modif related work de Michel

[Sensornets15.git] / Example.tex

diff --git a/Example.tex b/Example.tex
index 05e4f23e16a1e70a8bd5c8f0041401c26e62dc9e..ac05f6c8b3c28e7f090ca1386e6392f3defd0aeb 100644 (file)
--- a/Example.tex
+++ b/Example.tex
@@ -27,7 +27,7 @@
 \title{Distributed Lifetime Coverage Optimization Protocol \\in Wireless Sensor Networks}
 
 \author{\authorname{Ali Kadhum Idrees, Karine Deschinkel, Michel Salomon, and Rapha\"el Couturier}
 \title{Distributed Lifetime Coverage Optimization Protocol \\in Wireless Sensor Networks}
 
 \author{\authorname{Ali Kadhum Idrees, Karine Deschinkel, Michel Salomon, and Rapha\"el Couturier}

-\affiliation{FEMTO-ST Institute, UMR 6174 CNRS, University of Franche-Comte, Belfort, France}
+\affiliation{FEMTO-ST Institute, UMR 6174 CNRS, University of Franche-Comt\'e, Belfort, France}

 %\affiliation{\sup{2}Department of Computing, Main University, MySecondTown, MyCountry}
 \email{ali.idness@edu.univ-fcomte.fr, $\lbrace$karine.deschinkel, michel.salomon, raphael.couturier$\rbrace$@univ-fcomte.fr}
 %\email{\{f\_author, s\_author\}@ips.xyz.edu, t\_author@dc.mu.edu}
 %\affiliation{\sup{2}Department of Computing, Main University, MySecondTown, MyCountry}
 \email{ali.idness@edu.univ-fcomte.fr, $\lbrace$karine.deschinkel, michel.salomon, raphael.couturier$\rbrace$@univ-fcomte.fr}
 %\email{\{f\_author, s\_author\}@ips.xyz.edu, t\_author@dc.mu.edu}


@@ -53,7 +53,7 @@ Optimization, Scheduling.}
   to ensure  coverage at  a low  energy cost, allowing  to optimize  the network
   lifetime. More  precisely, a period  consists of four  phases: (i)~Information
   Exchange,  (ii)~Leader   Election,  (iii)~Decision,  and   (iv)~Sensing.   The
   to ensure  coverage at  a low  energy cost, allowing  to optimize  the network
   lifetime. More  precisely, a period  consists of four  phases: (i)~Information
   Exchange,  (ii)~Leader   Election,  (iii)~Decision,  and   (iv)~Sensing.   The

-  decision process,  which result in  an activity scheduling vector,  is carried
+  decision process, which results in  an activity scheduling vector,  is carried

   out by a leader node through  the solving of an integer program. In comparison
   with  some other  protocols, the  simulations  done using  the discrete  event
   simulator OMNeT++ show that our approach  is able to increase the WSN lifetime
   out by a leader node through  the solving of an integer program. In comparison
   with  some other  protocols, the  simulations  done using  the discrete  event
   simulator OMNeT++ show that our approach  is able to increase the WSN lifetime


@@ -63,37 +63,39 @@ Optimization, Scheduling.}
 
 \section{\uppercase{Introduction}}
 \label{sec:introduction}
 
 \section{\uppercase{Introduction}}
 \label{sec:introduction}

+

 \noindent 
 Energy efficiency is  a crucial issue in wireless  sensor networks since sensory
 \noindent 
 Energy efficiency is  a crucial issue in wireless  sensor networks since sensory

-consumption,  in order  to maximize  the network  lifetime, represent  the major
+consumption, in  order to  maximize the network  lifetime, represents  the major

 difficulty when designing WSNs. As a consequence, one of the scientific research
 challenges in  WSNs, which has  been addressed by  a large amount  of literature
 during the  last few  years, is  the design of  energy efficient  approaches for
 coverage and connectivity~\cite{conti2014mobile}.   Coverage reflects how well a
 difficulty when designing WSNs. As a consequence, one of the scientific research
 challenges in  WSNs, which has  been addressed by  a large amount  of literature
 during the  last few  years, is  the design of  energy efficient  approaches for
 coverage and connectivity~\cite{conti2014mobile}.   Coverage reflects how well a

-sensor field is  monitored. On the one  hand we want to monitor the area  of interest in the most
-efficient way~\cite{Nayak04}. On the other hand we want to use as less energy as
-possible.  Sensor nodes  are  battery-powered  with no  means  of recharging  or
-replacing, usually due to environmental (hostile or unpractical environments) or
-cost reasons.  Therefore, it  is desired  that the WSNs  are deployed  with high
-densities so as to exploit the  overlapping sensing regions of some sensor nodes
-to save energy by  turning off some of them during the  sensing phase to prolong
-the network lifetime.
+sensor  field is  monitored. On  the one  hand we  want to  monitor the  area of
+interest in the most efficient way~\cite{Nayak04}.  On the other hand we want to
+use as less energy as possible.   Sensor nodes are battery-powered with no means
+of recharging or replacing, usually due to environmental (hostile or unpractical
+environments)  or cost  reasons.  Therefore,  it is  desired that  the  WSNs are
+deployed with high densities so as to exploit the overlapping sensing regions of
+some sensor nodes to save energy by  turning off some of them during the sensing
+phase to prolong the network lifetime.

 
 In this  paper we design  a protocol that  focuses on the area  coverage problem
 with  the objective  of maximizing  the network  lifetime. Our  proposition, the
 
 In this  paper we design  a protocol that  focuses on the area  coverage problem
 with  the objective  of maximizing  the network  lifetime. Our  proposition, the

-DiLCO protocol,  maintains the coverage and  improves the lifetime  in WSNs. The
-area of  interest is  first divided into  subregions using  a divide-and-conquer
-algorithm and  an activity scheduling  for sensor nodes  is then planned  by the
-elected leader in each subregion. In fact,  the nodes in a subregion can be seen
-as a cluster where each node sends  sensing data to the cluster head or the sink
-node.  Furthermore, the  activities in a subregion/cluster can  continue even if
-another cluster  stops due to too  many node failures.  Our Distributed Lifetime
-Coverage Optimization (DILCO) protocol  considers periods, where a period starts
-with  a  discovery phase  to  exchange information  between  sensors  of a  same
+Distributed  Lifetime  Coverage  Optimization  (DILCO) protocol,  maintains  the
+coverage  and improves  the lifetime  in  WSNs. The  area of  interest is  first
+divided  into subregions using  a divide-and-conquer  algorithm and  an activity
+scheduling  for sensor  nodes is  then  planned by  the elected  leader in  each
+subregion. In fact, the nodes in a subregion can be seen as a cluster where each
+node sends sensing data to the  cluster head or the sink node.  Furthermore, the
+activities in a subregion/cluster can continue even if another cluster stops due
+to too many node failures.  Our DiLCO protocol considers periods, where a period
+starts with a discovery phase to  exchange information between sensors of a same

 subregion, in order to choose in a suitable manner a sensor node (the leader) to
 carry out the coverage strategy. In each subregion the activation of the sensors
 for the  sensing phase of the current  period is obtained by  solving an integer
 subregion, in order to choose in a suitable manner a sensor node (the leader) to
 carry out the coverage strategy. In each subregion the activation of the sensors
 for the  sensing phase of the current  period is obtained by  solving an integer

-program.
+program. The  resulting activation  vector is broadcasted  by a leader  to every
+node of its subregion.

 
 The remainder  of the  paper continues with  Section~\ref{sec:Literature Review}
 where a  review of some related  works is presented. The  next section describes
 
 The remainder  of the  paper continues with  Section~\ref{sec:Literature Review}
 where a  review of some related  works is presented. The  next section describes


@@ -105,65 +107,124 @@ in Section~\ref{sec:Conclusion and Future Works}.
 
 \section{\uppercase{Literature Review}}
 \label{sec:Literature Review}
 
 \section{\uppercase{Literature Review}}
 \label{sec:Literature Review}

-\noindent In this section, we summarize some related works regarding coverage problem , and distinguish our DiLCO protocol from the works presented in the literature.\\
+
+\noindent In  this section, we  summarize some related works  regarding coverage
+problem  and distinguish  our DiLCO  protocol from  the works  presented  in the
+literature.
+

 The most discussed coverage  problems in literature
 The most discussed coverage  problems in literature

-can  be classified into  three types  \cite{li2013survey}: area  coverage (where
-every point inside an area is  to be monitored), target coverage (where the main
-objective is to  cover only a finite number of  discrete points called targets),
-and  barrier coverage (to  prevent intruders  from entering  into the  region of
-interest). 
-{\it In DiLCO protocol, the area coverage, ie the coverage
-of every point in the sensing region, is transformed to the coverage of a fraction of points called primary points. }
-
-The major approach to extend network lifetime while preserving coverage is to divide/organize the sensors into a suitable number of set covers (disjoint or non-disjoint) where each set completely covers an interest region and to activate these set covers successively. The network activity can be planned in advance and scheduled for the entire network lifetime  or organized in periods, and the set of
-active sensor nodes is decided at the beginning of each period.
-Active node selection is determined based on the problem
-requirements (e.g. area monitoring, connectivity, power
-efficiency). Different methods has been proposed in literature.
-
-{\it DiLCO protocol works in periods, each period contains a preliminary phase for information exchange and decisions, followed by a sensing phase  where 
-one  cover  set  is in charge  of  the  sensing  task.}
-
-Various approaches, including centralised, distributed and localized algorithms, have been proposed to extend the network lifetime. 
+can  be classified into  three types  \cite{li2013survey}: area  coverage \cite{Misra} where
+every point inside an area is  to be monitored, target coverage  \cite{yang2014novel} where the main
+objective is to  cover only a finite number of  discrete points called targets,
+and  barrier coverage \cite{Kumar:2005}\cite{kim2013maximum} to  prevent intruders  from entering  into the  region of interest. In \cite{Deng2012} authors transform the area coverage problem to the target coverage problem taking into account the intersection points among disks of sensors nodes or between disk of sensor nodes and boundaries. 
+{\it In DiLCO  protocol, the area coverage, i.e. the coverage  of every point in
+  the sensing  region, is transformed  to the coverage  of a fraction  of points
+  called primary points. }
+
+
+The major  approach to extend network  lifetime while preserving  coverage is to
+divide/organize the  sensors into a suitable  number of set  covers (disjoint or
+non-disjoint),  where each  set completely  covers a  region of  interest,  and to
+activate these set  covers successively. The network activity  can be planned in
+advance and scheduled  for the entire network lifetime  or organized in periods,
+and the set of  active sensor nodes is decided at the  beginning of each period \cite{ling2009energy}.
+Active node selection is determined based on the problem requirements (e.g. area
+monitoring,  connectivity,  power   efficiency). For instance, Jaggi et al. \cite{jaggi2006}
+address the problem of maximizing network lifetime by dividing sensors into the maximum number of disjoint subsets such that each subset can ensure both coverage and connectivity. A greedy algorithm is applied once to solve this problem and the computed sets are activated in succession to achieve the desired network lifetime. 
+Vu \cite{chin2007}, Padmatvathy et al. \cite{pc10}, propose algorithms working in a periodic fashion where a cover set is computed at the beginning of each period.
+{\it Motivated by these works, DiLCO protocol  works in periods, where each  period contains a preliminary
+  phase  for information  exchange and  decisions, followed  by a  sensing phase
+  where one cover set is in charge of the sensing task.}
+
+Various   approaches,   including   centralized,  or distributed
+algorithms, have been proposed to extend the network lifetime.

 %For instance, in order to hide the occurrence of faults, or the sudden unavailability of
 %sensor nodes, some distributed algorithms have been developed in~\cite{Gallais06,Tian02,Ye03,Zhang05,HeinzelmanCB02}. 
 %For instance, in order to hide the occurrence of faults, or the sudden unavailability of
 %sensor nodes, some distributed algorithms have been developed in~\cite{Gallais06,Tian02,Ye03,Zhang05,HeinzelmanCB02}. 

+In       distributed      algorithms~\cite{yangnovel,ChinhVu,qu2013distributed},
+information  is   disseminated  throughout   the  network  and   sensors  decide
+cooperatively by communicating with their neighbors which of them will remain in
+sleep    mode   for    a   certain    period   of    time.     The   centralized
+algorithms~\cite{cardei2005improving,zorbas2010solving,pujari2011high}     always
+provide nearly or close to optimal  solution since the algorithm has global view
+of the whole  network. But such a method has the  disadvantage of requiring high
+communication costs,  since the  node (located at  the base station)  making the
+decision needs information from all the sensor nodes in the area and the amount of information can be huge.
+{\it  In order to be suitable for large-scale network,  in the DiLCO  protocol,  the area  coverage  is divided  into several  smaller
+  subregions, and in  each of one, a  node called the leader is  in charge for
+  selecting the active sensors for the current period.}
+
+A large  variety of coverage scheduling  algorithms have been  developed. Many of
+the existing  algorithms, dealing with the  maximization of the  number of cover
+sets, are heuristics.  These heuristics  involve the construction of a cover set
+by including in priority the sensor  nodes which cover critical targets, that is
+to  say targets  that  are covered  by  the smallest  number  of sensors \cite{berman04,zorbas2010solving}.  Other
+approaches  are based  on  mathematical programming  formulations~\cite{cardei2005energy,5714480,pujari2011high,Yang2014} and  dedicated
+techniques (solving with a branch-and-bound algorithms available in optimization
+solver).  The problem is formulated  as an optimization problem (maximization of
+the  lifetime  or  number  of  cover  sets) under  target  coverage  and  energy
+constraints.   Column  generation techniques,  well-known  and widely  practiced
+techniques for solving  linear programs with too many  variables, have been also
+used~\cite{castano2013column,rossi2012exact,deschinkel2012column}. {\it  In DiLCO  protocol,  each leader,  in  each subregion,  solves an  integer
+  program with a double objective  consisting in minimizing the overcoverage and
+  limiting  the  undercoverage.  This  program  is inspired  from  the  work  of
+  \cite{pedraza2006}  where the  objective is  to maximize  the number  of cover
+  sets.}
+
+% ***** Part which must be rewritten - Start
+
+% Start of Ali's papers catalog => there's no link between them or with our work
+% (use of subregions; optimization based method; etc.)
+\iffalse
+Diongue  and  Thiare~\cite{diongue2013alarm}  proposed  an  energy  aware  sleep
+scheduling  algorithm  for lifetime  maximization  in  wireless sensor  networks
+(ALARM).  The proposed approach permits to schedule redundant nodes according to
+the weibull distribution.  This work did not analyze the  ALARM scheme under the
+coverage problem.
+
+Shi et al.~\cite{shi2009} modeled the Area Coverage Problem (ACP), which will be
+changed  into a  set coverage  problem. By  using this  model, they  proposed an
+Energy-Efficient central-Scheduling  greedy algorithm, which  can reduces energy
+consumption and increases network lifetime, by selecting a appropriate subset of
+sensor nodes to support the networks periodically.
+
+In ~\cite{chenait2013distributed},  the authors presented  a coverage-guaranteed
+distributed  sleep/wake scheduling  scheme so  ass to  prolong  network lifetime
+while guaranteeing network coverage. This scheme mitigates scheduling process to
+be more stable by avoiding  useless transitions between states without affecting
+the coverage level required by the application.
+
+The work  in~\cite{cheng2014achieving} presented a  unified sensing architecture
+for duty  cycled sensor  networks, called uSense,  which comprises  three ideas:
+Asymmetric Architecture, Generic Switching  and Global Scheduling. The objective
+is to provide a flexible and efficient coverage in sensor networks.
+
+In~\cite{ling2009energy},  the  lifetime  of  a  sensor  node  is  divided  into
+epochs. At  each epoch,  the base station  deduces the current  sensing coverage
+requirement  from application  or user  request. It  then applies  the heuristic
+algorithm in order to produce the set  of active nodes which take the mission of
+sensing during the current epoch.  After  that, the produced schedule is sent to
+the sensor nodes in the network.
+
+% What is the link between the previous work and this paragraph about DiLCO ?
+
+
+
+Yang et al.~\cite{yang2014energy} investigated  full area coverage problem under
+the probabilistic  sensing model in the  sensor networks. They  have studied the
+relationship between the coverage of two adjacent points mathematically and then
+convert  the problem of  full area  coverage into  point coverage  problem. They
+proposed $\varepsilon$-full area coverage optimization (FCO) algorithm to select
+a subset of sensors to provide  probabilistic area coverage dynamically so as to
+extend the network lifetime.
+
+The work proposed by  \cite{qu2013distributed} considers the coverage problem in
+WSNs where  each sensor has variable  sensing radius. The final  objective is to
+maximize the network coverage lifetime in WSNs.
+\fi
+% Same remark, no link with the two previous citations...

 
 

-In distributed algorithms~\cite{yangnovel,ChinhVu,qu2013distributed}, information is disseminated throughout the network and sensors decide cooperatively by communicating with their neighbours which of them will remain in sleep mode for a certain period of time. 
-The centralized algorithms~\cite{cardei2005improving,zorbas2010solving,pujari2011high} always provide nearly
-or close  to optimal solution since the  algorithm has global view  of the whole
-network, but such a method has the disadvantage of requiring 
-high communication costs,  since the  node (located at the base station) making the decision  needs information from all the  sensor nodes in the area. 
-
-A large variety of coverage scheduling algorithms have been proposed in the literature. Many of the existing algorithms, dealing with the maximisation of the number of cover sets, are heuristics. These heuristics involve the construction of a cover set by including in priority the sensor nodes which cover critical targets, that is to say targets that are covered by the smallest number of sensors. Other approaches are based on mathematical programming formulations and dedicated techniques (solving with a branch-and-bound algorithms available in optimization solver). The problem is formulated as an optimization problem (maximization of the lifetime, of the number of cover sets) under target coverage and energy constraints. Column  generation techniques,  well-known and widely practiced techniques for solving linear programs with too many variables, have been also used~\cite{castano2013column,rossi2012exact,deschinkel2012column}.
-
-Diongue and Thiare~\cite{diongue2013alarm} proposed an energy aware sleep scheduling algorithm for lifetime maximization in wireless sensor networks (ALARM).  The proposed approach permits to schedule redundant nodes according to the weibull distribution. This work did not analyze the ALARM scheme under the coverage problem. 
-
-Shi et al.~\cite{shi2009} modeled the Area Coverage Problem (ACP), which will be changed into a set coverage
-problem. By using this model, they are proposed  an  Energy-Efficient central-Scheduling greedy algorithm, which can reduces energy consumption and increases network lifetime, by selecting a appropriate subset of sensor nodes to support the networks periodically.
-
-In ~\cite{chenait2013distributed}, the authors presented a coverage-guaranteed distributed sleep/wake scheduling
-scheme so ass to prolong network lifetime while guaranteeing network coverage. This scheme mitigates scheduling process to be more stable by avoiding useless transitions between states without affecting the coverage level required by the application.
-
-The work in~\cite{cheng2014achieving} presented a unified sensing architecture for duty cycled sensor networks, called uSense, which comprises three ideas: Asymmetric Architecture, Generic Switching and Global Scheduling. The objective is to  provide a flexible and efficient coverage in sensor networks.
-
-In~\cite{ling2009energy}, the lifetime of
-a sensor node is divided into epochs. At each epoch, the
-base station deduces the current sensing coverage requirement
-from application or user request. It then applies the heuristic algorithm in order to produce the set of active nodes which take the mission of sensing during the current epoch.  After that, the produced schedule is sent to the sensor nodes in the network. 
-
-{\it In DiLCO protocol, the area coverage is divided into several smaller subregions, and in each of which, a node called the leader is on charge for selecting the active sensors for the current period.} 
-
-Yang et al.~\cite{yang2014energy} investigated full area coverage problem
-under the probabilistic sensing model in the sensor networks. They have studied the relationship between the
-coverage of two adjacent points mathematically and then convert the problem of full area coverage into point coverage problem. They proposed $\varepsilon$-full area coverage optimization (FCO) algorithm to select a subset
-of sensors to provide probabilistic area coverage dynamically so as to extend the network lifetime.
-
-The work in~\cite{cheng2014achieving} presented a unified sensing architecture for duty cycled sensor networks, called uSense, which comprises three ideas: Asymmetric Architecture, Generic Switching and Global Scheduling. The objective is to  provide a flexible and efficient coverage in sensor networks.
-
-The work proposed by \cite{qu2013distributed} considers the coverage problem in WSNs where each sensor has variable sensing radius. The final objective is to maximize the network coverage lifetime in WSNs.
-
-{\it In DiLCO protocol, each leader, in each subregion, solves an integer program with a double objective consisting in minimizing the overcoverage and limiting the undercoverage. This program is inspired from the work of \cite{pedraza2006} where the objective is to maximize the number of cover sets.} 

  
  

+% ***** Part which must be rewritten - End

 
 \iffalse
 
 
 \iffalse
 


@@ -557,10 +618,9 @@ We define the Overcoverage variable $\Theta_{p}$ as:
 \end{array} \right.
 \label{eq13} 
 \end{equation}
 \end{array} \right.
 \label{eq13} 
 \end{equation}

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

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


@@ -711,15 +771,18 @@ ActiveSleep  packet. To  compute the  energy  needed by  a node  to transmit  or
 receive such  packets, we  use the equation  giving the  energy spent to  send a
 1-bit-content   message  defined   in~\cite{raghunathan2002energy}   (we  assume
 symmetric  communication costs), and  we set  their respective  size to  112 and
 receive such  packets, we  use the equation  giving the  energy spent to  send a
 1-bit-content   message  defined   in~\cite{raghunathan2002energy}   (we  assume
 symmetric  communication costs), and  we set  their respective  size to  112 and

-24~bits. The energy required to send or receive a 1-bit is equal to $0.2575 mW$.
+24~bits. The energy required to send  or receive a 1-bit-content message is thus
+is equal to 0.2575 mW.

 
 Each node has an initial energy level, in Joules, which is randomly drawn in the
 
 Each node has an initial energy level, in Joules, which is randomly drawn in the

-interval  $[500-700]$.   If  it's  energy   provision  reaches  a   value  below
-$E_{th}=36$~Joules, the minimum  energy needed for a node  to stay active during
-one period,  it will no  more participate in  the coverage task. This  value has
-been computed  by multiplying the energy  consumed in active state  (9.72 mW) by
-the time in  second for one round (3600 seconds).  According  to the interval of
-initial energy, a sensor may be active during at most 20 rounds.
+interval  $[500-700]$.  If  it's  energy  provision reaches  a  value below  the
+threshold  $E_{th}=36$~Joules, the  minimum energy  needed  for a  node to  stay
+active during one period, it will no more participate in the coverage task. This
+value  corresponds  to the  energy  needed by  the  sensing  phase, obtained  by
+multiplying the energy consumed in active  state (9.72 mW) by the time in seconds
+for one period (3600 seconds), and  adding the energy for the pre-sensing phases.
+According to  the interval of initial energy,  a sensor may be  active during at
+most 20 rounds.

 
 In the simulations,  we introduce the following performance  metrics to evaluate
 the efficiency of our approach:
 
 In the simulations,  we introduce the following performance  metrics to evaluate
 the efficiency of our approach:


@@ -790,14 +853,15 @@ by the whole network in the sensing phase (active and sleeping nodes).
 
 
 \iffalse 
 
 
 \iffalse 

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

  
  

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

 
 \fi
 
 
 \fi
 


@@ -831,11 +895,11 @@ compared to DiLCO  in the first thirty periods. This can  be easily explained by
 the number of  active nodes: the optimization process  of our protocol activates
 less nodes  than DESK  or GAF, resulting  in a  slight decrease of  the coverage
 ratio. In case of DiLCO-2  (respectively DiLCO-4), the coverage ratio exhibits a
 the number of  active nodes: the optimization process  of our protocol activates
 less nodes  than DESK  or GAF, resulting  in a  slight decrease of  the coverage
 ratio. In case of DiLCO-2  (respectively DiLCO-4), the coverage ratio exhibits a

-fast decrease with  the number of periods and reaches zero  value in period {\bf
-  18} (respectively {\bf 46}), whereas the  other versions of DiLCO, DESK, and GAF
-ensure a coverage  ratio above 50\% for subsequent periods.  We believe that the
-results obtained with  these two methods can be explained  by a high consumption
-of energy and we will check this assumption in the next subsection.
+fast decrease  with the number  of periods and  reaches zero value  in period~18
+(respectively 46), whereas  the other versions of DiLCO, DESK,  and GAF ensure a
+coverage ratio above  50\% for subsequent periods.  We  believe that the results
+obtained with these two methods can be explained by a high consumption of energy
+and we will check this assumption in the next subsection.

 
 Concerning  DiLCO-8, DiLCO-16,  and  DiLCO-32,  these methods  seem  to be  more
 efficient than DESK  and GAF, since they can provide the  same level of coverage
 
 Concerning  DiLCO-8, DiLCO-16,  and  DiLCO-32,  these methods  seem  to be  more
 efficient than DESK  and GAF, since they can provide the  same level of coverage


@@ -971,12 +1035,12 @@ the performance of our approach, we  compared it with two other approaches using
 many performance metrics  like coverage ratio or network  lifetime. We have also
 study the  impact of the  number of subregions  chosen to subdivide the  area of
 interest,  considering  different  network  sizes.  The  experiments  show  that
 many performance metrics  like coverage ratio or network  lifetime. We have also
 study the  impact of the  number of subregions  chosen to subdivide the  area of
 interest,  considering  different  network  sizes.  The  experiments  show  that

-increasing the  number of subregions allows  to improves the  lifetime. The more
-there  are   subregions,  the  more   the  network  is  robust   against  random
-disconnection resulting from dead nodes.  However, for a given sensing field and
-network size  there is an optimal  number of subregions.  Therefore,  in case of
-our simulation  context a  subdivision in $16$~subregions  seems to be  the most
-relevant. The optimal number of subregions will be investigated in the future.
+increasing the  number of subregions improves  the lifetime. The  more there are
+subregions,  the  more  the  network  is  robust  against  random  disconnection
+resulting from dead nodes.  However, for  a given sensing field and network size
+there is an optimal number of  subregions.  Therefore, in case of our simulation
+context  a subdivision in  $16$~subregions seems  to be  the most  relevant. The
+optimal number of subregions will be investigated in the future.

 
 \iffalse
 \noindent In this paper, we have  addressed the problem of the coverage and the lifetime
 
 \iffalse
 \noindent In this paper, we have  addressed the problem of the coverage and the lifetime