a distributed method is clearly required.
-
-\begin{figure}[h!]
+\begin{figure}[ht!]

 \centering
 \centering

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

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

-
-\subsection{Network Lifetime}
-In Figure~\ref{fig9} and in Figure~\ref{fig8}, network lifetime, $Lifetime95$ and $Lifetime50$ respectively, are illustrated for different  network sizes. 
-
-\begin{figure}[h!]
-\centering
-\includegraphics[scale=0.5]{R1/LT95.pdf}  
-\caption{The Network Lifetime for $Lifetime95$}
-\label{fig9}
-\end{figure} 
-
-
-As highlighted by Figure~\ref{fig9}, network lifetime obviously
-increases when the size of the network increases. MuDiLCO-T (whatever values of $T$) maximizes the lifetime of the network compared with other approaches. The gain in lifetime for a coverage over $95\%$ is greater than $38\%$ between GAF and MuDiLCO-3.
-
-% By choosing the best suited nodes, for each round, by optimizing the coverage and lifetime of the network to cover the area of interest with a maximum number rounds and by letting the other nodes sleep in order to be used later in next rounds, our MuDiLCO-T protocol efficiently prolonges the network lifetime. 
-
-%In Figure~\ref{fig8}, Comparison shows that our MuDiLCO-T protocol, which are used distributed optimization on the subregions with the ability of producing T rounds, is the best one because it is robust to network disconnection during the network lifetime as well as it consume less energy in comparison with other approaches. It also means that distributing the protocol in each sensor node and subdividing the sensing field into many subregions, which are managed independently and simultaneously, is the most relevant way to maximize the lifetime of a network.
-
-
-%We see that our MuDiLCO-7 protocol results in execution times that quickly become unsuitable for a sensor network as well as the energy consumption seems to be huge because it used a larger number of rounds T during performing the optimization decision in the subregions, which is led to decrease the network lifetime. On the other side, our MuDiLCO-1, 3, and 5 protocol seems to be more efficient in comparison with other approaches because they are prolonged the lifetime of the network more than DESK and GAF.
- 
-
-
-\begin{figure}[h!]
-\centering
-\includegraphics[scale=0.5]{R1/LT50.pdf}  
-\caption{The Network Lifetime for $Lifetime50$}
-\label{fig8}
+As expected,  the execution time increases  with the number of  rounds $T$ taken
+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, denoted by Unlimited-MuDiLCO-5, is also 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 $Lifetime_{95}$
+and  $Lifetime_{50}$.  Both  figures show  that the  network lifetime  increases
+together with the  number of sensor nodes, whatever the  protocol, thanks to the
+node  density  which results  in  more  and more  redundant  nodes  that can  be
+deactivated and thus save energy.  Compared to the other approaches, our MuDiLCO
+protocol  maximizes the  lifetime of  the network.   In particular  the gain  in
+lifetime for a coverage  over 95\%, and a network of  250~nodes, is greater than
+43\% 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 best  suited
+  method compared to MuDiLCO-7.}
+
+\begin{figure}[t!]
+  \centering
+  \begin{tabular}{cl}
+    \parbox{9.5cm}{\includegraphics[scale=0.5125]{F/LT95.pdf}} & (a) \\
+    \verb+ + \\
+    \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{figure} 
 

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

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

-In this paper, we have  addressed the problem of the coverage and the lifetime
-optimization in wireless  sensor networks. This is a key issue as
-sensor nodes have limited resources in terms of memory,  energy and
-computational power. To cope with this problem, the field of sensing
-is divided into smaller subregions using the concept of divide-and-conquer method, and then a MuDiLCO protocol optimizes  coverage and  lifetime performances in each subregion.
-The  proposed  protocol  combines  two efficient  techniques:  network
-leader election  and sensor activity scheduling.
-%,  where the challenges
-%include how to select the  most efficient leader in each subregion and
-%the best cover sets %of active nodes that will optimize the network lifetime
-%while taking the responsibility of covering the corresponding
-%subregion using more than one cover set during the sensing phase. 
-The activity scheduling in each subregion works in periods, each period consists  of four phases: (i) Information Exchange,
-(ii) Leader Election, (iii) Decision Phase
-to plan the activity of the sensors over $T$ rounds  (iv) Sensing Phase itself divided into T rounds. 
-
-Simulations results show the relevance  of the proposed MuDiLCO
-protocol in terms of lifetime, coverage ratio, active sensors ratio, energy consumption, execution time. Indeed, when
-dealing with large wireless sensor networks, a distributed
-approach like the one we propose allows to reduce the difficulty of a
-single global optimization problem by partitioning it in many smaller
-problems, one per subregion, that can be solved more easily. Nevertheless, results also show that it is not possible to plan the activity of sensors  over too many rounds because the resulting optimization problem leads to too high resolution time and thus to an excessive energy consumption.
+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
+approach, like the one  we propose, allows to reduce the  difficulty of a single
+global optimization problem by partitioning it in many smaller problems, one per
+subregion,  that  can be  solved  more  easily.  \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.
 % use section* for acknowledgement
 
 %In  future work, we plan  to study and propose adjustable sensing range coverage optimization protocol, which computes  all active sensor schedules in one time, by using
 %optimization  methods. This protocol can prolong the network lifetime by minimizing the number of the active sensor nodes near the borders by optimizing the sensing range of sensor nodes.
 % use section* for acknowledgement

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

 
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