First modifications (up to section 3.2)

diff --git a/article.tex b/article.tex
index 99b5e0e4f672f57254bb3cccf35f5b528cb9b9ea..192acf53c897b42d8a65d77d937c9407765a38a8 100644 (file)
--- a/article.tex
+++ b/article.tex
@@ -84,14 +84,11 @@
 %e-mail: ali.idness@edu.univ-fcomte.fr, \\
 %$\lbrace$karine.deschinkel, michel.salomon, raphael.couturier$\rbrace$@univ-fcomte.fr.}
 
 %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}}
-}  
-
+\author{Ali   Kadhum   Idrees$^{a,b}$,   Karine  Deschinkel$^{a}$,   \\   Michel
+  Salomon$^{a}$,   and  Rapha\"el   Couturier   $^{a}$  \\   $^{a}${\em{FEMTO-ST
+      Institute,  UMR  6174  CNRS,   \\  University  Bourgogne  Franche-Comt\'e,
+      Belfort, France}} \\ $^{b}${\em{Department of Computer Science, University
+      of Babylon, Babylon, Iraq}} }

 
 \begin{abstract}
 %One of  the fundamental challenges in Wireless Sensor Networks (WSNs)
 
 \begin{abstract}
 %One of  the fundamental challenges in Wireless Sensor Networks (WSNs)


@@ -99,31 +96,33 @@ Michel Salomon$^{a}$ and Rapha\"el Couturier $^{a}$ \\
 %continuously  and  effectively  when  monitoring a  certain  area  (or
 %region) of  interest. 
 Coverage and  lifetime are  two paramount problems  in Wireless  Sensor Networks
 %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
 Optimization  protocol (MuDiLCO)  is proposed  to maintain  the coverage  and to
 improve the lifetime in wireless sensor  networks. The area of interest is first

-divided  into subregions and  then the  MuDiLCO protocol  is distributed  on the
-sensor nodes in each subregion. The proposed MuDiLCO protocol works in periods
-during which sets of sensor nodes are scheduled to remain active for a number of
-rounds  during the  sensing phase,  to  ensure coverage  so as  to maximize  the
-lifetime of  WSN. \textcolor{green}{The decision process is  carried out by a  leader node, which
-solves an optimization problem to  produce the best  representative sets to  be used
-during the rounds  of the sensing phase. The optimization problem formulated as an integer program is solved to optimality through a branch-and-Bound method for small instances. For larger instances, the best feasible solution found by the solver after a given time limit threshold is considered. }
+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)}. 
 %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.
-
+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,
 Optimization, Scheduling, Distributed Computation.
 \end{abstract}
 
 \begin{keyword}
 Wireless   Sensor   Networks,   Area   Coverage,   Network   Lifetime,
 Optimization, Scheduling, Distributed Computation.

-

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


@@ -167,10 +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}
 
 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
 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}.
 
 concluding    remarks   and    some    suggestions   for    future   works    in
 Section~\ref{sec:conclusion}.
 


@@ -204,43 +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
 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
 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
 higher cost in communications, since the  node that will make the decision needs

-information from all the  sensor nodes. \textcolor{green} {Exact or heuristics approaches are designed to provide cover sets.
- %(Moreover, centralized approaches usually
+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.) 
 %suffer from the scalability problem, making them less competitive as the network
 %size increases.) 

-Contrary to exact methods, heuristic methods can handle very large and centralized problems. They are proposed to reduce computational overhead such as energy consumption, delay and generally increase in
-the network lifetime. }
+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
 
 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
 lifetime of the network in comparison  to the disjoint cover set algorithms, but

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

 order  of complexity.   Moreover, in  case of  a sensor's  failure, non-disjoint
 order  of complexity.   Moreover, in  case of  a sensor's  failure, non-disjoint

-scheduling  policies are less  resilient and  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}    
 
 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
 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
 al.~\cite{cheng2014energy} have defined a  heuristic algorithm called Cover Sets
 Balance  (CSB), which  chooses  a set  of  active nodes  using  the tuple  (data
 coverage range, residual  energy).  Then, they have introduced  a new Correlated

-Node Set Computing (CNSC) algorithm to  find the correlated node set for a given
-node.   After that,  they  proposed a  High  Residual Energy  First (HREF)  node
-selection algorithm to minimize the number  of active nodes so as to prolong the
-network  lifetime.  Various  centralized  methods  based  on  column  generation
-approaches                    have                   also                   been
-proposed~\cite{gentili2013,castano2013column,rossi2012exact,deschinkel2012column}. 
-\textcolor{green}{In~\cite{gentili2013}, authors highlight the trade-off between the network lifetime and the coverage percentage. They show that network lifetime can be hugely improved by decreasing the coverage ratio. }
+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}
 
 \subsection{Distributed approaches}
 %{\bf Distributed approaches}


@@ -297,16 +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
 \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
 in~\cite{idrees2014coverage}.   In~\cite{idrees2014coverage},  the  protocol  is

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

 more  interesting  to  divide  the  area  into  several  subregions,  given  the
 computation complexity. Compared to our previous paper, in this one we study the
 possibility of dividing  the sensing phase into multiple rounds  and we also add
 more  interesting  to  divide  the  area  into  several  subregions,  given  the
 computation complexity. Compared to our previous paper, in this one we study the
 possibility of dividing  the sensing phase into multiple rounds  and we also add

-an  improved  model  of energy  consumption  to  assess  the efficiency  of  our
+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
 approach. In fact, in this paper we make a multiround optimization, while it was

-a single round optimization in our previous work. \textcolor{green}{The idea is to take advantage of the pre-sensing phase
- to plan the sensor's activity for several rounds instead of one, thus saving energy. In addition, when the optimization problem becomes more complex, its resolution is stopped after a given time threshold}.
+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
    
 
 \iffalse
    


@@ -540,8 +546,16 @@ 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
+subsection~\ref{ch4:sec:04:06}.

 
 

-\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 (point  center: ($p_x,p_y$))  of  a wireless sensor node  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 coordinates of the primary points are the following :\\
+The coordinates of the primary points are defined as follows:\\

 %$(p_x,p_y)$ = point center of wireless sensor node\\  
 $X_1=(p_x,p_y)$ \\ 
 $X_2=( p_x + R_s * (1), p_y + R_s * (0) )$\\           
 %$(p_x,p_y)$ = point center of wireless sensor node\\  
 $X_1=(p_x,p_y)$ \\ 
 $X_2=( p_x + R_s * (1), p_y + R_s * (0) )$\\           


@@ -570,28 +584,29 @@ $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})) $.
 
 
 $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
+% \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
 
 %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


@@ -612,7 +627,7 @@ $X_{25}=( p_x + R_s * (\frac{1}{2}), p_y + R_s * (\frac{-\sqrt{3}}{2})) $.
 %smaller  areas,  called  subregions,  and  then our MuDiLCO  protocol will be
 %implemented in each subregion in a distributed way.
 
 %smaller  areas,  called  subregions,  and  then our MuDiLCO  protocol will be
 %implemented in each subregion in a distributed way.
 

-\textcolor{green}{The WSN area of  interest is, in a first step,  divided into regular homogeneous
+\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
 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


@@ -623,7 +638,7 @@ less than or equal to 3.}
 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
 As  can be seen  in Figure~\ref{fig2},  our protocol  works in  periods fashion,
 where  each is  divided  into 4  phases: Information~Exchange,  Leader~Election,
 Decision, and Sensing.  Each sensing phase may be itself divided into $T$ rounds

-\textcolor{green} {of equal duration} and for each round a set of sensors (a cover set) is responsible for the sensing
+\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.
 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.


@@ -646,7 +661,7 @@ running out of energy), because it works in periods.
 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
 decision, the node will not participate to this phase, and, on the other hand,
 if the node failure occurs after the decision, the sensing  task of the network
 will be temporarily affected:  only during  the period of sensing until a new

-period starts. \textcolor{green}{The duration of the rounds are predefined parameters. Round duration should be long enough to hide the system control overhead and short enough to minimize the negative effects in case of node failure.}
+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
 
 %%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


@@ -716,7 +731,7 @@ consumption due to the communications.
 
 \subsection{Decision phase}
 
 
 \subsection{Decision phase}
 

-Each  WSNL will \textcolor{green}{ solve an integer program to  select which  cover sets  will be
+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.  }
 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.  }


@@ -854,7 +869,7 @@ 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 $W_{\theta}$.
 
 In our simulations priority is given  to the coverage by choosing $W_{U}$ very
 large compared to $W_{\theta}$.
 

-\textcolor{green}{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 $A \subset J$, 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})).}
+\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 $A \subset J$, 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.
 
 
 %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.
 
 


@@ -1136,7 +1151,7 @@ $W_{U}$ & $|P|^2$ \\
 % is used to refer this table in the text
 \end{table}
 
 % is used to refer this table in the text
 \end{table}
 

-\textcolor{green}{The MuDilLCO protocol is declined into  four versions: MuDiLCO-1,  MuDiLCO-3, MuDiLCO-5,
+\textcolor{blue}{The MuDilLCO 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 prohibitif when the size of the problem increases, a time limit treshold  has been fixed to solve large instances.  In these cases, the solver returns the best solution found, which is not necessary the optimal solution.
  Table \ref{tl} shows time limit values. These time limit treshold 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 treshold 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. }. 
 
 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 prohibitif when the size of the problem increases, a time limit treshold  has been fixed to solve large instances.  In these cases, the solver returns the best solution found, which is not necessary the optimal solution.
  Table \ref{tl} shows time limit values. These time limit treshold 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 treshold 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. }. 
 


@@ -1352,8 +1367,7 @@ indicate the energy consumed by the whole network in round $t$.
 
 \end{enumerate}
 
 
 \end{enumerate}
 

-\section{Results and analysis}
-\subsection{Performance Analysis for Different Number of Primary Points}
+\subsection{Performance analysis for different number of primary points}

 \label{ch4:sec:04:06}
 
 In this section, we study the performance of MuDiLCO-1 approach for different numbers of primary points. The objective of this comparison is to select the suitable primary point model to be used by a MuDiLCO protocol. In this comparison, MuDiLCO-1 protocol is used with five models, which are called Model-5 (it uses 5 primary points), Model-9, Model-13, Model-17, and Model-21. 
 \label{ch4:sec:04:06}
 
 In this section, we study the performance of MuDiLCO-1 approach for different numbers of primary points. The objective of this comparison is to select the suitable primary point model to be used by a MuDiLCO protocol. In this comparison, MuDiLCO-1 protocol is used with five models, which are called Model-5 (it uses 5 primary points), Model-9, Model-13, Model-17, and Model-21. 


@@ -1362,7 +1376,7 @@ In this section, we study the performance of MuDiLCO-1 approach for different nu
 %\begin{enumerate}[i)]
 
 %\item {{\bf Coverage Ratio}}
 %\begin{enumerate}[i)]
 
 %\item {{\bf Coverage Ratio}}

-\subsubsection{Coverage Ratio} 
+\subsubsection{Coverage ratio} 

 
 Figure~\ref{Figures/ch4/R2/CR} shows the average coverage ratio for 150 deployed nodes.  
 \parskip 0pt    
 
 Figure~\ref{Figures/ch4/R2/CR} shows the average coverage ratio for 150 deployed nodes.  
 \parskip 0pt    


@@ -1379,7 +1393,7 @@ As shown in Figure ~\ref{Figures/ch4/R2/CR}, coverage ratio decreases when the n
 
 
 %\item {{\bf Network Lifetime}}
 
 
 %\item {{\bf Network Lifetime}}

-\subsubsection{Network Lifetime}
+\subsubsection{Network lifetime}

 
 Finally, we study the effect of increasing the primary points on the lifetime of the network. 
 %In Figure~\ref{Figures/ch4/R2/LT95} and in Figure~\ref{Figures/ch4/R2/LT50}, network lifetime, $Lifetime95$ and $Lifetime50$ respectively, are illustrated for different network sizes. 
 
 Finally, we study the effect of increasing the primary points on the lifetime of the network. 
 %In Figure~\ref{Figures/ch4/R2/LT95} and in Figure~\ref{Figures/ch4/R2/LT50}, network lifetime, $Lifetime95$ and $Lifetime50$ respectively, are illustrated for different network sizes. 


@@ -1400,8 +1414,7 @@ Comparison shows that Model-5, which uses less number of primary points, is the
 
 %\end{enumerate}
 
 
 %\end{enumerate}
 

-
-\subsection{Comparison Results}
+\subsection{Results and analysis}

 
 \subsubsection{Coverage ratio} 
 
 
 \subsubsection{Coverage ratio}