X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/JournalMultiPeriods.git/blobdiff_plain/ed874f7124de4037b1abb5073c85c4bc3cc698cd..ae6793487a83754a54348f1eb0adc66723f59b98:/article.tex?ds=sidebyside

diff --git a/article.tex b/article.tex
index 7b6a536..1c13bc7 100644
--- a/article.tex
+++ b/article.tex
@@ -73,15 +73,22 @@
 %% \author[label1,label2]{}
 %% \address[label1]{}
 %% \address[label2]{}
-\author{Ali Kadhum Idrees, Karine Deschinkel, \\
-Michel Salomon, and Rapha\"el Couturier}
+%\author{Ali Kadhum Idrees, Karine Deschinkel, \\
+%Michel Salomon, and Rapha\"el Couturier}
+
 %\thanks{are members in the AND team - DISC department - FEMTO-ST Institute, University of Franche-Comt\'e, Belfort, France.
 % e-mail: ali.idness@edu.univ-fcomte.fr, $\lbrace$karine.deschinkel, michel.salomon, raphael.couturier$\rbrace$@univ-fcomte.fr.}% <-this % stops a space
 %\thanks{}% <-this % stops a space
  
-\address{FEMTO-ST Institute, University of Franche-Comt\'e, Belfort, France. \\ 
-e-mail: ali.idness@edu.univ-fcomte.fr, \\
-$\lbrace$karine.deschinkel, michel.salomon, raphael.couturier$\rbrace$@univ-fcomte.fr.}
+%\address{FEMTO-ST Institute, University of Franche-Comt\'e, Belfort, France. \\ 
+%e-mail: ali.idness@edu.univ-fcomte.fr, \\
+%$\lbrace$karine.deschinkel, michel.salomon, raphael.couturier$\rbrace$@univ-fcomte.fr.}
+
+\author{Ali   Kadhum   Idrees$^{a,b}$,   Karine  Deschinkel$^{a}$,   \\   Michel
+  Salomon$^{a}$,   and  Rapha\"el   Couturier   $^{a}$  \\   $^{a}${\em{FEMTO-ST
+      Institute,  UMR  6174  CNRS,   \\  University  Bourgogne  Franche-Comt\'e,
+      Belfort, France}} \\ $^{b}${\em{Department of Computer Science, University
+      of Babylon, Babylon, Iraq}} }
 
 \begin{abstract}
 %One of  the fundamental challenges in Wireless Sensor Networks (WSNs)
@@ -89,26 +96,33 @@ $\lbrace$karine.deschinkel, michel.salomon, raphael.couturier$\rbrace$@univ-fcom
 %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 Multiround Distributed Lifetime Coverage
+(WSNs). In this paper, a  method called Multiround Distributed Lifetime Coverage
 Optimization  protocol (MuDiLCO)  is proposed  to maintain  the coverage  and to
 improve the lifetime in wireless sensor  networks. The area of interest is first
-divided  into subregions and  then the  MuDiLCO protocol  is distributed  on the
-sensor nodes in each subregion. The proposed MuDiLCO protocol works in periods
-during which sets of sensor nodes are scheduled to remain active for a number of
-rounds  during the  sensing phase,  to  ensure coverage  so as  to maximize  the
-lifetime of  WSN.  The decision process is  carried out by a  leader node, which
-solves an  integer program to  produce the best  representative sets to  be used
-during the rounds  of the sensing phase. Compared  with some existing protocols,
-simulation  results based  on  multiple criteria  (energy consumption,  coverage
-ratio, and  so on) show that  the proposed protocol can  prolong efficiently the
-network lifetime and improve the coverage performance.
-
+divided into  subregions and  then the  MuDiLCO protocol  is distributed  on the
+sensor nodes in  each subregion. The proposed MuDiLCO protocol  works in periods
+during which sets of sensor nodes are  scheduled, with one set for each round of
+a period, to remain active during the  sensing phase and thus ensure coverage so
+as  to maximize  the  WSN lifetime.   \textcolor{blue}{The  decision process  is
+  carried out by a leader node,  which solves an optimization problem to produce
+  the  best representative  sets to  be used  during the  rounds of  the sensing
+  phase. The optimization problem formulated as  an integer program is solved to
+  optimality through a Branch-and-Bound method  for small instances.  For larger
+  instances, the best  feasible solution found by the solver  after a given time
+  limit threshold is considered.}
+%The decision process is  carried out by a  leader node, which
+%solves an  integer program to  produce the best  representative sets to  be used
+%during the rounds  of the sensing phase. 
+%\textcolor{red}{The integer program is solved by either GLPK solver or Genetic Algorithm (GA)}. 
+Compared  with some  existing protocols,  simulation results  based on  multiple
+criteria (energy consumption, coverage ratio, and  so on) show that the proposed
+protocol can prolong  efficiently the network lifetime and  improve the coverage
+performance.
 \end{abstract}
 
 \begin{keyword}
-Wireless   Sensor   Networks,   Area   Coverage,   Network   lifetime,
+Wireless   Sensor   Networks,   Area   Coverage,   Network   Lifetime,
 Optimization, Scheduling, Distributed Computation.
-
 \end{keyword}
 
 \end{frontmatter}
@@ -152,10 +166,10 @@ the network lifetime by using an optimized multiround scheduling.
 
 The remainder of the paper is organized as follows. The next section
 % Section~\ref{rw}
-reviews  the related works  in the  field.  Section~\ref{pd}  is devoted  to the
+reviews the  related works  in the  field.  Section~\ref{pd}  is devoted  to the
 description of MuDiLCO protocol.  Section~\ref{exp} shows the simulation results
 obtained using  the discrete event  simulator OMNeT++ \cite{varga}.   They fully
-demonstrate  the  usefulness  of   the  proposed  approach.   Finally,  we  give
+demonstrate  the  usefulness  of  the   proposed  approach.   Finally,  we  give
 concluding    remarks   and    some    suggestions   for    future   works    in
 Section~\ref{sec:conclusion}.
 
@@ -173,7 +187,7 @@ algorithms in WSNs according to several design choices:
 \item  Sensors   scheduling  algorithm  implementation,   i.e.   centralized  or
   distributed/localized algorithms.
 \item The objective of sensor coverage, i.e. to maximize the network lifetime or
-  to minimize the number of sensors during a sensing round.
+  to minimize the number of active sensors during a sensing round.
 \item The homogeneous or heterogeneous nature  of the nodes, in terms of sensing
   or communication capabilities.
 \item The node deployment method, which may be random or deterministic.
@@ -189,39 +203,47 @@ many cover sets) can be added to the above list.
 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
+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
+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. Moreover, centralized approaches usually
-suffer from the scalability problem, making them less competitive as the network
-size increases.
+information from  all the sensor  nodes.  \textcolor{blue} {Exact  or heuristics
+  approaches are designed to provide cover sets.
+%(Moreover, centralized approaches usually
+%suffer from the scalability problem, making them less competitive as the network
+%size increases.) 
+Contrary to exact methods, heuristic 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
+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
+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
+scheduling policies  are less  resilient and  reliable because  a sensor  may be
 involved in more than one cover sets.
 %For instance, the proposed work in ~\cite{cardei2005energy, berman04}    
 
-In~\cite{yang2014maximum},  the  authors have  considered  a linear  programming
+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
+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{castano2013column,rossi2012exact,deschinkel2012column}.
+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}
 %{\bf Distributed approaches}
@@ -278,15 +300,19 @@ 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
+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
+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
+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.
+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}.
 
 \iffalse
    
@@ -518,11 +544,69 @@ Zhou~\cite{Zhang05} proved that if  the transmission range fulfills the previous
 hypothesis, a complete coverage of  a convex area implies connectivity among the
 active nodes.
 
-Instead  of working  with a  continuous coverage  area, we  make it  discrete by
-considering for each sensor a set of points called primary points. Consequently,
-we assume  that the sensing disk  defined by a sensor  is covered if  all of its
-primary points are covered. The choice of number and locations of primary points
-is the subject of another study not presented here.
+%Instead  of working  with a  continuous coverage  area, we  make it  discrete by considering for each sensor a set of points called primary points. Consequently, we assume  that the sensing disk  defined by a sensor  is covered if  all of its primary points are covered. The choice of number and locations of primary points is the subject of another study not presented here.
+
+\indent Instead of working with the coverage area, we consider for each sensor a
+set of  points called  primary points~\cite{idrees2014coverage}. We  assume that
+the sensing  disk defined by a  sensor is covered  if all the primary  points of
+this  sensor are  covered.   By knowing  the position  of  wireless sensor  node
+(centered at  the the  position $\left(p_x,p_y\right)$)  and it's  sensing range
+$R_s$,  we define  up to  25 primary  points $X_1$  to $X_{25}$  as decribed  on
+Figure~\ref{fig1}. The optimal number of primary points is investigated in
+section~\ref{ch4:sec:04:06}.
+
+The coordinates of the primary points are defined as follows:\\
+%$(p_x,p_y)$ = point center of wireless sensor node\\  
+$X_1=(p_x,p_y)$ \\ 
+$X_2=( p_x + R_s * (1), p_y + R_s * (0) )$\\           
+$X_3=( p_x + R_s * (-1), p_y + R_s * (0)) $\\
+$X_4=( p_x + R_s * (0), p_y + R_s * (1) )$\\
+$X_5=( p_x + R_s * (0), p_y + R_s * (-1 )) $\\
+$X_6=( p_x + R_s * (\frac{-\sqrt{2}}{2}), p_y + R_s * (\frac{\sqrt{2}}{2})) $\\
+$X_7=( p_x + R_s * (\frac{\sqrt{2}}{2}), p_y + R_s * (\frac{\sqrt{2}}{2})) $\\
+$X_8=( p_x + R_s * (\frac{-\sqrt{2}}{2}), p_y + R_s * (\frac{-\sqrt{2}}{2})) $\\
+$X_9=( p_x + R_s * (\frac{\sqrt{2}}{2}), p_y + R_s * (\frac{-\sqrt{2}}{2})) $\\
+$X_{10}= ( p_x + R_s * (\frac{-\sqrt{2}}{2}), p_y + R_s * (0)) $\\
+$X_{11}=( p_x + R_s *  (\frac{\sqrt{2}}{2}), p_y + R_s * (0))$\\
+$X_{12}=( p_x + R_s * (0), p_y + R_s * (\frac{\sqrt{2}}{2})) $\\
+$X_{13}=( p_x + R_s * (0), p_y + R_s * (\frac{-\sqrt{2}}{2})) $\\
+$X_{14}=( p_x + R_s * (\frac{\sqrt{3}}{2}), p_y + R_s * (\frac{1}{2})) $\\
+$X_{15}=( p_x + R_s * (\frac{-\sqrt{3}}{2}), p_y + R_s * (\frac{1}{2})) $\\
+$X_{16}=( p_x + R_s * (\frac{\sqrt{3}}{2}), p_y + R_s * (\frac{- 1}{2})) $\\
+$X_{17}=( p_x + R_s * (\frac{-\sqrt{3}}{2}), p_y + R_s * (\frac{- 1}{2})) $\\
+$X_{18}=( p_x + R_s * (\frac{\sqrt{3}}{2}), p_y + R_s * (0)) $\\
+$X_{19}=( p_x + R_s * (\frac{-\sqrt{3}}{2}), p_y + R_s * (0)) $\\
+$X_{20}=( p_x + R_s * (0), p_y + R_s * (\frac{1}{2})) $\\
+$X_{21}=( p_x + R_s * (0), p_y + R_s * (-\frac{1}{2})) $\\
+$X_{22}=( p_x + R_s * (\frac{1}{2}), p_y + R_s * (\frac{\sqrt{3}}{2})) $\\
+$X_{23}=( p_x + R_s * (\frac{- 1}{2}), p_y + R_s * (\frac{\sqrt{3}}{2})) $\\
+$X_{24}=( p_x + R_s * (\frac{- 1}{2}), p_y + R_s * (\frac{-\sqrt{3}}{2})) $\\
+$X_{25}=( p_x + R_s * (\frac{1}{2}), p_y + R_s * (\frac{-\sqrt{3}}{2})) $.
+
+
+%\begin{figure} %[h!]
+%\centering
+% \begin{multicols}{2}
+%\centering
+%\includegraphics[scale=0.28]{fig21.pdf}\\~ (a)
+%\includegraphics[scale=0.28]{principles13.pdf}\\~(c) 
+%\hfill \hfill
+%\includegraphics[scale=0.28]{fig25.pdf}\\~(e)
+%\includegraphics[scale=0.28]{fig22.pdf}\\~(b)
+%\hfill \hfill
+%\includegraphics[scale=0.28]{fig24.pdf}\\~(d)
+%\includegraphics[scale=0.28]{fig26.pdf}\\~(f)
+%\end{multicols} 
+%\caption{Wireless Sensor Node represented by (a) 5, (b) 9, (c) 13, (d) 17, (e) 21 and (f) 25 primary points respectively}
+%\label{fig1}
+%\end{figure}
+    
+\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}
 
 %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
@@ -539,19 +623,29 @@ is the subject of another study not presented here.
 
 \subsection{Background idea}
 %%RC : we need to clarify the difference between round and period. Currently it seems to be the same (for me at least).
-The area of  interest can be divided using  the divide-and-conquer strategy into
-smaller  areas,  called  subregions,  and  then our MuDiLCO  protocol will be
-implemented in each subregion in a distributed way.
-
-As  can be seen  in Figure~\ref{fig2},  our protocol  works in  periods fashion,
-where  each is  divided  into 4  phases: Information~Exchange,  Leader~Election,
-Decision, and Sensing.  Each sensing phase may be itself divided into $T$ rounds
-and for each round a set of sensors (a cover set) is responsible for the sensing
-task. In  this way  a multiround optimization  process is performed  during each
-period  after  Information~Exchange  and  Leader~Election phases,  in  order  to
-produce $T$ cover sets that will take the mission of sensing for $T$ rounds.
-\begin{figure}[ht!]
-\centering \includegraphics[width=100mm]{Modelgeneral.pdf} % 70mm
+%The area of  interest can be divided using  the divide-and-conquer strategy into
+%smaller  areas,  called  subregions,  and  then our MuDiLCO  protocol will be
+%implemented in each subregion in a distributed way.
+
+\textcolor{blue}{The WSN  area of  interest is,  in a  first step,  divided into
+  regular  homogeneous subregions  using  a divide-and-conquer  algorithm. In  a
+  second  step our  protocol  will be  executed  in a  distributed  way in  each
+  subregion  simultaneously  to  schedule  nodes'  activities  for  one  sensing
+  period. Sensor nodes are assumed to be deployed almost uniformly and with high
+  density over the region. The regular  subdivision is made such that the number
+  of hops between any pairs of sensors  inside a subregion is less than or equal
+  to 3.}
+
+As can  be seen  in Figure~\ref{fig2},  our protocol  works in  periods fashion,
+where   each   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} 
@@ -561,15 +655,19 @@ produce $T$ cover sets that will take the mission of sensing for $T$ 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 minimizes the impact of unexpected node failure (not due to batteries
-running out of energy), because it works in periods. 
+This  protocol minimizes  the  impact of  unexpected node  failure  (not due  to
+batteries running out of energy), because it works in periods.
 %This protocol is reliable against an unexpected node failure, because it works in periods. 
 %%RC : why? I am not convinced
- On the one hand, if a node failure is detected before  making the
-decision, the node will not participate to this phase, and, on the other hand,
-if the node failure occurs after the decision, the sensing  task of the network
-will be temporarily affected:  only during  the period of sensing until a new
-period starts.
+ 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   are   predefined
+   parameters. Round duration  should be long enough to hide  the system control
+   overhead and  short enough to minimize  the negative effects in  case of node
+   failure.}
+
 %%RC so if there are at least one failure per period, the coverage is bad...
 %%MS if we want to be reliable against many node failures we need to have an
 %% overcoverage...  
@@ -620,16 +718,16 @@ 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
+This step  consists in choosing  the Wireless  Sensor Node Leader  (WSNL), which
 will be responsible for executing the coverage algorithm.  Each subregion in the
 area of  interest will select its  own WSNL independently for  each period.  All
-the sensor  nodes cooperate to  elect a WSNL.   The nodes in the  same subregion
-will select the  leader based on the received information  from all other nodes
-in  the same subregion.   The selection  criteria are,  in order  of importance:
-larger  number  of neighbors,  larger  remaining energy,  and  then  in case  of
-equality, larger index. Observations on  previous simulations suggest to use the
-number  of  one-hop  neighbors  as   the  primary  criterion  to  reduce  energy
-consumption due to the communications.
+the sensor  nodes cooperate to  elect a WSNL.  The  nodes in the  same subregion
+will select the leader based on the received information from all other nodes in
+the same subregion.  The selection criteria  are, in order of importance: larger
+number of  neighbors, larger  remaining energy,  and then  in case  of equality,
+larger index. Observations on previous simulations  suggest to use the number of
+one-hop neighbors as  the primary criterion to reduce energy  consumption due to
+the communications.
 
 %the more priority selection factor is the number of $1-hop$ neighbors, $NBR j$, which can  minimize the energy consumption during the communication Significantly.  
 %The pseudo-code for leader election phase is provided in Algorithm~1.
@@ -638,20 +736,38 @@ consumption due to the communications.
 
 \subsection{Decision phase}
 
-Each  WSNL will solve  an integer  program to  select which  cover sets  will be
-activated in  the following  sensing phase  to cover the  subregion to  which it
-belongs.  The integer  program will produce $T$ cover sets,  one for each round.
-The WSNL will send an Active-Sleep  packet to each sensor in the subregion based
-on the algorithm's results, indicating if  the sensor should be active or not in
-each round  of the  sensing phase.  The  integer program  is based on  the model
-proposed by  \cite{pedraza2006} with some modifications, where  the objective is
-to find  a maximum  number of disjoint  cover sets.   To fulfill this  goal, the
-authors proposed an integer  program which forces undercoverage and overcoverage
+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.}
+%Each  WSNL will \textcolor{red}{ execute an optimization algorithm (see section \ref{oa})} to  select which  cover sets  will be
+%activated in  the following  sensing phase  to cover the  subregion to  which it
+%belongs.  The \textcolor{red}{optimization algorithm} will produce $T$ cover sets,  one for each round. The WSNL will send an Active-Sleep  packet to each sensor in the subregion based on the algorithm's results, indicating if  the sensor should be active or not in
+%each round  of the  sensing phase.  
+
+
+%solve  an integer  program
+
+
+
+
+
+
+
+%\section{\textcolor{red}{ Optimization Algorithm for Multiround Lifetime Coverage Optimization}}
+%\label{oa}
+As shown in Algorithm~\ref{alg:MuDiLCO}, the leader will execute an optimization
+algorithm based on an integer program. The integer program is based on the model
+proposed by \cite{pedraza2006}  with some modifications, where  the objective is
+to find  a maximum  number of disjoint  cover sets.  To  fulfill this  goal, the
+authors proposed an integer program  which forces undercoverage and overcoverage
 of  targets to  become minimal  at  the same  time.  They  use binary  variables
 $x_{jl}$ to indicate if  sensor $j$ belongs to cover set $l$.   In our model, we
-consider binary  variables $X_{t,j}$ to determine the  possibility of activating
-sensor $j$ during round $t$ of  a given sensing phase.  We also consider primary
-points as targets.  The  set of primary points is denoted by  $P$ and the set of
+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.
 
@@ -703,7 +819,7 @@ U_{t,p} = \left \{
 
 Our coverage optimization problem can then be formulated as follows:
 \begin{equation}
- \min \sum_{t=1}^{T} \sum_{p=1}^{P} \left(W_{\theta}* \Theta_{t,p} + W_{U} * U_{t,p}  \right)  \label{eq15} 
+ \min \sum_{t=1}^{T} \sum_{p=1}^{|P|} \left(W_{\theta}* \Theta_{t,p} + W_{U} * U_{t,p}  \right)  \label{eq15} 
 \end{equation}
 
 Subject to
@@ -712,7 +828,7 @@ Subject to
 \end{equation}
 
 \begin{equation}
-  \sum_{t=1}^{T}  X_{t,j}   \leq  \floor*{RE_{j}/E_{R}} \hspace{6 mm} \forall j \in J, t = 1,\dots,T
+  \sum_{t=1}^{T}  X_{t,j}   \leq  \floor*{RE_{j}/E_{R}} \hspace{10 mm}\forall j \in J\hspace{6 mm} 
   \label{eq144} 
 \end{equation}
 
@@ -746,29 +862,37 @@ U_{t,p} \in \lbrace0,1\rbrace, \hspace{10 mm}\forall p \in P, t = 1,\dots,T  \la
 
 The first group  of constraints indicates that some primary  point $p$ should be
 covered by at least  one sensor and, if it is not  always the case, overcoverage
-and undercoverage  variables help balancing the restriction  equations by taking
+and undercoverage variables  help balancing the restriction  equations by taking
 positive values. The constraint  given by equation~(\ref{eq144}) guarantees that
 the sensor has enough energy ($RE_j$  corresponds to its remaining energy) to be
 alive during  the selected rounds knowing  that $E_{R}$ is the  amount of energy
 required to be alive during one round.
 
-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. 
+There are  two main  objectives.  First,  we limit  the overcoverage  of primary
+points in order to activate a minimum  number of sensors.  Second we prevent the
+absence  of  monitoring  on  some  parts of  the  subregion  by  minimizing  the
+undercoverage.  The weights  $W_\theta$ and $W_U$ must be properly  chosen so as
+to guarantee that the maximum number of points are covered during each round.
 %% MS W_theta is smaller than W_u => problem with the following sentence
-In our simulations priority is given  to the coverage by choosing $W_{U}$ very
+In our simulations,  priority is given to the coverage  by choosing $W_{U}$ very
 large compared to $W_{\theta}$.
-%The Active-Sleep packet includes the schedule vector with the number of rounds that should be applied by the receiving sensor node during the sensing phase.
+
+\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})).}
+%The Active-Sleep packet includes the schedule vector with the number of rounds that should be applied by the receiving sensor node during the sensing phase
 
 \subsection{Sensing phase}
 
 The sensing phase consists of $T$ rounds. Each sensor node in the subregion will
 receive an Active-Sleep packet from WSNL, informing it to stay awake or to go to
-sleep for  each round of the sensing  phase.  Algorithm~\ref{alg:MuDiLCO}, which
-will be  executed by each node  at the beginning  of a period, explains  how the
-Active-Sleep packet is obtained.
+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.
 
 % 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. 
 
@@ -792,7 +916,7 @@ Active-Sleep packet is obtained.
         \emph{$\left\{\left(X_{1,k},\dots,X_{T,k}\right)\right\}_{k \in J}$ =
           Execute Integer Program Algorithm($T,J$)}\;
         \emph{$s_j.status$ = COMMUNICATION}\;
-        \emph{Send $ActiveSleep()$ to each node $k$ in subregion a packet \\
+        \emph{Send $ActiveSleep()$ packet to each node $k$ in subregion: a packet \\
           with vector of activity scheduling $(X_{1,k},\dots,X_{T,k})$}\;
         \emph{Update $RE_j $}\;
       }	  
@@ -812,22 +936,190 @@ Active-Sleep packet is obtained.
 
 \end{algorithm}
 
+\iffalse
+\textcolor{red}{This integer program can be solved using two approaches:}
+
+\subsection{\textcolor{red}{Optimization solver for Multiround Lifetime Coverage Optimization}}
+\label{glpk}
+\textcolor{red}{The modeling language for Mathematical Programming (AMPL)~\cite{AMPL} is  employed to generate the integer program instance  in a  standard format, which  is then read  and solved  by the optimization solver  GLPK (GNU  linear Programming Kit  available in  the public domain) \cite{glpk} through a Branch-and-Bound method. We named the protocol which is based on GLPK solver in the decision phase as MuDiLCO.}
+\fi
+
+\iffalse
+
+\subsection{\textcolor{red}{Genetic Algorithm for Multiround Lifetime Coverage Optimization}}
+\label{GA}
+\textcolor{red}{Metaheuristics  are a generic search strategies for exploring search spaces for solving the complex problems. These strategies have to dynamically balance between the exploitation of the accumulated search experience and the exploration of the search space. On one hand, this balance can find regions in the search space with high-quality solutions. On the other hand, it prevents waste too much time in regions of the search space which are either already explored or donât provide high-quality solutions. Therefore,  metaheuristic provides an enough good solution to an optimization problem, especially with incomplete  information or limited computation capacity \cite{bianchi2009survey}. Genetic Algorithm (GA) is one of the population-based metaheuristic methods that simulates the process of natural selection \cite{hassanien2015applications}.  GA starts with a population of random candidate solutions (called individuals or phenotypes) . GA uses genetic operators inspired by natural evolution, such as selection, mutation, evaluation, crossover, and replacement so as to improve the initial population of candidate solutions. This process repeated until a stopping criterion is satisfied. In comparison with GLPK optimization solver, GA provides a near optimal solution with acceptable execution time, as well as it requires a less amount of memory especially for large size problems. GLPK provides optimal solution, but it requires higher execution time and amount of memory for large problem.}
+
+\textcolor{red}{In this section, we present a metaheuristic based GA to solve our multiround lifetime coverage optimization problem. The proposed GA provides a near optimal sechedule for multiround sensing per period. The proposed GA is based on the mathematical model which is presented in Section \ref{oa}. Algorithm \ref{alg:GA} shows the proposed GA to solve the coverage lifetime optimization problem. We named the new protocol which is based on GA in the decision phase as GA-MuDiLCO. The proposed GA can be explained in more details as follow:}
+
+\begin{algorithm}[h!]    
+       
+ \small
+ \SetKwInput{Input}{\textcolor{red}{Input}}
+ \SetKwInput{Output}{\textcolor{red}{Output}}
+ \Input{ \textcolor{red}{$ P, J, T, S_{pop}, \alpha_{j,p}^{ind}, X_{t,j}^{ind}, \Theta_{t,p}^{ind}, U_{t,p}^{ind}, Child_{t,j}^{ind}, Ch.\Theta_{t,p}^{ind}, Ch.U_{t,p}^{ind_1}$}}
+ \Output{\textcolor{red}{$\left\{\left(X_{1,1},\dots, X_{t,j}, \dots, X_{T,J}\right)\right\}_{t \in T, j \in J}$}}
+
+  \BlankLine
+  %\emph{Initialize the sensor node and determine it's position and subregion} \; 
+  \ForEach {\textcolor{red}{Individual $ind$ $\in$ $S_{pop}$}} {
+     \emph{\textcolor{red}{Generate Randomly Chromosome $\left\{\left(X_{1,1},\dots, X_{t,j}, \dots, X_{T,J}\right)\right\}_{t \in T, j \in J}$}}\;
+     
+     \emph{\textcolor{red}{Update O-U-Coverage $\left\{(P, J, \alpha_{j,p}^{ind}, X_{t,j}^{ind}, \Theta_{t,p}^{ind}, U_{t,p}^{ind})\right\}_{p \in P}$}}\;
+     
+  
+     \emph{\textcolor{red}{Evaluate Individual $(P, J, X_{t,j}^{ind}, \Theta_{t,p}^{ind}, U_{t,p}^{ind})$}}\;  
+  }
+  
+  \While{\textcolor{red}{ Stopping criteria is not satisfied} }{
+  
+  \emph{\textcolor{red}{Selection $(ind_1, ind_2)$}}\;
+    \emph{\textcolor{red}{Crossover $(P_c, X_{t,j}^{ind_1}, X_{t,j}^{ind_2}, Child_{t,j}^{ind_1}, Child_{t,j}^{ind_2})$}}\;
+    \emph{\textcolor{red}{Mutation $(P_m, Child_{t,j}^{ind_1}, Child_{t,j}^{ind_2})$}}\;
+   
+   
+   \emph{\textcolor{red}{Update O-U-Coverage $(P, J, \alpha_{j,p}^{ind}, Child_{t,j}^{ind_1}, Ch.\Theta_{t,p}^{ind_1}, Ch.U_{t,p}^{ind_1})$}}\;
+  \emph{\textcolor{red}{Update O-U-Coverage $(P, J, \alpha_{j,p}^{ind}, Child_{t,j}^{ind_2}, Ch.\Theta_{t,p}^{ind_2}, Ch.U_{t,p}^{ind_2})$}}\;  
+ 
+\emph{\textcolor{red}{Evaluate New Individual$(P, J, Child_{t,j}^{ind_1}, Ch.\Theta_{t,p}^{ind_1}, Ch.U_{t,p}^{ind_1})$}}\;  
+ \emph{\textcolor{red}{Replacement $(P, J, T, Child_{t,j}^{ind_1}, Ch.\Theta_{t,p}^{ind_1}, Ch.U_{t,p}^{ind_1}, X_{t,j}^{ind}, \Theta_{t,p}^{ind}, U_{t,p}^{ind}  )$ }}\;
+ 
+ \emph{\textcolor{red}{Evaluate New Individual$(P, J, Child_{t,j}^{ind_2}, Ch.\Theta_{t,p}^{ind_2}, Ch.U_{t,p}^{ind_2})$}}\;  
+  
+ \emph{\textcolor{red}{Replacement $(P, J, T, Child_{t,j}^{ind_2}, Ch.\Theta_{t,p}^{ind_2}, Ch.U_{t,p}^{ind_2}, X_{t,j}^{ind}, \Theta_{t,p}^{ind}, U_{t,p}^{ind}  )$ }}\;
+  
+      
+  }
+  \emph{\textcolor{red}{$\left\{\left(X_{1,1},\dots,X_{t,j},\dots,X_{T,J}\right)\right\}$ =
+            Select Best Solution ($S_{pop}$)}}\;
+ \emph{\textcolor{red}{return X}} \;
+\caption{\textcolor{red}{GA($T, J$)}}
+\label{alg:GA}
+
+\end{algorithm}
+
+
+\begin{enumerate} [I)]
+
+\item \textcolor{red}{\textbf{Representation:} Since the proposed GA's goal is to find the optimal schedule of the sensor nodes which take the responsibility of monitoring the subregion for $T$ rounds in the sensing phase, the chromosome is defined as a schedule for alive  sensors and each chromosome contains $T$ rounds. The proposed GA uses binary representation, where each round in the schedule includes J genes, the total alive sensors in the subregion. Therefore, the gene of such a chromosome is a schedule of a sensor. In other words, The genes corresponding to active nodes have the value of one, the others are zero. Figure \ref{chromo} shows solution representation in the proposed GA.}
+%[scale=0.3]
+\begin{figure}[h!]
+\centering
+ \includegraphics [scale=0.35] {rep.pdf} 
+\caption{Candidate Solution representation by the proposed GA. }
+\label{chromo}
+\end{figure} 
+
+
+
+\item \textcolor{red}{\textbf{Initialize Population:} The initial population is randomly generated and each chromosome  in the GA population represents a possible sensors schedule solution to cover the entire subregion for $T$ rounds during current period. Each sensor in the chromosome is given a random value (0 or  1) for all rounds. If the random value is 1, the remaining  energy of this sensor should be adequate to activate this sensor during the current round. Otherwise, the value is set to 0. The energy constraint is applied for each sensor during all rounds. }
+
+
+\item \textcolor{red}{\textbf{Update O-U-Coverage:} 
+After creating the initial population, The overcoverage $\Theta_{t,p}$ and undercoverage $U_{t,p}$ for each candidate solution are computed (see Algorithm \ref{OU}) so as to use them in the next step.}
+
+\begin{algorithm}[h!]                
+  
+ \SetKwInput{Input}{\textcolor{red}{Input}}
+ \SetKwInput{Output}{\textcolor{red}{Output}}
+ \Input{ \textcolor{red}{parameters $P, J, ind, \alpha_{j,p}^{ind}, X_{t,j}^{ind}$}}
+ \Output{\textcolor{red}{$U^{ind} = \left\lbrace U_{1,1}^{ind}, \dots, U_{t,p}^{ind}, \dots, U_{T,P}^{ind} \right\rbrace$ and $\Theta^{ind} = \left\lbrace \Theta_{1,1}^{ind}, \dots, \Theta_{t,p}^{ind}, \dots, \Theta_{T,P}^{ind} \right\rbrace$}}
+
+  \BlankLine
+
+  \For{\textcolor{red}{$t\leftarrow 1$ \KwTo $T$}}{
+  \For{\textcolor{red}{$p\leftarrow 1$ \KwTo $P$}}{
+     
+ %    \For{$i\leftarrow 0$ \KwTo $I_j$}{
+     	\emph{\textcolor{red}{$SUM\leftarrow 0$}}\;
+         \For{\textcolor{red}{$j\leftarrow 1$ \KwTo $J$}}{
+              \emph{\textcolor{red}{$SUM \leftarrow SUM + (\alpha_{j,p}^{ind} \times X_{t,j}^{ind})$ }}\;
+         }
+         
+         \If { \textcolor{red}{SUM = 0}} {
+         \emph{\textcolor{red}{$U_{t,p}^{ind} \leftarrow 0$}}\;
+         \emph{\textcolor{red}{$\Theta_{t,p}^{ind} \leftarrow 1$}}\;
+         }
+         \Else{
+         \emph{\textcolor{red}{$U_{t,p}^{ind} \leftarrow SUM -1$}}\;
+         \emph{\textcolor{red}{$\Theta_{t,p}^{ind} \leftarrow 0$}}\;
+         }
+     
+     }
+     
+  }
+\emph{\textcolor{red}{return $U^{ind}, \Theta^{ind}$ }} \;
+\caption{O-U-Coverage}
+\label{OU}
+
+\end{algorithm}
+
+
+
+\item \textcolor{red}{\textbf{Evaluate Population:}
+After creating the initial population, each individual is evaluated and assigned a fitness value according to the fitness function is illustrated in Eq. \eqref{eqf}. In the proposed GA, the optimal (or near optimal) candidate solution, is the one with the minimum value for the fitness function. The lower the fitness values been assigned to an individual, the better opportunity it gets survived.  In our works, the function rewards  the decrease in the sensor nodes which cover the same primary point and penalizes the decrease to zero in the sensor nodes which cover the primary point. }
+
+\begin{equation}
+ F^{ind} \leftarrow  \sum_{t=1}^{T} \sum_{p=1}^{P} \left(W_{\theta}* \Theta_{t,p} + W_{U} * U_{t,p}  \right)    \label{eqf} 
+\end{equation}
+
+
+\item \textcolor{red}{\textbf{Selection:} In order to generate a new generation, a portion of the existing population is elected based on a fitness function that ranks the fitness of each candidate solution and preferentially select the best solutions. Two parents should be selected to the mating pool.  In the proposed GA-MuDiLCO algorithm, the first parent is selected by using binary tournament selection to select one of the parents \cite{goldberg1991comparative}. In this method,  two individuals are chosen at random from the population and the better of the two
+individuals is selected. If they have similar fitness values, one of them will be selected randomly. The best individual in the population is selected as a second parent.}
+
+
+
+\item \textcolor{red}{\textbf{Crossover:} Crossover is a genetic operator used to take more than one parent solutions and produce a child solution from them. If crossover probability $P_c$ is 100$\%$, then the crossover operation takes place between two individuals. If it is 0$\%$, the  two selected individuals in the mating pool will be the new chromosomes without crossover. In the proposed GA, a two-point crossover is used. Figure \ref{cross} gives an example for a two-point crossover for 8 sensors in the subregion and the schedule for 3 rounds.}
+
+
+\begin{figure}[h!]
+\centering
+ \includegraphics [scale = 0.3] {crossover.pdf} 
+\caption{Two-point crossover. }
+\label{cross}
+\end{figure} 
+
+
+\item \textcolor{red}{\textbf{Mutation:}
+Mutation is a divergence operation which introduces random modifications.  The purpose of the mutation is to maintain diversity within the population and prevent premature convergence. Mutation is used to add new genetic information (divergence) in order to achieve a global search over the solution search space and avoid to fall in local optima. The mutation operator in the proposed GA-MuDiLCO works as follow: If mutation probability $P_m$ is 100$\%$, then the mutation operation takes place on the new individual. The round number is selected randomly within (1..T) in the schedule solution. After that one sensor within this round is selected randomly within (1..J). If the sensor is scheduled as active "1", it should be rescheduled to sleep "0". If the sensor is scheduled as sleep, it rescheduled to active only if it has adequate remaining energy.}
+
+
+\item \textcolor{red}{\textbf{Update O-U-Coverage for children:}
+Before evaluating each new individual, Algorithm \ref{OU} is called for each new individual to compute the new undercoverage $Ch.U$ and overcoverage $Ch.\Theta$ parameters. }
+ 
+\item \textcolor{red}{\textbf{Evaluate New Individuals:}
+Each new individual is evaluated using Eq. \ref{eqf} but with using the new undercoverage $Ch.U$ and overcoverage $Ch.\Theta$ parameters of the new children.}
+
+\item \textcolor{red}{\textbf{Replacement:}
+After evaluation of new children, Triple Tournament Replacement (TTR) will be applied for each new individual. In TTR strategy, three individuals are selected
+randomly from the population. Find the worst from them and then check its fitness with the new individual fitness. If the fitness of the new individual is better than the fitness of  the worst individual, replace the new individual with the worst individual. Otherwise, the replacement is not done. }
+
+ 
+\item \textcolor{red}{\textbf{Stopping criteria:}
+The proposed GA-MuDiLCO stops when the stopping criteria is met. It stops after running for an amount of time in seconds equal to \textbf{Time limit}. The \textbf{Time limit} is the execution time obtained by the optimization solver GLPK for solving the same size of problem. The best solution will be selected as a schedule of sensors for $T$ rounds during the sensing phase in the current period.}
+
+
+
+\end{enumerate} 
+
+\fi
+
+%% EXPERIMENTAL STUDY
+
 \section{Experimental study}
 \label{exp}
 \subsection{Simulation setup}
 
-We  conducted  a  series of  simulations  to  evaluate  the efficiency  and  the
-relevance  of   our  approach,  using  the  discrete   event  simulator  OMNeT++
-\cite{varga}.     The     simulation     parameters    are     summarized     in
-Table~\ref{table3}.  Each experiment  for  a network  is  run over  25~different
-random topologies and  the results presented hereafter are  the average of these
-25 runs.
+We  conducted  a series  of  simulations  to  evaluate  the efficiency  and  the
+relevance  of  our   approach,  using  the  discrete   event  simulator  OMNeT++
+\cite{varga}.  The  simulation parameters are summarized  in Table~\ref{table3}.
+Each experiment for a network is run over 25~different random topologies and the
+results presented hereafter are the average of these 25 runs.
 %Based on the results of our proposed work in~\cite{idrees2014coverage}, we found as the region of interest are divided into larger subregions as the network lifetime increased. In this simulation, the network are divided into 16 subregions. 
 We  performed  simulations for  five  different  densities  varying from  50  to
-250~nodes deployed  over  a  $50 \times  25~m^2  $  sensing field.  More
-precisely, the  deployment is controlled  at a coarse  scale in order  to ensure
-that  the deployed  nodes can  cover the  sensing field  with the  given sensing
-range.
+250~nodes deployed  over a $50 \times  25~m^2 $ sensing field.   More precisely,
+the deployment  is controlled  at a  coarse scale  in order  to ensure  that the
+deployed nodes can cover the sensing field with the given sensing range.
 
 %%RC these parameters are realistic?
 %% maybe we can increase the field and sensing range. 5mfor Rs it seems very small... what do the other good papers consider ?
@@ -859,24 +1151,67 @@ Sensing time for one round & 60 Minutes \\
 $E_{R}$ & 36 Joules\\
 $R_s$ & 5~m   \\     
 %\hline
-$W_{\Theta}$ & 1   \\
+$W_{\theta}$ & 1   \\
 % [1ex] adds vertical space
 %\hline
-$W_{U}$ & $|P|^2$
+$W_{U}$ & $|P|^2$ \\
+%$P_c$ & 0.95   \\ 
+%$P_m$ & 0.6 \\
+%$S_{pop}$ & 50
 %inserts single line
 \end{tabular}
 \label{table3}
 % is used to refer this table in the text
 \end{table}
-  
-Our protocol  is declined into  four versions: MuDiLCO-1,  MuDiLCO-3, MuDiLCO-5,
-and  MuDiLCO-7, corresponding  respectively to  $T=1,3,5,7$ ($T$  the  number of
-rounds in one sensing period).  In  the following, we will make comparisons with
-two other methods. The first method, called DESK and proposed by \cite{ChinhVu},
-is  a   full  distributed  coverage   algorithm.   The  second   method,  called
-GAF~\cite{xu2001geography}, consists in dividing  the region into fixed squares.
-During the decision  phase, in each square, one sensor is  then chosen to remain
-active during the sensing phase time.
+
+\textcolor{blue}{Our  protocol  is  declined   into  four  versions:  MuDiLCO-1,
+  MuDiLCO-3, MuDiLCO-5, and MuDiLCO-7, corresponding respectively to $T=1,3,5,7$
+  ($T$ the  number of rounds in  one sensing period). Since  the time resolution
+  may  be prohibitive  when the  size  of the  problem increases,  a time  limit
+  threshold has  been fixed when  solving large  instances. In these  cases, the
+  solver returns  the best solution  found, which  is not necessary  the optimal
+  one. In practice, we only set time  limit values for the three largest network
+  sizes when $T=7$, using the following  respective values (in second): 0.03 for
+  150~nodes, 0.06 for 200~nodes, and 0.08 for 250~nodes.
+% Table \ref{tl} shows time limit values.
+  These time limit threshold have been  set empirically. The basic idea consists
+  in considering  the average execution  time to  solve the integer  programs to
+  optimality, then by  dividing this average time by three  to set the threshold
+  value.  After that,  this threshold value is increased if  necessary such that
+  the solver is able  to deliver a feasible solution within  the time limit.  In
+  fact, selecting the optimal values for the time limits will be investigated in
+  future.}
+%In Table \ref{tl},  "NO" indicates  that  the  problem has  been  solved to  optimality without time limit.}
+
+%\begin{table}[ht]
+%\caption{Time limit values for MuDiLCO protocol versions }
+%\centering
+%\begin{tabular}{|c|c|c|c|c|}
+% \hline
+% WSN size & MuDiLCO-1 & MuDiLCO-3 & MuDiLCO-5 & MuDiLCO-7 \\ [0.5ex]
+%\hline
+% 50 & NO & NO & NO & NO \\
+% \hline
+%100 & NO & NO & NO & NO \\
+%\hline
+%150 & NO & NO & NO & 0.03 \\
+%\hline
+%200 & NO & NO & NO & 0.06 \\
+% \hline
+% 250 & NO & NO & NO & 0.08 \\
+% \hline
+%\end{tabular}
+
+%\label{tl}
+
+%\end{table}
+
+ In the  following, we will make  comparisons with two other  methods. The first
+ method,  called DESK  and proposed  by  \cite{ChinhVu}, is  a full  distributed
+ coverage  algorithm.   The  second method,  called  GAF~\cite{xu2001geography},
+ consists in dividing the region into fixed squares.  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
@@ -938,24 +1273,24 @@ COMPUTATION & on & on & on & 26.83 \\
 % is used to refer this table in the text
 \end{table}
 
-For the sake of simplicity we ignore  the energy needed to turn on the radio, to
+For the sake of simplicity we ignore the  energy needed to turn on the radio, to
 start up the sensor node, to move from one status to another, etc.
 %We also do not consider the need of collecting sensing data. PAS COMPRIS
-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
+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 $0.2575~mW$.
+calculate the  energy cost  for transmitting  messages and  we propose  the same
+value for receiving  the packets. The energy  needed to send or  receive a 1-bit
+packet is equal to 0.2575~mW.
 
-The initial energy of each node  is randomly set in the interval $[500;700]$.  A
-sensor node  will not participate in the  next round if its  remaining energy is
+The initial energy of each node is  randomly set in the interval $[500;700]$.  A
+sensor node will  not participate in the  next round if its  remaining energy is
 less than  $E_{R}=36~\mbox{Joules}$, the minimum  energy needed for the  node to
-stay alive  during one round.  This value has  been computed by  multiplying the
+stay alive  during one round.  This  value has been computed  by multiplying the
 energy consumed in  active state (9.72 mW)  by the time in second  for one round
-(3600 seconds).  According to the  interval of initial  energy, a sensor  may be
+(3600 seconds).   According to the interval  of initial energy, a  sensor may be
 alive during at most 20 rounds.
 
 \subsection{Metrics}
@@ -1011,7 +1346,7 @@ network, and $R$ is the total number of subregions in the network.
   % New version with global loops on period
   \begin{equation*}
     \scriptsize
-    \mbox{EC} = \frac{\sum\limits_{m=1}^{M} \left[ \left( E^{\mbox{com}}_m+E^{\mbox{list}}_m+E^{\mbox{comp}}_m \right) +\sum\limits_{t=1}^{T_m} \left( E^{a}_t+E^{s}_t \right) \right]}{\sum\limits_{m=1}^{M_L} T_m},
+    \mbox{EC} = \frac{\sum\limits_{m=1}^{M} \left[ \left( E^{\mbox{com}}_m+E^{\mbox{list}}_m+E^{\mbox{comp}}_m \right) +\sum\limits_{t=1}^{T_m} \left( E^{a}_t+E^{s}_t \right) \right]}{\sum\limits_{m=1}^{M} T_m},
   \end{equation*}
 
 
@@ -1030,14 +1365,14 @@ network, and $R$ is the total number of subregions in the network.
 % Old version -> where $M_L$ and  $T_L$ are respectively the number of  periods and rounds during
 %$Lifetime_{95}$ or  $Lifetime_{50}$. 
 % New version
-where  $M$ is  the number  of periods  and  $T_m$ the  number of  rounds in  a
+where  $M$ is  the  number  of periods  and  $T_m$ the  number  of  rounds in  a
 period~$m$, both  during $Lifetime_{95}$  or $Lifetime_{50}$.  The  total energy
-consumed by the  sensors (EC) comes through taking  into consideration four main
+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  $m$.
+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 $t$.
@@ -1057,9 +1392,75 @@ indicate the energy consumed by the whole network in round $t$.
 
 \end{enumerate}
 
-\section{Results and analysis}
+\subsection{Performance analysis for different number of primary points}
+\label{ch4:sec:04:06}
+
+In this  section, we study the  performance of MuDiLCO-1 approach  for different
+numbers of  primary points. The  objective of this  comparison is to  select the
+suitable 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.
+
+%\begin{enumerate}[i)]
+
+%\item {{\bf Coverage Ratio}}
+\subsubsection{Coverage ratio} 
+
+Figure~\ref{Figures/ch4/R2/CR} shows the average coverage ratio for 150 deployed
+nodes.  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, coverage ratio produced by Model-9, Model-13, Model-17, and Model-21 decreases in comparison with Model-5 due to a larger time computation for the decision process for larger number of primary points.
+Moreover, when the  number of periods increases, 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.
+% smaller time computation of decision process for a smaller number of primary points.
+Overall this  model is slightly more  efficient than the other  ones, because it
+offers a good coverage ratio for a larger number of periods.
+%\parskip 0pt
+\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} 
+
+
+%\item {{\bf Network Lifetime}}
+\subsubsection{Network lifetime}
+
+Finally, we study the effect of increasing the number of primary points on the lifetime of the network. 
+%In Figure~\ref{Figures/ch4/R2/LT95} and in Figure~\ref{Figures/ch4/R2/LT50}, network lifetime, $Lifetime95$ and $Lifetime50$ respectively, are illustrated for different network sizes. 
+As       highlighted       by       Figures~\ref{Figures/ch4/R2/LT}(a)       and
+\ref{Figures/ch4/R2/LT}(b), the  network lifetime  obviously increases  when the
+size of the network increases, with  Model-5 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.
 
-\subsection{Coverage ratio} 
+%\end{enumerate}
+
+% MICHEL => TO BE CONTINUED
+
+\subsection{Experimental results and analysis}
+
+\subsubsection{Coverage ratio} 
 
 Figure~\ref{fig3} shows  the average coverage  ratio for 150 deployed  nodes. We
 can notice that for the first thirty rounds both DESK and GAF provide a coverage
@@ -1080,55 +1481,62 @@ rounds, and thus should extend the network lifetime.
 
 \begin{figure}[ht!]
 \centering
- \includegraphics[scale=0.5] {R1/CR.pdf} 
+ \includegraphics[scale=0.5] {F/CR.pdf} 
 \caption{Average coverage ratio for 150 deployed nodes}
 \label{fig3}
 \end{figure} 
 
-\subsection{Active sensors ratio} 
+\iffalse
+\textcolor{red}{ We
+can see that for the first thirty nine rounds GA-MuDiLCO provides a little bit better coverage ratio  than MuDiLCO. Both DESK and GAF provide a coverage
+which is a little bit better than the one of MuDiLCO and GA-MuDiLCO for the first thirty rounds because they activate a larger number of nodes during sensing phase. After that GA-MuDiLCO provides a coverage ratio near to the  MuDiLCO and better than DESK and GAF. GA-MuDiLCO gives approximate solution with activation a larger number of nodes than MuDiLCO during sensing phase while it activates a less number of nodes in comparison with both DESK and GAF. MuDiLCO and GA-MuDiLCO clearly outperform DESK and GAF for
+a number of periods between 31 and 103. This is because they optimize the coverage and the lifetime in a wireless sensor network by selecting the best representative sensor nodes to take the responsibility of coverage during the sensing phase.}
+\fi
+
+
+\subsubsection{Active sensors ratio} 
 
 It is crucial to have as few active nodes as possible in each round, in order to
-minimize    the    communication    overhead    and   maximize    the    network
-lifetime. Figure~\ref{fig4}  presents the active  sensor ratio for  150 deployed
+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.  After the
-thirty-fifth round, MuDiLCO exhibits larger numbers of active nodes, which agrees
-with  the  dual  observation  of  higher  level  of  coverage  made  previously.
-Obviously, in  that case DESK  and GAF have  less active nodes, since  they have
-activated many nodes  at the beginning. Anyway, MuDiLCO  activates the available
-nodes in a more efficient manner.
+MuDiLCO clearly outperforms them  with only 24.8\%  of active nodes. 
+%\textcolor{red}{GA-MuDiLCO activates a number of sensor nodes larger than MuDiLCO but lower than both DESK and GAF. GA-MuDiLCO-1, GA-MuDiLCO-3, and GA-MuDiLCO-5 continue in providing a larger number of active sensors until the forty-sixth round after that it provides less number of active nodes due to the died nodes. GA-MuDiLCO-7 provides a larger number of sensor nodes and maintains a better coverage ratio compared to MuDiLCO-7 until the fifty-seventh round.  After the thirty-fifth round, MuDiLCO exhibits larger numbers of active nodes compared with DESK  and GAF, which agrees with  the  dual  observation  of  higher  level  of  coverage  made  previously}.
+Obviously, in that case DESK  and GAF have less active nodes, since  they have activated many nodes  at the beginning. Anyway, MuDiLCO  activates the available nodes in a more efficient manner. 
+%\textcolor{red}{GA-MuDiLCO activates near optimal number of sensor nodes also in efficient manner compared with both DESK  and GAF}.
 
 \begin{figure}[ht!]
 \centering
-\includegraphics[scale=0.5]{R1/ASR.pdf}  
+\includegraphics[scale=0.5]{F/ASR.pdf}  
 \caption{Active sensors ratio for 150 deployed nodes}
 \label{fig4}
 \end{figure} 
 
-\subsection{Stopped simulation runs}
+%\textcolor{red}{GA-MuDiLCO activates a sensor nodes larger than MuDiLCO but lower than both DESK and GAF }
+
+
+\subsubsection{Stopped simulation runs}
 %The results presented in this experiment, is to show the comparison of our MuDiLCO protocol with other two approaches from the point of view the stopped simulation runs per round. Figure~\ref{fig6} illustrates the percentage of stopped simulation
 %runs per round for 150 deployed nodes. 
 
 Figure~\ref{fig6} reports the cumulative  percentage of stopped simulations runs
 per round for  150 deployed nodes. This figure gives the  breakpoint for each method.  DESK stops first,  after approximately 45~rounds, because it consumes the
 more energy by  turning on a large number of redundant  nodes during the sensing
-phase. GAF  stops secondly for the  same reason than  DESK.  MuDiLCO overcomes
-DESK and GAF because the  optimization process distributed on several subregions
-leads  to coverage  preservation and  so extends  the network  lifetime.  Let us
-emphasize that the  simulation continues as long as a network  in a subregion is
-still connected.
+phase. GAF  stops secondly for the  same reason than  DESK. 
+%\textcolor{red}{GA-MuDiLCO  stops thirdly for the  same reason than  DESK and GAF.} \textcolor{red}{MuDiLCO and GA-MuDiLCO overcome}
+%DESK and GAF because \textcolor{red}{they activate less number of sensor nodes, as well as }the optimization process distributed on several subregions leads to coverage  preservation and  so extends  the network  lifetime.  
+Let us emphasize that the  simulation continues as long as a network  in a subregion is still connected. 
 
 %%% The optimization effectively continues as long as a network in a subregion is still connected. A VOIR %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 \begin{figure}[ht!]
 \centering
-\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} 
 
-\subsection{Energy consumption} \label{subsec:EC}
+\subsubsection{Energy consumption} \label{subsec:EC}
 
 We  measure  the  energy  consumed  by the  sensors  during  the  communication,
 listening, computation, active, and sleep status for different network densities
@@ -1139,9 +1547,9 @@ network sizes, for $Lifetime_{95}$ and $Lifetime_{50}$.
 \begin{figure}[h!]
   \centering
   \begin{tabular}{cl}
-    \parbox{9.5cm}{\includegraphics[scale=0.5]{R1/EC95.pdf}} & (a) \\
+    \parbox{9.5cm}{\includegraphics[scale=0.5]{F/EC95.pdf}} & (a) \\
     \verb+ + \\
-    \parbox{9.5cm}{\includegraphics[scale=0.5]{R1/EC50.pdf}} & (b)
+    \parbox{9.5cm}{\includegraphics[scale=0.5]{F/EC50.pdf}} & (b)
   \end{tabular}
   \caption{Energy consumption for (a) $Lifetime_{95}$ and 
     (b) $Lifetime_{50}$}
@@ -1150,23 +1558,20 @@ network sizes, for $Lifetime_{95}$ and $Lifetime_{50}$.
 
 The  results  show  that  MuDiLCO  is  the  most  competitive  from  the  energy
 consumption point of view.  The  other approaches have a high energy consumption
-due  to activating a  larger number  of redundant  nodes as  well as  the energy
-consumed during  the different  status of the  sensor node. Among  the different
-versions of our protocol, the MuDiLCO-7  one consumes more energy than the other
-versions. This is  easy to understand since the bigger the  number of rounds and
-the number of  sensors involved in the integer program are,  the larger the time
-computation to solve the optimization problem is. To improve the performances of
-MuDiLCO-7, we  should increase the  number of subregions  in order to  have less
-sensors to consider in the integer program.
-
+due  to activating a  larger number  of redundant  nodes as  well as  the energy consumed during  the different  status of the  sensor node.
+% Among  the different versions of our protocol, the MuDiLCO-7  one consumes more energy than the other
+%versions. This is  easy to understand since the bigger the  number of rounds and the number of  sensors involved in the integer program are,  the larger the time computation to solve the optimization problem is. To improve the performances of MuDiLCO-7, we  should increase the  number of subregions  in order to  have less sensors to consider in the integer program.
+%\textcolor{red}{As shown in Figure~\ref{fig7}, GA-MuDiLCO consumes less energy than both DESK and GAF, but a little bit higher than MuDiLCO  because it provides a near optimal solution by activating a larger number of nodes during the sensing phase.  GA-MuDiLCO consumes less energy in comparison with MuDiLCO-7 version, especially for the dense networks. However, MuDiLCO protocol and GA-MuDiLCO protocol are the most competitive from the energy
+%consumption point of view. The other approaches have a high energy consumption
+%due to activating a larger number of redundant nodes.}
 %In fact,  a distributed optimization decision, which produces T rounds, on the subregions is  greatly reduced the cost of communications and the time of listening as well as the energy needed for sensing phase and computation so thanks to the partitioning of the initial network into several independent subnetworks and producing T rounds for each subregion periodically. 
 
 
-\subsection{Execution time}
-
+\subsubsection{Execution time}
+\label{et}
 We observe  the impact of the  network size and of  the number of  rounds on the
 computation  time.   Figure~\ref{fig77} gives  the  average  execution times  in
-seconds (needed to solve optimization problem) for different values of $T$.  The
+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
@@ -1177,7 +1582,7 @@ for different network sizes.
 
 \begin{figure}[ht!]
 \centering
-\includegraphics[scale=0.5]{R1/T.pdf}  
+\includegraphics[scale=0.5]{F/T.pdf}  
 \caption{Execution Time (in seconds)}
 \label{fig77}
 \end{figure} 
@@ -1195,7 +1600,7 @@ optimization problem.
 
 %While MuDiLCO-1, 3, and 5 solves the optimization process with suitable execution times to be used on wireless sensor network because it distributed on larger number of small subregions as well as it is used acceptable number of round(s) T.  We think that in distributed fashion the solving of the optimization problem to produce T rounds in a subregion can be tackled by sensor nodes. Overall, to be able to deal with very large networks, a distributed method is clearly required.
 
-\subsection{Network lifetime}
+\subsubsection{Network lifetime}
 
 The next  two figures,  Figures~\ref{fig8}(a) and \ref{fig8}(b),  illustrate the
 network lifetime  for different network sizes,  respectively for $Lifetime_{95}$
@@ -1210,14 +1615,15 @@ of  $Lifetime_{95}$  with  large  wireless  sensor  networks  results  from  the
 difficulty  of the optimization  problem to  be solved  by the  integer program.
 This  point was  already noticed  in subsection  \ref{subsec:EC} devoted  to the
 energy consumption,  since network lifetime and energy  consumption are directly
-linked.
-
+linked. 
+%\textcolor{red}{As can be seen in these figures, the lifetime increases with the size of the network, and it is clearly largest for the MuDiLCO
+%and the GA-MuDiLCO protocols. GA-MuDiLCO prolongs the network lifetime obviously in comparison with both DESK and GAF, as well as the MuDiLCO-7 version for $lifetime_{95}$.  However, comparison shows that MuDiLCO protocol and GA-MuDiLCO protocol, which use distributed optimization over the subregions are the best ones because they are robust to network disconnection during the network lifetime as well as they consume less energy in comparison with other approaches.}
 \begin{figure}[t!]
   \centering
   \begin{tabular}{cl}
-    \parbox{9.5cm}{\includegraphics[scale=0.5]{R1/LT95.pdf}} & (a) \\
+    \parbox{9.5cm}{\includegraphics[scale=0.5]{F/LT95.pdf}} & (a) \\
     \verb+ + \\
-    \parbox{9.5cm}{\includegraphics[scale=0.5]{R1/LT50.pdf}} & (b)
+    \parbox{9.5cm}{\includegraphics[scale=0.5]{F/LT50.pdf}} & (b)
   \end{tabular}
   \caption{Network lifetime for (a) $Lifetime_{95}$ and 
     (b) $Lifetime_{50}$}
@@ -1252,7 +1658,7 @@ 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.
+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
