New modifications up to section 4.5

[JournalMultiPeriods.git] / article.tex

diff --git a/article.tex b/article.tex
index 192acf53c897b42d8a65d77d937c9407765a38a8..1c13bc7ae9769e1303adf95998da1a35afac6f40 100644 (file)
--- a/article.tex
+++ b/article.tex
@@ -553,7 +553,7 @@ 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
 (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}.
+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\\  
 
 The coordinates of the primary points are defined as follows:\\
 %$(p_x,p_y)$ = point center of wireless sensor node\\  


@@ -562,12 +562,12 @@ $X_2=( p_x + R_s * (1), p_y + R_s * (0) )$\\
 $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_3=( p_x + R_s * (-1), p_y + R_s * (0)) $\\
 $X_4=( p_x + R_s * (0), p_y + R_s * (1) )$\\
 $X_5=( p_x + R_s * (0), p_y + R_s * (-1 )) $\\

-$X_6= ( p_x + R_s * (\frac{-\sqrt{2}}{2}), p_y + R_s * (0)) $\\
-$X_7=( p_x + R_s *  (\frac{\sqrt{2}}{2}), p_y + R_s * (0))$\\
+$X_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_8=( p_x + R_s * (\frac{-\sqrt{2}}{2}), p_y + R_s * (\frac{-\sqrt{2}}{2})) $\\
 $X_9=( p_x + R_s * (\frac{\sqrt{2}}{2}), p_y + R_s * (\frac{-\sqrt{2}}{2})) $\\

-$X_{10}=( p_x + R_s * (\frac{-\sqrt{2}}{2}), p_y + R_s * (\frac{\sqrt{2}}{2})) $\\
-$X_{11}=( p_x + R_s * (\frac{\sqrt{2}}{2}), p_y + R_s * (\frac{\sqrt{2}}{2})) $\\
+$X_{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_{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})) $\\


@@ -627,23 +627,25 @@ $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{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 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 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}[ht!]
-\centering \includegraphics[width=100mm]{Modelgeneral.pdf} % 70mm
+\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} 
 \caption{The MuDiLCO protocol scheme executed on each node}
 \label{fig2}
 \end{figure} 


@@ -653,15 +655,18 @@ 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.
 
 % 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
 %This protocol is reliable against an unexpected node failure, because it works in periods. 
 %%RC : why? I am not convinced

- On the one hand, if a node failure is detected before  making the
-decision, the node will not participate to this phase, and, on the other hand,
-if the node failure occurs after the decision, the sensing  task of the network
-will be temporarily affected:  only during  the period of sensing until a new
-period starts. \textcolor{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.}
+ 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
 
 %%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


@@ -713,16 +718,16 @@ corresponds to the time that a sensor can live in the active mode.
 
 \subsection{Leader Election phase}
 
 
 \subsection{Leader Election phase}
 

-This step  consists in  choosing the Wireless  Sensor Node Leader  (WSNL), which
+This step  consists in choosing  the Wireless  Sensor Node Leader  (WSNL), which

 will be responsible for executing the coverage algorithm.  Each subregion in the
 area of  interest will select its  own WSNL independently for  each period.  All
 will be responsible for executing the coverage algorithm.  Each subregion in the
 area of  interest will select its  own WSNL independently for  each period.  All

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


@@ -731,10 +736,12 @@ consumption due to the communications.
 
 \subsection{Decision phase}
 
 
 \subsection{Decision phase}
 

-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{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  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


@@ -751,15 +758,16 @@ each round  of the  sensing phase.  }
 
 %\section{\textcolor{red}{ Optimization Algorithm for Multiround Lifetime Coverage Optimization}}
 %\label{oa}
 
 %\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
+As shown in Algorithm~\ref{alg:MuDiLCO}, the leader will execute an optimization
+algorithm based on an integer program. The integer program is based on the model
+proposed by \cite{pedraza2006}  with some modifications, where  the objective is
+to find  a maximum  number of disjoint  cover sets.  To  fulfill this  goal, the
+authors proposed an integer program  which forces undercoverage and overcoverage

 of  targets to  become minimal  at  the same  time.  They  use binary  variables
 $x_{jl}$ to indicate if  sensor $j$ belongs to cover set $l$.   In our model, we
 of  targets to  become minimal  at  the same  time.  They  use binary  variables
 $x_{jl}$ to indicate if  sensor $j$ belongs to cover set $l$.   In our model, we

-consider binary  variables $X_{t,j}$ to determine the  possibility of 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.
 
 sensors by  $J$. Only sensors  able to  be alive during  at least one  round are
 involved in the integer program.
 


@@ -811,7 +819,7 @@ U_{t,p} = \left \{
 
 Our coverage optimization problem can then be formulated as follows:
 \begin{equation}
 
 Our coverage optimization problem can then be formulated as follows:
 \begin{equation}

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

 \end{equation}
 
 Subject to
 \end{equation}
 
 Subject to


@@ -854,32 +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
 
 The first group  of constraints indicates that some primary  point $p$ should be
 covered by at least  one sensor and, if it is not  always the case, overcoverage

-and undercoverage  variables help balancing the restriction  equations by taking
+and undercoverage variables  help balancing the restriction  equations by taking

 positive values. The constraint  given by equation~(\ref{eq144}) guarantees that
 the sensor has enough energy ($RE_j$  corresponds to its remaining energy) to be
 alive during  the selected rounds knowing  that $E_{R}$ is the  amount of energy
 required to be alive during one round.
 
 positive values. The constraint  given by equation~(\ref{eq144}) guarantees that
 the sensor has enough energy ($RE_j$  corresponds to its remaining energy) to be
 alive during  the selected rounds knowing  that $E_{R}$ is the  amount of energy
 required to be alive during one round.
 

-There  are two main  objectives.  First,  we limit  the overcoverage  of primary
-points in order to activate a  minimum number of sensors.  Second we prevent the
-absence  of  monitoring  on  some  parts  of the  subregion  by  minimizing  the
-undercoverage.  The weights  $W_\theta$ and $W_U$ must be  properly chosen so as
-to guarantee that the maximum number of points are covered during each round. 
+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
 %% 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 $W_{\theta}$.
 
 large compared to $W_{\theta}$.
 

-\textcolor{blue}{The size of the problem depends on the number of variables and constraints. The number of variables is linked to the number of alive sensors $A \subset J$, 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.
-
+\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
 
 \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. 
 
 
 % 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. 
 


@@ -901,9 +914,9 @@ Active-Sleep packet is obtained.
       \If{$ s_j.ID = LeaderID $}{
         \emph{$s_j.status$ = COMPUTATION}\;
         \emph{$\left\{\left(X_{1,k},\dots,X_{T,k}\right)\right\}_{k \in J}$ =
       \If{$ s_j.ID = LeaderID $}{
         \emph{$s_j.status$ = COMPUTATION}\;
         \emph{$\left\{\left(X_{1,k},\dots,X_{T,k}\right)\right\}_{k \in J}$ =

-          Execute \textcolor{red}{Optimization Algorithm}($T,J$)}\;
+          Execute Integer Program Algorithm($T,J$)}\;

         \emph{$s_j.status$ = COMMUNICATION}\;
         \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 $}\;
       }          
           with vector of activity scheduling $(X_{1,k},\dots,X_{T,k})$}\;
         \emph{Update $RE_j $}\;
       }          


@@ -1091,22 +1104,22 @@ The proposed GA-MuDiLCO stops when the stopping criteria is met. It stops after
 
 \fi
 
 
 \fi
 

+%% EXPERIMENTAL STUDY
+

 \section{Experimental study}
 \label{exp}
 \subsection{Simulation setup}
 
 \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
 %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 ?
 
 %%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 ?


@@ -1151,42 +1164,54 @@ $W_{U}$ & $|P|^2$ \\
 % is used to refer this table in the text
 \end{table}
 
 % is used to refer this table in the text
 \end{table}
 

-\textcolor{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. }. 
-
-\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}
-
+\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.
+ 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
 
 Some preliminary experiments were performed to study the choice of the number of
 subregions  which subdivides  the  sensing field,  considering different  network


@@ -1248,24 +1273,24 @@ COMPUTATION & on & on & on & 26.83 \\
 % is used to refer this table in the text
 \end{table}
 
 % is used to refer this table in the text
 \end{table}
 

-For 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
 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
 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
+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.
 
 packet is equal to 0.2575~mW.
 

-The initial energy of each node  is randomly set in the interval $[500;700]$.  A
-sensor node  will not participate in the  next round if its  remaining energy is
+The initial energy of each node is  randomly set in the interval $[500;700]$.  A
+sensor node will  not participate in the  next round if its  remaining energy is

 less than  $E_{R}=36~\mbox{Joules}$, the minimum  energy needed for the  node to
 less than  $E_{R}=36~\mbox{Joules}$, the minimum  energy needed for the  node to

-stay alive  during one round.  This value has  been computed by  multiplying the
+stay alive  during one round.  This  value has been computed  by multiplying the

 energy consumed in  active state (9.72 mW)  by the time in second  for one round
 energy consumed in  active state (9.72 mW)  by the time in second  for one round

-(3600 seconds).  According to the  interval of initial  energy, a sensor  may be
+(3600 seconds).   According to the interval  of initial energy, a  sensor may be

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


@@ -1340,14 +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
 % 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
 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  $E^{\scriptsize  \mbox{com}}_m$,
 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 $t$.
 $E^{\scriptsize \mbox{comp}}_m$  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 $t$.


@@ -1370,34 +1395,47 @@ indicate the energy consumed by the whole network in round $t$.
 \subsection{Performance analysis for different number of primary points}
 \label{ch4:sec:04:06}
 
 \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. 
-
+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} 
 
 
 %\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.  
-\parskip 0pt    
-\begin{figure}[h!]
+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} 
 \centering
  \includegraphics[scale=0.5] {R2/CR.pdf} 
 \caption{Coverage ratio for 150 deployed nodes}
 \label{Figures/ch4/R2/CR}
 \end{figure} 

-As can be seen in Figure~\ref{Figures/ch4/R2/CR}, at the beginning the models which use a larger number of primary points provide slightly better coverage ratios, but latter they are the worst. 
-%Moreover, when the number of periods increases, coverage ratio produced by Model-9, Model-13, Model-17, and Model-21 decreases in comparison with Model-5 due to a larger time computation for the decision process for larger number of primary points.
-Moreover, when the number of periods increases, coverage ratio produced by all models decrease, but Model-5 is the one with the slowest decrease due to a smaller time computation of decision process for a smaller number of primary points. 
-As shown in Figure ~\ref{Figures/ch4/R2/CR}, coverage ratio decreases when the number of periods increases due to dead nodes. Model-5 is slightly more efficient than other models, because it offers a good coverage ratio for a larger number of periods in comparison with other models.

 
 
 %\item {{\bf Network Lifetime}}
 \subsubsection{Network lifetime}
 
 
 
 %\item {{\bf Network Lifetime}}
 \subsubsection{Network lifetime}
 

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

-As highlighted by Figures~\ref{Figures/ch4/R2/LT}(a) and \ref{Figures/ch4/R2/LT}(b), the network lifetime obviously increases when the size of the network increases, with  Model-5 that leads to the larger lifetime improvement. 
+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
 
 \begin{figure}[h!]
 \centering


@@ -1410,11 +1448,17 @@ As highlighted by Figures~\ref{Figures/ch4/R2/LT}(a) and \ref{Figures/ch4/R2/LT}
   \label{Figures/ch4/R2/LT}
 \end{figure}
 
   \label{Figures/ch4/R2/LT}
 \end{figure}
 

-Comparison shows that Model-5, which uses less number of primary points, is the best one because it is less energy consuming during the network lifetime. It is also the better one from the point of view of coverage ratio. Our proposed Model-5 efficiently prolongs the network lifetime with a good coverage ratio in comparison with other models. Therefore, we have chosen the model with five primary points for all the experiments presented thereafter. 
+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.

 
 %\end{enumerate}
 
 
 %\end{enumerate}
 

-\subsection{Results and analysis}
+% MICHEL => TO BE CONTINUED
+
+\subsection{Experimental results and analysis}

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