Relecture apres ajouts Ali

diff --git a/article.tex b/article.tex
index 85045d384353349751f573e4fef41c883419ec71..36c5dcfffed59a26d6d8cf35911ac4e859c16a81 100644 (file)
--- a/article.tex
+++ b/article.tex
@@ -152,9 +152,9 @@ the network lifetime by using an optimized multirounds 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
 description of MuDiLCO protocol.  Section~\ref{exp} shows the simulation results

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

 demonstrate  the  usefulness  of   the  proposed  approach.   Finally,  we  give
 concluding    remarks   and    some    suggestions   for    future   works    in
 Section~\ref{sec:conclusion}.
 demonstrate  the  usefulness  of   the  proposed  approach.   Finally,  we  give
 concluding    remarks   and    some    suggestions   for    future   works    in
 Section~\ref{sec:conclusion}.


@@ -199,20 +199,20 @@ size increases.
 
 The first algorithms proposed in the literature consider that the cover sets are
 disjoint: a sensor node appears in exactly one of the generated cover sets.  For
 
 The first algorithms proposed in the literature consider that the cover sets are
 disjoint: a sensor node appears in exactly one of the generated cover sets.  For

-instance,  Slijepcevic and Potkonjak \cite{Slijepcevic01powerefficient} proposed
+instance, Slijepcevic  and Potkonjak \cite{Slijepcevic01powerefficient} proposed

 an  algorithm, which  allocates sensor  nodes  in mutually  independent sets  to
 monitor an area divided into several fields.  Their algorithm builds a cover set
 by including in  priority the sensor nodes which cover  critical fields, that is
 an  algorithm, which  allocates sensor  nodes  in mutually  independent sets  to
 monitor an area divided into several fields.  Their algorithm builds a cover set
 by including in  priority the sensor nodes which cover  critical fields, that is

-to  say fields that  are covered  by the  smallest number  of sensors.  The time
-complexity  of  their  heuristic  is   $O(n^2)$  where  $n$  is  the  number  of
-sensors.   Abrams  et   al.~\cite{abrams2004set}  designed  three  approximation
-algorithms for a variation of the set k-cover problem, where the objective is to
-partition the sensors into covers such that the number of covers that include an
-area, summed over all areas, is maximized.  Their work builds upon previous work
+to say  fields that  are covered by  the smallest  number of sensors.   The time
+complexity of  their heuristic is $O(n^2)$  where $n$ is the  number of sensors.
+Abrams et al.~\cite{abrams2004set} designed three approximation algorithms for a
+variation of  the set k-cover problem,  where the objective is  to partition the
+sensors into covers such that the  number of covers that include an area, summed
+over  all   areas,  is  maximized.    Their  work  builds  upon   previous  work

 in~\cite{Slijepcevic01powerefficient}  and  the  generated  cover  sets  do  not
 provide complete coverage of the monitoring zone.
 
 in~\cite{Slijepcevic01powerefficient}  and  the  generated  cover  sets  do  not
 provide complete coverage of the monitoring zone.
 

-\cite{cardei2005improving} proposed a method to efficiently  compute the maximum
+\cite{cardei2005improving} proposed a method  to efficiently compute the maximum

 number of disjoint  set covers such that each set can  monitor all targets. They
 first transform the problem into a  maximum flow problem, which is formulated as
 a mixed integer  programming (MIP). Then their heuristic uses  the output of the
 number of disjoint  set covers such that each set can  monitor all targets. They
 first transform the problem into a  maximum flow problem, which is formulated as
 a mixed integer  programming (MIP). Then their heuristic uses  the output of the


@@ -222,15 +222,15 @@ number      of     set      covers     slightly      larger      compared     to
 complexity of the mixed integer programming resolution.
 
 Zorbas et al.  \cite{zorbas2010solving} presented a centralized greedy algorithm
 complexity of the mixed integer programming resolution.
 
 Zorbas et al.  \cite{zorbas2010solving} presented a centralized greedy algorithm

-for  the efficient  production  of  both node  disjoint  and non-disjoint  cover
-sets.   Compared   to  algorithm's   results   of   Slijepcevic  and   Potkonjak
+for the efficient production of  both node disjoint and non-disjoint cover sets.
+Compared    to    algorithm's    results    of   Slijepcevic    and    Potkonjak

 \cite{Slijepcevic01powerefficient}, their heuristic produces more disjoint cover
 \cite{Slijepcevic01powerefficient}, their heuristic produces more disjoint cover

-sets with  a slight growth rate  in execution time.  When producing non-disjoint
+sets with a  slight growth rate in execution  time.  When producing non-disjoint

 cover sets,  both Static-CCF  and Dynamic-CCF algorithms,  where CCF  means that
 they  use a cost  function called  Critical Control  Factor, provide  cover sets
 cover sets,  both Static-CCF  and Dynamic-CCF algorithms,  where CCF  means that
 they  use a cost  function called  Critical Control  Factor, provide  cover sets

-offering     longer    network     lifetime    than     those     produced    by
-\cite{cardei2005energy}.   Also,  they   require  a   smaller  number   of  node
-participations in order to achieve these results.
+offering longer network lifetime than those produced by \cite{cardei2005energy}.
+Also, they require  a smaller number of node participations  in order to achieve
+these results.

 
 In  the  case  of  non-disjoint algorithms  \cite{pujari2011high},  sensors  may
 participate in  more than one  cover set.  In  some cases, this may  prolong the
 
 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


@@ -241,26 +241,34 @@ scheduling policies are less resilient and less reliable because a sensor may be
 involved   in   more  than   one   cover   sets.    For  instance,   Cardei   et
 al.~\cite{cardei2005energy}  present a  linear programming  (LP) solution  and a
 greedy approach to extend the  sensor network lifetime by organizing the sensors
 involved   in   more  than   one   cover   sets.    For  instance,   Cardei   et
 al.~\cite{cardei2005energy}  present a  linear programming  (LP) solution  and a
 greedy approach to extend the  sensor network lifetime by organizing the sensors

-into a maximal  number of non-disjoint cover sets.  Simulation results show that
+into a maximal number of  non-disjoint cover sets.  Simulation results show that

 by  allowing sensors  to  participate  in multiple  sets,  the network  lifetime
 increases     compared     with     related     work~\cite{cardei2005improving}.
 In~\cite{berman04},  the  authors  have  formulated  the  lifetime  problem  and
 by  allowing sensors  to  participate  in multiple  sets,  the network  lifetime
 increases     compared     with     related     work~\cite{cardei2005improving}.
 In~\cite{berman04},  the  authors  have  formulated  the  lifetime  problem  and

-suggested another (LP)  technique to solve this problem.  A centralized solution
+suggested another (LP) technique to  solve this problem.  A centralized solution

 based  on  the  Garg-K\"{o}nemann  algorithm~\cite{garg98},  provably  near  the
 optimal solution, is also proposed.
 
 based  on  the  Garg-K\"{o}nemann  algorithm~\cite{garg98},  provably  near  the
 optimal solution, is also proposed.
 

-In~\cite{yang2014maximum}, The authors are proposed a linear programming approach for selecting the minimum number of sensor nodes in working station so as to preserve a maximum coverage and extend lifetime of the network. Cheng et al.~\cite{cheng2014energy} are proposed a heuristic algorithm called Cover Sets Balance (CSB) algorithm to choose a set of active nodes using the tuple (data coverage range, residual energy). Then, they are introduced a new Correlated Node Set Computing (CNSC) algorithm to find the correlated node set for a given node.  After that, they are proposed a High Residual Energy First (HREF) node selection algorithm to minimize the number of active nodes so as to prolong the network lifetime. 
-In~\cite{castano2013column,rossi2012exact,deschinkel2012column}, The authors are proposed a centralized methods based on column generation approach to extend lifetime in wireless sensor networks while coverage preservation.
-
+In~\cite{yang2014maximum},  the  authors  have  proposed  a  linear  programming
+approach for selecting  the minimum number of working sensor  nodes, in order to
+as to preserve  a maximum coverage and extend lifetime of  the network. Cheng et
+al.~\cite{cheng2014energy} have defined a  heuristic algorithm called Cover Sets
+Balance (CSB), which choose a set of active nodes using the tuple (data coverage
+range, residual energy).   Then, they have introduced a  new Correlated Node Set
+Computing (CNSC)  algorithm to find  the correlated node  set for a  given node.
+After that,  they proposed  a High Residual  Energy First (HREF)  node selection
+algorithm to  minimize the number of active  nodes so as to  prolong the network
+lifetime. Various centralized methods based on column generation approaches have
+also been proposed~\cite{castano2013column,rossi2012exact,deschinkel2012column}.

 
 \subsection{Distributed approaches}
 %{\bf Distributed approaches}
 In distributed  and localized coverage  algorithms, the required  computation to
 schedule the  activity of  sensor nodes  will be done  by the  cooperation among
 neighboring nodes. These  algorithms may require more computation  power for the
 
 \subsection{Distributed approaches}
 %{\bf Distributed approaches}
 In distributed  and localized coverage  algorithms, the required  computation to
 schedule the  activity of  sensor nodes  will be done  by the  cooperation among
 neighboring nodes. These  algorithms may require more computation  power for the

-processing  by the cooperating  sensor nodes,  but they  are more  scalable for
-large  WSNs.    Localized  and   distributed  algorithms  generally   result  in
-non-disjoint set covers.
+processing by the cooperating sensor nodes, but they are more scalable for large
+WSNs.  Localized and distributed algorithms generally result in non-disjoint set
+covers.

 
 Some        distributed       algorithms        have        been       developed
 in~\cite{Gallais06,Tian02,Ye03,Zhang05,HeinzelmanCB02,    yardibi2010distributed}
 
 Some        distributed       algorithms        have        been       developed
 in~\cite{Gallais06,Tian02,Ye03,Zhang05,HeinzelmanCB02,    yardibi2010distributed}


@@ -273,7 +281,7 @@ area    \cite{Berman05efficientenergy}     or    maximum    uncovered    targets
 \cite{lu2003coverage}.  In \cite{Tian02}, the  scheduling scheme is divided into
 rounds,  where each  round has  a self-scheduling  phase followed  by  a sensing
 phase.  Each  sensor broadcasts  a message containing  the node~ID and  the node
 \cite{lu2003coverage}.  In \cite{Tian02}, the  scheduling scheme is divided into
 rounds,  where each  round has  a self-scheduling  phase followed  by  a sensing
 phase.  Each  sensor broadcasts  a message containing  the node~ID and  the node

-location to  its neighbors at the  beginning of each round.  A sensor determines
+location to its  neighbors at the beginning of each  round.  A sensor determines

 its status by a  rule named off-duty eligible rule, which tells  him to turn off
 if its sensing area is covered by its neighbors. A back-off scheme is introduced
 to let each sensor  delay the decision process with a random  period of time, in
 its status by a  rule named off-duty eligible rule, which tells  him to turn off
 if its sensing area is covered by its neighbors. A back-off scheme is introduced
 to let each sensor  delay the decision process with a random  period of time, in


@@ -291,7 +299,7 @@ not require  location information of sensors while  maintaining connectivity and
 satisfying a  user defined  coverage target.  In  DASSA, nodes use  the residual
 energy levels  and feedback from the  sink for scheduling the  activity of their
 neighbors.  This  feedback mechanism reduces  the randomness in  scheduling that
 satisfying a  user defined  coverage target.  In  DASSA, nodes use  the residual
 energy levels  and feedback from the  sink for scheduling the  activity of their
 neighbors.  This  feedback mechanism reduces  the randomness in  scheduling that

-would  otherwise  occur   due  to  the  absence  of   location  information.  In
+would  otherwise  occur  due  to   the  absence  of  location  information.   In

 \cite{ChinhVu},  the  author  proposed  a novel  distributed  heuristic,  called
 Distributed  Energy-efficient Scheduling  for k-coverage  (DESK),  which ensures
 that  the energy  consumption among  the sensors  is balanced  and  the lifetime
 \cite{ChinhVu},  the  author  proposed  a novel  distributed  heuristic,  called
 Distributed  Energy-efficient Scheduling  for k-coverage  (DESK),  which ensures
 that  the energy  consumption among  the sensors  is balanced  and  the lifetime


@@ -315,7 +323,7 @@ More recently,  Shibo et  al. \cite{Shibo} expressed  the coverage problem  as a
 minimum weight submodular set cover problem and proposed a Distributed Truncated
 Greedy Algorithm (DTGA) to solve it.  They take advantage from both temporal and
 spatial  correlations between  data sensed  by different  sensors,  and leverage
 minimum weight submodular set cover problem and proposed a Distributed Truncated
 Greedy Algorithm (DTGA) to solve it.  They take advantage from both temporal and
 spatial  correlations between  data sensed  by different  sensors,  and leverage

-prediction,  to   improve  the   lifetime.  In  \cite{xu2001geography},   Xu  et
+prediction,  to   improve  the  lifetime.   In   \cite{xu2001geography},  Xu  et

 al. proposed  an algorithm, called  Geographical Adaptive Fidelity  (GAF), which
 uses geographic location  information to divide the area  of interest into fixed
 square grids. Within each grid, it keeps only one node staying awake to take the
 al. proposed  an algorithm, called  Geographical Adaptive Fidelity  (GAF), which
 uses geographic location  information to divide the area  of interest into fixed
 square grids. Within each grid, it keeps only one node staying awake to take the


@@ -328,7 +336,7 @@ randomized \cite{Ye03} or regulated \cite{cardei2005maximum} over time.
 
 The MuDiLCO protocol (for Multiperiod Distributed Lifetime Coverage Optimization
 protocol) presented  in this  paper is an  extension of the  approach introduced
 
 The MuDiLCO protocol (for Multiperiod Distributed Lifetime Coverage Optimization
 protocol) presented  in this  paper is an  extension of the  approach introduced

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

 deployed over  only two  subregions. Simulation results  have shown that  it was
 more  interesting  to  divide  the  area  into  several  subregions,  given  the
 computation complexity. Compared to our previous paper, in this one we study the
 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


@@ -418,18 +426,17 @@ is the subject of another study not presented here.
 
 \subsection{Background idea}
 
 
 \subsection{Background idea}
 

-The  area of  interest  can be  divided  using the  divide-and-conquer
-strategy into  smaller areas, called subregions, and  then our MuDiLCO
-protocol will be implemented in each subregion in a distributed way.
+The area of  interest can be divided using  the divide-and-conquer strategy into
+smaller  areas,  called  subregions,  and  then our  MuDiLCO  protocol  will  be
+implemented in each subregion in a distributed way.

 
 

-As can  be seen  in Figure~\ref{fig2}, our  protocol works  in periods
-fashion, where  each is  divided into 4  phases: Information~Exchange,
-Leader~Election,  Decision, and  Sensing.  Each  sensing phase  may be
-itself divided  into $T$ rounds  and for each  round a set  of sensors
-(said a cover set) is responsible for the sensing task.
+As  can be seen  in Figure~\ref{fig2},  our protocol  works in  periods fashion,
+where  each is  divided  into 4  phases: Information~Exchange,  Leader~Election,
+Decision, and Sensing.  Each sensing phase may be itself divided into $T$ rounds
+and for each  round a set of sensors  (said a cover set) is  responsible for the
+sensing task.

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

-\centering
-\includegraphics[width=95mm]{Modelgeneral.pdf} % 70mm
+\centering \includegraphics[width=100mm]{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} 


@@ -439,38 +446,36 @@ itself divided  into $T$ rounds  and for each  round a set  of sensors
 % 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 is reliable  against an unexpected node failure, because
-it works  in periods. On the one  hand, if a node  failure is detected
-before  making the  decision, the  node will  not participate  to this
-phase, and,  on the other hand,  if the node failure  occurs after the
-decision,  the  sensing  task  of  the  network  will  be  temporarily
-affected: only during the period of sensing until a new period starts.
+This protocol is  reliable against an unexpected node  failure, because it works
+in periods.  On the one hand,  if a node  failure is detected before  making the
+decision, the node  will not participate to this phase, and,  on the other hand,
+if the node  failure occurs after the decision, the sensing  task of the network
+will be  temporarily affected:  only during  the period of  sensing until  a new
+period starts.

 
 

-The energy consumption and some  other constraints can easily be taken
-into account,  since the  sensors can update  and then  exchange their
-information (including their residual energy) at the beginning of each
-period.  However, the pre-sensing phases (Information Exchange, Leader
-Election, and Decision) are energy consuming for some nodes, even when
-they do not join the network to monitor the area.
+The  energy consumption  and some  other constraints  can easily  be  taken into
+account,  since the  sensors  can  update and  then  exchange their  information
+(including their residual energy) at the beginning of each period.  However, the
+pre-sensing  phases (Information  Exchange, Leader  Election, and  Decision) are
+energy  consuming for some  nodes, even  when they  do not  join the  network to
+monitor the area.

 
 %%%%%%%%%%%%%%%%%parler optimisation%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
 %%%%%%%%%%%%%%%%%parler optimisation%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 

-We  define two  types of  packets that  will be  used by  the proposed
-protocol:
+We define two types of packets that will be used by the proposed protocol:

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

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

 \end{enumerate}
 
 There are five status for each sensor node in the network:
 \begin{enumerate}[(a)] 
 \end{enumerate}
 
 There are five status for each sensor node in the network:
 \begin{enumerate}[(a)] 

-\item LISTENING: sensor  node is waiting for a  decision (to be active
-  or not);
-\item COMPUTATION: sensor node has  been elected as leader and applies
-  the optimization process;
+\item LISTENING: sensor node is waiting for a decision (to be active or not);
+\item  COMPUTATION: sensor  node  has been  elected  as leader  and applies  the
+  optimization process;

 \item ACTIVE: sensor node participate to the monitoring of the area;
 \item SLEEP: sensor node is turned off to save energy;
 \item COMMUNICATION: sensor node is transmitting or receiving packet.
 \item ACTIVE: sensor node participate to the monitoring of the area;
 \item SLEEP: sensor node is turned off to save energy;
 \item COMMUNICATION: sensor node is transmitting or receiving packet.


@@ -494,17 +499,16 @@ 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 will be responsible  for executing the coverage algorithm.  Each
-subregion  in  the   area  of  interest  will  select   its  own  WSNL
-independently  for each  period.  All  the sensor  nodes  cooperate to
-elect a WSNL.  The nodes in  the same subregion will select the leader
-based on  the received informations from  all other nodes  in the same
-subregion.  The selection criteria are, in order of importance: larger
-number of  neighbors, larger  remaining energy, and  then in  case of
-equality, larger  index. Observations on  previous simulations suggest
-to use  the number of one-hop  neighbors as the  primary criterion to
-reduce energy consumption due to the communications.
+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 informations  from all other nodes
+in  the same subregion.   The selection  criteria are,  in order  of importance:
+larger  number  of neighbors,  larger  remaining energy,  and  then  in case  of
+equality, larger index. Observations on  previous simulations suggest to use the
+number  of  one-hop  neighbors  as   the  primary  criterion  to  reduce  energy
+consumption due to the communications.

 
 %the more priority selection factor is the number of $1-hop$ neighbors, $NBR j$, which can  minimize the energy consumption during the communication Significantly.  
 %The pseudo-code for leader election phase is provided in Algorithm~1.
 
 %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.


@@ -513,29 +517,27 @@ reduce energy consumption due to the communications.
 
 \subsection{Decision phase}
 
 
 \subsection{Decision phase}
 

-Each WSNL  will solve  an integer program  to select which  cover sets
-will  be  activated  in  the  following sensing  phase  to  cover  the
-subregion to which  it belongs.  The integer program  will produce $T$
-cover sets,  one for each round.   The WSNL will  send an Active-Sleep
-packet  to each  sensor  in  the subregion  based  on the  algorithm's
-results,  indicating if the  sensor should  be active  or not  in each
-round of the sensing phase. The  integer program is based on the model
-proposed  by  \cite{pedraza2006}  with  some modification,  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 activation  of
-sensor $j$  during the  round $t$  of a given  sensing phase.  We also
-consider  primary points  as targets.   The set  of primary  points is
-denoted by $P$ and the set of  sensors by $J$. Only sensors able to be
-alive during at least one round are involved in the integer program.
+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 modification, 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 activation of sensor
+$j$ during  the round $t$  of a given  sensing phase.  We also  consider primary
+points as targets.  The  set of primary points is denoted by  $P$ and the set of
+sensors by  $J$. Only sensors  able to  be alive during  at least one  round are
+involved in the integer program.

 
 %parler de la limite en energie Et pour un round
 
 
 %parler de la limite en energie Et pour un round
 

-For  a primary  point  $p$, let  $\alpha_{j,p}$  denote the  indicator
-function of whether the point $p$ is covered, that is:
+For a  primary point  $p$, let $\alpha_{j,p}$  denote the indicator  function of
+whether the point $p$ is covered, that is:

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


@@ -565,10 +567,10 @@ We define the Overcoverage variable $\Theta_{t,p}$ as:
 \end{array} \right.
 \label{eq13} 
 \end{equation}
 \end{array} \right.
 \label{eq13} 
 \end{equation}

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

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


@@ -609,41 +611,38 @@ U_{t,p} \in \lbrace0,1\rbrace, \hspace{10 mm}\forall p \in P, t = 1,\dots,T  \la
 %(W_{\theta}+W_{\psi} = P)    \label{eq19} 
 %\end{equation}
 
 %(W_{\theta}+W_{\psi} = P)    \label{eq19} 
 %\end{equation}
 

-

 \begin{itemize}
 \begin{itemize}

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

 \end{itemize}
 
 \end{itemize}
 

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

 %The Active-Sleep packet includes the schedule vector with the number of rounds that should be applied by the receiving sensor node during the sensing phase.
 
 \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
 %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
+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.
 
 will be  executed by each node  at the beginning  of a period, explains  how the
 Active-Sleep packet is obtained.
 


@@ -744,17 +743,25 @@ $w_{U}$ & $|P^2|$
 % is used to refer this table in the text
 \end{table}
   
 % is used to refer this table in the text
 \end{table}
   

-Our protocol is declined into four versions: MuDiLCO-1,  MuDiLCO-3, MuDiLCO-5,
+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
 and  MuDiLCO-7, corresponding  respectively to  $T=1,3,5,7$ ($T$  the  number of

-rounds  in one  sensing period).  In  the following,  the general  case will  be
-denoted by MuDiLCO-T. We are studied the impact of dividing the sensing feild (using Divide and Conquer method) on the performance of our  MuDiLCO-T protocol with different network sizes, and we are found that as the number of subregions increase, the network lifetime increase and the MuDiLCO-T protocol become more powerful against the network disconnection.
-This subdivision should be stopped when there is no benefit from the optimization, therefore Our MuDiLCO-T protocol is distributed over 16 rather than 32 subregions because there is a balance between the benefit from the optimization and the execution time is needed to sove it.    We compare MuDiLCO-T with two  other methods.  The first
-method,  called  DESK and  proposed  by  \cite{ChinhVu}  is a  full  distributed
+rounds  in one  sensing period).   In the  following, the  general case  will be
+denoted by  MuDiLCO-T and we will  make comparisons with two  other methods. The
+first method, called DESK and  proposed by \cite{ChinhVu}, is a full distributed

 coverage  algorithm.   The  second  method,  called  GAF~\cite{xu2001geography},
 consists in dividing the region  into fixed squares.  During the decision phase,
 in each  square, one sensor is then  chosen to remain active  during the sensing
 phase time.
 
 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 subdivide  the  sensing field,  considering different  network
+sizes. They show that as the number of subregions increases, so does the network
+lifetime. Moreover, it  makes the MuDiLCO-T protocol more  robust against random
+network  disconnection due  to  node failures.  However,  too much  subdivisions
+reduces the advantage  of the optimization. In fact, there  is a balance between
+the  benefit  from the  optimization  and the  execution  time  needed to  solve
+it. Therefore, we  have set the number  of subregions to 16 rather  than 32. 
+

 \subsection{Energy Model}
 
 We  use an  energy consumption  model  proposed by~\cite{ChinhVu}  and based  on
 \subsection{Energy Model}
 
 We  use an  energy consumption  model  proposed by~\cite{ChinhVu}  and based  on


@@ -826,7 +833,6 @@ energy consumed in  active state (9.72 mW)  by the time in second  for one round
 (3600 seconds).  According to the  interval of initial  energy, a sensor  may be
 alive during at most 20 rounds.
 
 (3600 seconds).  According to the  interval of initial  energy, a sensor  may be
 alive during at most 20 rounds.
 

-

 \subsection{Metrics}
 
 To evaluate our approach we consider the following performance metrics:
 \subsection{Metrics}
 
 To evaluate our approach we consider the following performance metrics:


@@ -843,7 +849,8 @@ To evaluate our approach we consider the following performance metrics:
 \end{equation*}
 where $n^t$ is  the number of covered  grid points by the active  sensors of all
 subregions during round $t$ in the current sensing phase and $N$ is total number
 \end{equation*}
 where $n^t$ is  the number of covered  grid points by the active  sensors of all
 subregions during round $t$ in the current sensing phase and $N$ is total number

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

 %The accuracy of this method depends on the distance between grids. In our
 %simulations, the sensing field has been divided into 50 by 25 grid points, which means
 %there are $51 \times 26~ = ~ 1326$ points in total.
 %The accuracy of this method depends on the distance between grids. In our
 %simulations, the sensing field has been divided into 50 by 25 grid points, which means
 %there are $51 \times 26~ = ~ 1326$ points in total.


@@ -924,16 +931,16 @@ can notice that for the first thirty rounds both DESK and GAF provide a coverage
 which is a little bit better than the  one of MuDiLCO-T. This is due to the fact
 that in  comparison with MuDiLCO that  uses optimization to put  in SLEEP status
 redundant sensors,  more sensor  nodes remain  active with DESK  and GAF.   As a
 which is a little bit better than the  one of MuDiLCO-T. This is due to the fact
 that in  comparison with MuDiLCO that  uses optimization to put  in SLEEP status
 redundant sensors,  more sensor  nodes remain  active with DESK  and GAF.   As a

-consequence,  when the  number of  rounds increases,  a larger  number  of nodes
+consequence,  when the  number  of rounds  increases,  a larger  number of  node

 failures can be observed in DESK and  GAF, resulting in a faster decrease of the
 coverage ratio.  Furthermore,  our protocol allows to maintain  a coverage ratio
 failures can be observed in DESK and  GAF, resulting in a faster decrease of the
 coverage ratio.  Furthermore,  our protocol allows to maintain  a coverage ratio

-greater than  50\% for  far more rounds.  Overall, the proposed  sensor activity
+greater than  50\% for far more  rounds.  Overall, the  proposed sensor activity

 scheduling based on optimization in  MuDiLCO maintains higher coverage ratios of
 the area of interest for a larger number of rounds. It also means that MuDiLCO-T
 scheduling based on optimization in  MuDiLCO maintains higher coverage ratios of
 the area of interest for a larger number of rounds. It also means that MuDiLCO-T

-save more  energy, with less  dead nodes, at  most for several rounds,  and thus
+saves more  energy, with less dead nodes,  at most for several  rounds, and thus

 should extend the network lifetime.
 
 should extend the network lifetime.
 

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

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


@@ -954,7 +961,7 @@ Obviously, in  that case DESK  and GAF have  less active nodes, since  they have
 activated many nodes at the beginning. Anyway, MuDiLCO-T activates the available
 nodes in a more efficient manner.
 
 activated many nodes at the beginning. Anyway, MuDiLCO-T activates the available
 nodes in a more efficient manner.
 

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

 \centering
 \includegraphics[scale=0.5]{R1/ASR.pdf}  
 \caption{Active sensors ratio for 150 deployed nodes}
 \centering
 \includegraphics[scale=0.5]{R1/ASR.pdf}  
 \caption{Active sensors ratio for 150 deployed nodes}


@@ -977,7 +984,7 @@ still connected.
 
 %%% The optimization effectively continues as long as a network in a subregion is still connected. A VOIR %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
 %%% The optimization effectively continues as long as a network in a subregion is still connected. A VOIR %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 

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

 \centering
 \includegraphics[scale=0.5]{R1/SR.pdf} 
 \caption{Cumulative percentage of stopped simulation runs for 150 deployed nodes }
 \centering
 \includegraphics[scale=0.5]{R1/SR.pdf} 
 \caption{Cumulative percentage of stopped simulation runs for 150 deployed nodes }


@@ -1022,55 +1029,53 @@ sensors to consider in the integer program.
 
 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
 
 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 (times  needed to  solve optimization problem)  for different  values of
-$T$.   The original  execution time  is  computed on  a laptop  DELL with  Intel
-Core~i3~2370~M (2.4 GHz) processor (2  cores) and the MIPS (Million Instructions
-Per Second) rate equal to 35330. To  be consistent with the use of a sensor node
-with Atmels AVR ATmega103L microcontroller (6 MHz) and a MIPS rate equal to 6 to
-run  the optimization  resolution, this  time  is multiplied  by 2944.2  $\left(
+seconds (needed to solve optimization problem) for different values of $T$.  The
+original execution time  is computed on a laptop  DELL with Intel Core~i3~2370~M
+(2.4 GHz)  processor (2  cores) and the  MIPS (Million Instructions  Per Second)
+rate equal to 35330. To be consistent  with the use of a sensor node with Atmels
+AVR ATmega103L  microcontroller (6 MHz) and  a MIPS rate  equal to 6 to  run the
+optimization   resolution,   this  time   is   multiplied   by  2944.2   $\left(

 \frac{35330}{2} \times  \frac{1}{6} \right)$ and  reported on Figure~\ref{fig77}
 \frac{35330}{2} \times  \frac{1}{6} \right)$ and  reported on Figure~\ref{fig77}

-for different network sizes. 
+for different network sizes.

 
 

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

 \centering
 \includegraphics[scale=0.5]{R1/T.pdf}  
 \caption{Execution Time (in seconds)}
 \label{fig77}
 \end{figure} 
 
 \centering
 \includegraphics[scale=0.5]{R1/T.pdf}  
 \caption{Execution Time (in seconds)}
 \label{fig77}
 \end{figure} 
 

-As expected,  the execution time  increases with the number  of rounds
-$T$ taken into account for  scheduling of the sensing phase. The times
-obtained for $T=1,3$ or $5$  seems bearable, but for $T=7$ they become
-quickly  unsuitable for  a  sensor node,  especially  when the  sensor
-network  size increases.  Again,  we can  notice  that if  we want  to
-schedule the nodes activities for a large number of rounds, we need to
-choose a relevant number of  subregion in order to avoid a complicated
-and cumbersome  optimization. On the one  hand, a large  value for $T$
-permits  to reduce the  energy-overhead due  to the  three pre-sensing
-phases,  on the  other hand  a leader  node may  waste  a considerable
-amount of energy to solve the optimization problem.
+As expected,  the execution time increases  with the number of  rounds $T$ taken
+into account for scheduling of the sensing phase. The times obtained for $T=1,3$
+or $5$ seems bearable, but for $T=7$ they become quickly unsuitable for a sensor
+node, especially when  the sensor network size increases.   Again, we can notice
+that if we want  to schedule the nodes activities for a  large number of rounds,
+we need to choose a relevant number of subregion in order to avoid a complicated
+and cumbersome optimization.  On the one hand, a large value  for $T$ permits to
+reduce the  energy-overhead due  to the three  pre-sensing phases, on  the other
+hand  a leader  node may  waste a  considerable amount  of energy  to  solve the
+optimization problem.

 
 %While MuDiLCO-1, 3, and 5 solves the optimization process with suitable execution times to be used on wireless sensor network because it distributed on larger number of small subregions as well as it is used acceptable number of round(s) T.  We think that in distributed fashion the solving of the optimization problem to produce T rounds in a subregion can be tackled by sensor nodes. Overall, to be able to deal with very large networks, a distributed method is clearly required.
 
 \subsection{Network Lifetime}
 
 
 %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}
 

-The  next   two  figures,  Figures~\ref{fig8}(a)   and  \ref{fig8}(b),
-illustrate   the  network  lifetime   for  different   network  sizes,
-respectively  for $Lifetime_{95}$  and $Lifetime_{50}$.   Both figures
-show that the  network lifetime increases together with  the number of
-sensor nodes, whatever the protocol,  thanks to the node density which
-result in  more and more redundant  nodes that can  be deactivated and
-thus  save energy.  Compared  to the  other approaches,  our MuDiLCO-T
-protocol  maximizes the lifetime  of the  network.  In  particular the
-gain in  lifetime for a coverage  over 95\% is greater  than 38\% when
-switching from  GAF to  MuDiLCO-3.  The slight  decrease that  can bee
-observed for MuDiLCO-7 in  case of $Lifetime_{95}$ with large wireless
-sensor networks result from the difficulty of the optimization problem
-to be solved  by the integer program.  This  point was already noticed
-in subsection \ref{subsec:EC} devoted to the energy consumption, since
-network lifetime and energy consumption are directly linked.
-
-\begin{figure}[h!]
+The next  two figures,  Figures~\ref{fig8}(a) and \ref{fig8}(b),  illustrate the
+network lifetime  for different network sizes,  respectively for $Lifetime_{95}$
+and  $Lifetime_{50}$.  Both  figures show  that the  network  lifetime increases
+together with the  number of sensor nodes, whatever the  protocol, thanks to the
+node  density  which  result in  more  and  more  redundant  nodes that  can  be
+deactivated  and  thus save  energy.   Compared  to  the other  approaches,  our
+MuDiLCO-T protocol  maximizes the  lifetime of the  network.  In  particular the
+gain in  lifetime for a coverage over  95\% is greater than  38\% when switching
+from GAF to MuDiLCO-3.  The slight  decrease that can bee observed for MuDiLCO-7
+in case of  $Lifetime_{95}$ with large wireless sensor  networks result from the
+difficulty  of the optimization  problem to  be solved  by the  integer program.
+This  point was  already noticed  in subsection  \ref{subsec:EC} devoted  to the
+energy consumption,  since network lifetime and energy  consumption are directly
+linked.
+
+\begin{figure}[t!]

   \centering
   \begin{tabular}{cl}
     \parbox{9.5cm}{\includegraphics[scale=0.5]{R1/LT95.pdf}} & (a) \\
   \centering
   \begin{tabular}{cl}
     \parbox{9.5cm}{\includegraphics[scale=0.5]{R1/LT95.pdf}} & (a) \\


@@ -1093,43 +1098,45 @@ network lifetime and energy consumption are directly linked.
 \section{Conclusion and Future Works}
 \label{sec:conclusion}
 
 \section{Conclusion and Future Works}
 \label{sec:conclusion}
 

-In this paper,  we have addressed the problem of  the coverage and the
-lifetime optimization in wireless sensor networks. This is a key issue
-as sensor nodes have limited resources in terms of memory, energy, and
-computational power. To  cope with this problem, the  field of sensing
-is   divided   into   smaller   subregions  using   the   concept   of
-divide-and-conquer  method,  and  then  we propose  a  protocol  which
-optimizes coverage  and lifetime  performances in each  subregion. Our
-protocol,  called MuDiLCO  (Multiperiod Distributed  Lifetime Coverage
-Optimization)  combines  two   efficient  techniques:  network  leader
-election and sensor activity scheduling.
+In this  paper, we have addressed the  problem of the coverage  and the lifetime
+optimization in  wireless sensor networks. This  is a key issue  as sensor nodes
+have limited resources  in terms of memory, energy,  and computational power. To
+cope with this problem, the field  of sensing is divided into smaller subregions
+using the concept  of divide-and-conquer method, and then  we propose a protocol
+which  optimizes  coverage and  lifetime  performances  in  each subregion.  Our
+protocol,   called   MuDiLCO    (Multiperiod   Distributed   Lifetime   Coverage
+Optimization)  combines two  efficient techniques:  network leader  election and
+sensor activity scheduling.

 %,  where the challenges
 %include how to select the  most efficient leader in each subregion and
 %the best cover sets %of active nodes that will optimize the network lifetime
 %while taking the responsibility of covering the corresponding
 %subregion using more than one cover set during the sensing phase. 
 %,  where the challenges
 %include how to select the  most efficient leader in each subregion and
 %the best cover sets %of active nodes that will optimize the network lifetime
 %while taking the responsibility of covering the corresponding
 %subregion using more than one cover set during the sensing phase. 

-The activity scheduling in each subregion works in periods, where each
-period consists of four  phases: (i) Information Exchange, (ii) Leader
-Election, (iii)  Decision Phase  to plan the  activity of  the sensors
-over $T$ rounds (iv) Sensing Phase itself divided into T rounds.
-
-Simulations  results show the  relevance of  the proposed  protocol in
-terms  of  lifetime,  coverage  ratio, active  sensors  ratio,  energy
-consumption, execution time. Indeed,  when dealing with large wireless
-sensor networks, a distributed approach like the one we propose allows
-to reduce  the difficulty of  a single global optimization  problem by
-partitioning it in many smaller  problems, one per subregion, that can
-be solved more easily. Nevertheless,  results also show that it is not
-possible to plan the activity of sensors over too many rounds, because
-the resulting  optimization problem leads to too  high resolution time
-and thus to an excessive energy consumption.
+The activity  scheduling in each subregion  works in periods,  where each period
+consists of four  phases: (i) Information Exchange, (ii)  Leader Election, (iii)
+Decision Phase to plan the activity  of the sensors over $T$ rounds (iv) Sensing
+Phase itself divided into T rounds.
+
+Simulations  results show the  relevance of  the proposed  protocol in  terms of
+lifetime, coverage  ratio, active  sensors ratio, energy  consumption, execution
+time. Indeed,  when dealing with  large wireless sensor networks,  a distributed
+approach like  the one we  propose allows to  reduce the difficulty of  a single
+global optimization problem by partitioning it in many smaller problems, one per
+subregion, that can be solved  more easily. Nevertheless, results also show that
+it is not possible to plan the activity of sensors over too many rounds, because
+the resulting optimization problem leads to too high resolution time and thus to
+an excessive energy consumption.

 
 %In  future work, we plan  to study and propose adjustable sensing range coverage optimization protocol, which computes  all active sensor schedules in one time, by using
 %optimization  methods. This protocol can prolong the network lifetime by minimizing the number of the active sensor nodes near the borders by optimizing the sensing range of sensor nodes.
 % use section* for acknowledgement
 
 \section*{Acknowledgment}
 
 %In  future work, we plan  to study and propose adjustable sensing range coverage optimization protocol, which computes  all active sensor schedules in one time, by using
 %optimization  methods. This protocol can prolong the network lifetime by minimizing the number of the active sensor nodes near the borders by optimizing the sensing range of sensor nodes.
 % use section* for acknowledgement
 
 \section*{Acknowledgment}

-As a Ph.D. student, Ali Kadhum IDREES would like to gratefully acknowledge the University of Babylon - IRAQ for the financial support and in the same time would like to acknowledge Campus France (The French national agency for the promotion of higher education, international student services, and international mobility) and University of Franche-Comt\'e - FRANCE for all the support in FRANCE.
+As a Ph.D.  student, Ali Kadhum IDREES would like  to gratefully acknowledge the
+University  of Babylon  - Iraq  for the  financial support,  Campus  France (The
+French  national agency  for the  promotion of  higher  education, international
+student   services,  and   international  mobility),   and  the   University  of
+Franche-Comt\'e - France for all the support in France.

 
 %% \linenumbers
 
 
 %% \linenumbers