Last Version after correction

[UIC2013.git] / bare_conf.tex

diff --git a/bare_conf.tex b/bare_conf.tex
old mode 100755 (executable)
new mode 100644 (file)
index b9ed86e..80629a9
--- a/bare_conf.tex
+++ b/bare_conf.tex
@@ -1,4 +1,3 @@
- 

 \documentclass[conference]{IEEEtran}
 
 \ifCLASSINFOpdf
 \documentclass[conference]{IEEEtran}
 
 \ifCLASSINFOpdf


@@ -9,7 +8,7 @@
 
 \hyphenation{op-tical net-works semi-conduc-tor}
 
 
 \hyphenation{op-tical net-works semi-conduc-tor}
 

-\usepackage{float}
+\usepackage{float} 

 \usepackage{epsfig}
 \usepackage{calc}
  \usepackage{times,amssymb,amsmath,latexsym}
 \usepackage{epsfig}
 \usepackage{calc}
  \usepackage{times,amssymb,amsmath,latexsym}


@@ -30,17 +29,25 @@
 \usepackage{epsfig}
 \usepackage{caption}
 \usepackage{multicol}
 \usepackage{epsfig}
 \usepackage{caption}
 \usepackage{multicol}

-
+\usepackage{times}
+\usepackage{graphicx,epstopdf}
+\epstopdfsetup{suffix=}
+\DeclareGraphicsExtensions{.ps}
+\DeclareGraphicsRule{.ps}{pdf}{.pdf}{`ps2pdf -dEPSCrop -dNOSAFER #1 \noexpand\OutputFile}

 
 \begin{document}
 
 
 \begin{document}
 

-\title{Energy-Efficient Activity Scheduling in Heterogeneous Energy Wireless Sensor Networks}
+%\title{ Coverage and Lifetime Optimization in Heterogeneous Energy Wireless Sensor Networks} 
+\title{Coverage and Lifetime Optimization in Heterogeneous Energy Wireless Sensor Networks}
+%Activity Scheduling for Coverage and Lifetime Optimization in  Wireless Sensor Networks}

 
 % author names and affiliations
 % use a multiple column layout for up to three different
 % affiliations
 
 % author names and affiliations
 % use a multiple column layout for up to three different
 % affiliations

-\author{\IEEEauthorblockN{Ali Kadhum Idrees, Karine Deschinkel, Michel Salomon, and Rapha\"el Couturier }
-\IEEEauthorblockA{FEMTO-ST Institute, UMR 6174 CNRS, University of Franche-Comte, Belfort, France \\
+\author{\IEEEauthorblockN{Ali Kadhum Idrees, Karine Deschinkel, Michel Salomon, and Rapha\"el Couturier}
+\IEEEauthorblockA{FEMTO-ST Institute, UMR 6174 CNRS \\ 
+University of Franche-Comt\'e \\
+Belfort, France \\

 Email: ali.idness@edu.univ-fcomte.fr, $\lbrace$karine.deschinkel, michel.salomon, raphael.couturier$\rbrace$@univ-fcomte.fr}
 %\email{\{ali.idness, karine.deschinkel, michel.salomon, raphael.couturier\}@univ-fcomte.fr}
 %\and
 Email: ali.idness@edu.univ-fcomte.fr, $\lbrace$karine.deschinkel, michel.salomon, raphael.couturier$\rbrace$@univ-fcomte.fr}
 %\email{\{ali.idness, karine.deschinkel, michel.salomon, raphael.couturier\}@univ-fcomte.fr}
 %\and


@@ -55,94 +62,72 @@ Email: ali.idness@edu.univ-fcomte.fr, $\lbrace$karine.deschinkel, michel.salomon
 
 \begin{abstract}
 One of  the fundamental challenges in Wireless  Sensor Networks (WSNs)
 
 \begin{abstract}
 One of  the fundamental challenges in Wireless  Sensor Networks (WSNs)

-is  coverage  preservation  and  extension  of  the  network  lifetime
+is  the coverage  preservation  and  the extension  of  the  network  lifetime

 continuously  and  effectively  when  monitoring a  certain  area  (or
 continuously  and  effectively  when  monitoring a  certain  area  (or

-region) of interest. In this paper a coverage optimization protocol to
+region) of interest. In this paper, a coverage optimization protocol to

 improve the lifetime in  heterogeneous energy wireless sensor networks
 is proposed.   The area of  interest is first divided  into subregions
 improve the lifetime in  heterogeneous energy wireless sensor networks
 is proposed.   The area of  interest is first divided  into subregions

-using a  divide-and-conquer method and then scheduling  of sensor node
+using a  divide-and-conquer method and then the scheduling  of sensor node

 activity  is  planned for  each  subregion.   The proposed  scheduling
 considers  rounds during  which  a small  number  of nodes,  remaining
 active  for  sensing, is  selected  to  ensure  coverage.  Each  round
 consists  of   four  phases:  (i)~Information   Exchange,  (ii)~Leader
 Election, (iii)~Decision,  and (iv)~Sensing.  The  decision process is
 activity  is  planned for  each  subregion.   The proposed  scheduling
 considers  rounds during  which  a small  number  of nodes,  remaining
 active  for  sensing, is  selected  to  ensure  coverage.  Each  round
 consists  of   four  phases:  (i)~Information   Exchange,  (ii)~Leader
 Election, (iii)~Decision,  and (iv)~Sensing.  The  decision process is

-carried  out  by  a  leader  node which  solves  an  integer  program.
+carried  out  by  a  leader  node, which  solves  an  integer  program.

 Simulation  results show that  the proposed  approach can  prolong the
 network lifetime and improve the coverage performance.
 \end{abstract}
 
 Simulation  results show that  the proposed  approach can  prolong the
 network lifetime and improve the coverage performance.
 \end{abstract}
 

-%\keywords{Area Coverage, Wireless Sensor Networks, lifetime Optimization, Distributed Protocol.}
+\begin{IEEEkeywords}
+Area Coverage, Network lifetime, Optimization, Scheduling, Distributed Protocol.
+\end{IEEEkeywords}
+%\keywords{Area Coverage, Network lifetime, Optimization, Distributed Protocol}

  
 \IEEEpeerreviewmaketitle
 
 \section{Introduction}
 
  
 \IEEEpeerreviewmaketitle
 
 \section{Introduction}
 

-\noindent Recent years have witnessed significant advances in wireless
-communications and embedded micro-sensing MEMS technologies which have
-made  emerge wireless  sensor networks  as one  of the  most promising
-technologies~\cite{asc02}.   In fact, they  present huge  potential in
-several  domains ranging  from  health care  applications to  military
-applications.  A sensor network is  composed of a large number of tiny
-sensing  devices deployed  in a  region of  interest. Each  device has
-processing  and wireless communication  capabilities, which  enable to
-sense its environment, to compute, to store information and to deliver
-report messages to a base station.
-%These sensor nodes run on batteries with limited capacities. To achieve a long life of the network, it is important to conserve battery power. Therefore, lifetime optimisation is one of the most critical issues in wireless sensor networks.
-One of the main design issues in Wireless Sensor Networks (WSNs) is to
-prolong the  network lifetime,  while achieving acceptable  quality of
-service for applications.  Indeed, sensor nodes have limited resources
-in terms of memory, energy and computational power.
-
-Since sensor nodes have limited battery life and without being able to
-replace batteries,  especially in remote and  hostile environments, it
-is desirable that  a WSN should be deployed  with high density because
-spatial redundancy can  then be exploited to increase  the lifetime of
-the network. In such a high  density network, if all sensor nodes were
-to be  activated at the same  time, the lifetime would  be reduced. To
-extend the lifetime  of the network, the main idea  is to take benefit
-from  the overlapping  sensing regions  of some  sensor nodes  to save
-energy  by  turning  off  some  of  them  during  the  sensing  phase.
-Obviously, the deactivation of nodes  is only relevant if the coverage
-of the monitored area  is not affected.  Consequently, future software
-may  need to  adapt  appropriately to  achieve  acceptable quality  of
-service  for  applications.  In  this  paper  we  concentrate on  area
+\noindent The fast developments in the low-cost sensor devices and wireless communications have allowed the emergence the WSNs. WSN includes a large number of small , limited-power sensors  that can sense, process and transmit
+ data over a wireless communication . They communicate with each other by using multi-hop wireless communications , cooperate together to monitor the area of interest, and the measured data can be reported
+ to a monitoring center
+called, sink, for analysis it~\cite{Ammari01, Sudip03}. There are several applications used the WSN including health, home, environmental, military,and industrial applications~\cite{Akyildiz02}.
+The coverage  problem is one of the fundamental challenges in WSNs~\cite{Nayak04} that consists in monitoring efficiently and  continuously the area of interest. The limited energy of sensors represents the main challenge in the WSNs design~\cite{Ammari01}, where it is difficult to replace and/or
+ recharge their batteries because the the area of interest nature (such as hostile environments) and the cost. So, it is necessary that a WSN deployed with high density  because spatial redundancy can then be exploited to increase the lifetime of the network . However, turn on all the sensor nodes, which monitor the same region at the same time leads to decrease the lifetime of the network. To extend the lifetime of the network, the main idea is to take advantage of the overlapping sensing regions of some sensor nodes to save energy by turning off some of them during the sensing phase~\cite{Misra05}. WSNs require energy-efficient solutions to improve the network lifetime that is constrained by the limited power of each sensor node ~\cite{Akyildiz02}. 
+In  this  paper,  we  concentrate on  the area

 coverage  problem,  with  the  objective  of  maximizing  the  network
 lifetime  by using  an adaptive  scheduling. The  area of  interest is
 divided into subregions and an activity scheduling for sensor nodes is
 planned for  each subregion. 
 coverage  problem,  with  the  objective  of  maximizing  the  network
 lifetime  by using  an adaptive  scheduling. The  area of  interest is
 divided into subregions and an activity scheduling for sensor nodes is
 planned for  each subregion. 

-% DEBUT AJOUT
-{\bf In fact, the nodes in a  subregion can be seen as a cluster where
+ 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
   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 much node failures.}
-% FIN AJOUT
+  if another cluster stops due to too many node failures.

 Our scheduling  scheme considers rounds,  where a round starts  with a
 discovery  phase  to  exchange  information  between  sensors  of  the
 Our scheduling  scheme considers rounds,  where a round starts  with a
 discovery  phase  to  exchange  information  between  sensors  of  the

-subregion,  in order to  choose in  suitable manner  a sensor  node to
+subregion,  in order to  choose in  a suitable manner  a sensor  node to

 carry  out a coverage  strategy. This  coverage strategy  involves the
 carry  out a coverage  strategy. This  coverage strategy  involves the

-solving of  an integer  program which provides  the activation  of the
+solving of  an integer  program, which provides  the activation  of the

 sensors for the sensing phase of the current round.
 
 The remainder of the paper is organized as follows.  The next section
 % Section~\ref{rw}
 reviews the related work in the field.  Section~\ref{pd} is devoted to
 the    scheduling     strategy    for    energy-efficient    coverage.
 sensors for the sensing phase of the current round.
 
 The remainder of the paper is organized as follows.  The next section
 % Section~\ref{rw}
 reviews the related work in the field.  Section~\ref{pd} is devoted to
 the    scheduling     strategy    for    energy-efficient    coverage.

-Section~\ref{cp} gives the coverage model formulation which is used to
+Section~\ref{cp} gives the coverage model formulation, which is used to

 schedule  the  activation  of  sensors.  Section~\ref{exp}  shows  the
 schedule  the  activation  of  sensors.  Section~\ref{exp}  shows  the

-simulation  results obtained  using  the discrete  event simulator  on
-OMNET++  \cite{varga}. They  fully demonstrate  the usefulness  of the
+simulation  results 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}.
 
 proposed  approach.   Finally, we  give  concluding  remarks and  some
 suggestions for future works in Section~\ref{sec:conclusion}.
 

-\section{Related Works}
+\section{Related works}

 \label{rw}
 
 \noindent This section is dedicated to the various approaches proposed
 in  the literature  for  the coverage  lifetime maximization  problem,
 where the  objective is to  optimally schedule sensors'  activities in
 order to  extend network lifetime  in a randomly deployed  network. As
 \label{rw}
 
 \noindent This section is dedicated to the various approaches proposed
 in  the literature  for  the coverage  lifetime maximization  problem,
 where the  objective is to  optimally schedule sensors'  activities in
 order to  extend network lifetime  in a randomly deployed  network. As

-this problem is subject to a wide range of interpretations, we suggest
-to recall main definitions and assumptions related to our work.
+this problem is subject to a wide range of interpretations, we have chosen
+to recall the main definitions and assumptions related to our work.

 
 %\begin{itemize}
 %\item Area Coverage: The main objective is to cover an area. The area coverage requires
 
 %\begin{itemize}
 %\item Area Coverage: The main objective is to cover an area. The area coverage requires


@@ -153,20 +138,21 @@ to recall main definitions and assumptions related to our work.
 
 %\item Barrier Coverage An objective to determine the maximal support/breach paths that traverse a sensor field. Barrier coverage is expressed as finding one or more routes with starting position and ending position when the targets pass through the area deployed with sensor nodes~\cite{Santosh04,Ai05}.
 %\end{itemize}
 
 %\item Barrier Coverage An objective to determine the maximal support/breach paths that traverse a sensor field. Barrier coverage is expressed as finding one or more routes with starting position and ending position when the targets pass through the area deployed with sensor nodes~\cite{Santosh04,Ai05}.
 %\end{itemize}

-{\bf Coverage}
+\subsection{Coverage} 
+%{\bf Coverage}

 
 The most  discussed coverage problems in literature  can be classified
 into two types \cite{ma10}: area coverage (also called full or blanket
 coverage) and target coverage.  An  area coverage problem is to find a
 
 The most  discussed coverage problems in literature  can be classified
 into two types \cite{ma10}: area coverage (also called full or blanket
 coverage) and target coverage.  An  area coverage problem is to find a

-minimum number of sensors to work such that each physical point in the
+minimum number of sensors to work, such that each physical point in the

 area is within the sensing range  of at least one working sensor node.
 Target coverage problem  is to cover only a  finite number of discrete
 points  called targets.   This type  of coverage  has  mainly military
 applications. Our work will concentrate on the area coverage by design
 area is within the sensing range  of at least one working sensor node.
 Target coverage problem  is to cover only a  finite number of discrete
 points  called targets.   This type  of coverage  has  mainly military
 applications. Our work will concentrate on the area coverage by design

-and implementation of a  strategy which efficiently selects the active
+and implementation of a  strategy, which efficiently selects the active

 nodes   that  must   maintain  both   sensing  coverage   and  network
 nodes   that  must   maintain  both   sensing  coverage   and  network

-connectivity and in the same time improve the lifetime of the wireless
-sensor  network.   But  requiring  that  all physical  points  of  the
+connectivity and at the same time improve the lifetime of the wireless
+sensor  network.   But,  requiring  that  all physical  points  of  the

 considered region are covered may  be too strict, especially where the
 sensor network is not dense.   Our approach represents an area covered
 by a sensor as a set of primary points and tries to maximize the total
 considered region are covered may  be too strict, especially where the
 sensor network is not dense.   Our approach represents an area covered
 by a sensor as a set of primary points and tries to maximize the total


@@ -174,13 +160,14 @@ number  of  primary points  that  are  covered  in each  round,  while
 minimizing  overcoverage (points  covered by  multiple  active sensors
 simultaneously).
 
 minimizing  overcoverage (points  covered by  multiple  active sensors
 simultaneously).
 

-{\bf Lifetime}
+\subsection{Lifetime} 
+%{\bf Lifetime}

 
 Various   definitions   exist   for   the   lifetime   of   a   sensor
 
 Various   definitions   exist   for   the   lifetime   of   a   sensor

-network~\cite{die09}.  Main definitions proposed in the literature are
-related to the  remaining energy of the nodes or  to the percentage of
-coverage. The lifetime of the  network is mainly defined as the amount
-of  time that  the network  can  satisfy its  coverage objective  (the
+network~\cite{die09}.  The main definitions proposed in the literature are
+related to the  remaining energy of the nodes or  to the coverage percentage. 
+The lifetime of the  network is mainly defined as the amount
+of  time during which  the network  can  satisfy its  coverage objective  (the

 amount of  time that the network  can cover a given  percentage of its
 area or targets of interest). In this work, we assume that the network
 is alive  until all  nodes have  been drained of  their energy  or the
 amount of  time that the network  can cover a given  percentage of its
 area or targets of interest). In this work, we assume that the network
 is alive  until all  nodes have  been drained of  their energy  or the


@@ -189,59 +176,60 @@ during the WSN lifetime.  Network connectivity is important because an
 active sensor node without  connectivity towards a base station cannot
 transmit information on an event in the area that it monitors.
 
 active sensor node without  connectivity towards a base station cannot
 transmit information on an event in the area that it monitors.
 

-{\bf Activity scheduling}
+\subsection{Activity scheduling} 
+%{\bf Activity scheduling}

 
 Activity scheduling is to  schedule the activation and deactivation of
 sensor nodes.  The  basic objective is to decide  which sensors are in
 
 Activity scheduling is to  schedule the activation and deactivation of
 sensor nodes.  The  basic objective is to decide  which sensors are in

-what states (active or sleeping mode)  and for how long, such that the
+what states (active or sleeping mode)  and for how long, so that the

 application  coverage requirement  can be  guaranteed and  the network
 lifetime can be  prolonged. Various approaches, including centralized,
 distributed, and localized algorithms, have been proposed for activity
 application  coverage requirement  can be  guaranteed and  the network
 lifetime can be  prolonged. Various approaches, including centralized,
 distributed, and localized algorithms, have been proposed for activity

-scheduling.  In  the distributed algorithms, each node  in the network
+scheduling.  In  distributed algorithms, each node  in the network

 autonomously makes decisions on whether  to turn on or turn off itself
 only using  local neighbor information. In  centralized algorithms, a
 central controller  (a node or  base station) informs every  sensors of
 the time intervals to be activated.
 
 autonomously makes decisions on whether  to turn on or turn off itself
 only using  local neighbor information. In  centralized algorithms, a
 central controller  (a node or  base station) informs every  sensors of
 the time intervals to be activated.
 

-{\bf Distributed approaches}
+\subsection{Distributed approaches}
+%{\bf Distributed approaches}

 
 Some      distributed     algorithms      have      been     developed
 in~\cite{Gallais06,Tian02,Ye03,Zhang05,HeinzelmanCB02}  to perform the
 scheduling.   Distributed algorithms typically  operate in  rounds for
 
 Some      distributed     algorithms      have      been     developed
 in~\cite{Gallais06,Tian02,Ye03,Zhang05,HeinzelmanCB02}  to perform the
 scheduling.   Distributed algorithms typically  operate in  rounds for

-predetermined  duration. At  the  beginning of  each  round, a  sensor
-exchange information with its neighbors and makes a decision to either
+a predetermined  duration. At  the  beginning of  each  round, a  sensor
+exchanges information with its neighbors and makes a decision to either

 remain turned  on or to  go to sleep  for the round. This  decision is
 remain turned  on or to  go to sleep  for the round. This  decision is

-basically based  on simple greedy criteria like  the largest uncovered
+basically made on simple greedy criteria like  the largest uncovered

 area   \cite{Berman05efficientenergy},   maximum   uncovered   targets
 \cite{1240799}.   In \cite{Tian02}, the  scheduling scheme  is divided
 into rounds, where each round  has a self-scheduling phase followed by
 area   \cite{Berman05efficientenergy},   maximum   uncovered   targets
 \cite{1240799}.   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 node ID
-and node location  to its neighbors at the beginning  of each round. A
-sensor determines  its status by  a rule named off-duty  eligible rule
+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  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 order to avoid
 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 order to avoid

-that nodes  make conflicting decisions simultaneously and  that a part
-of       the       area        is       no       longer       covered.
+simultaneous conflicting decisions between nodes and  lack of coverage on any area.

 \cite{Prasad:2007:DAL:1782174.1782218}  defines a model  for capturing
 the dependencies  between different cover sets  and proposes localized
 heuristic  based on this  dependency.  The  algorithm consists  of two
 phases, an initial  setup phase during which each  sensor computes and
 \cite{Prasad:2007:DAL:1782174.1782218}  defines a model  for capturing
 the dependencies  between different cover sets  and proposes localized
 heuristic  based on this  dependency.  The  algorithm consists  of two
 phases, an initial  setup phase during which each  sensor computes and

-prioritize the  covers and  a sensing phase  during which  each sensor
+prioritizes the  covers and  a sensing phase  during which  each sensor

 first decides  its on/off status, and  then remains on or  off for the
 rest  of the  duration.  Authors  in \cite{chin2007}  propose  a novel
 distributed  heuristic named  Distributed  Energy-efficient Scheduling
 for k-coverage  (DESK) so  that the energy  consumption among  all the
 sensors  is balanced,  and  network lifetime  is  maximized while  the
 first decides  its on/off status, and  then remains on or  off for the
 rest  of the  duration.  Authors  in \cite{chin2007}  propose  a novel
 distributed  heuristic named  Distributed  Energy-efficient Scheduling
 for k-coverage  (DESK) so  that the energy  consumption among  all the
 sensors  is balanced,  and  network lifetime  is  maximized while  the

-coverage requirements  is being  maintained.  This algorithm  works in
+coverage requirement  is being  maintained.  This algorithm  works in

 round, requires only  1-sensing-hop-neighbor information, and a sensor
 decides  its status  (active/sleep)  based on  its perimeter  coverage
 computed  through the k-Non-Unit-disk  coverage algorithm  proposed in
 \cite{Huang:2003:CPW:941350.941367}.
 
 round, requires only  1-sensing-hop-neighbor information, and a sensor
 decides  its status  (active/sleep)  based on  its perimeter  coverage
 computed  through the k-Non-Unit-disk  coverage algorithm  proposed in
 \cite{Huang:2003:CPW:941350.941367}.
 

-Some others approaches do  not consider synchronized and predetermined
+Some other approaches do  not consider a synchronized and predetermined

 period  of time  where the  sensors are  active or  not.  Indeed, each
 period  of time  where the  sensors are  active or  not.  Indeed, each

-sensor  maintains its  own timer  and its  time wake-up  is randomized
+sensor  maintains its  own timer  and its  wake-up time is randomized

 \cite{Ye03} or regulated \cite{cardei05} over time.
 %A ecrire \cite{Abrams:2004:SKA:984622.984684}p33
 
 \cite{Ye03} or regulated \cite{cardei05} over time.
 %A ecrire \cite{Abrams:2004:SKA:984622.984684}p33
 


@@ -253,7 +241,8 @@ sensor  maintains its  own timer  and its  time wake-up  is randomized
 
 %In this paper we focus on centralized algorithms because distributed algorithms are outside the scope of our work. Note that centralized coverage algorithms have the advantage of requiring very low processing power from the sensor nodes which have usually limited processing capabilities. Moreover, a recent study conducted in \cite{pc10} concludes that there is a threshold in terms of network size to switch from a localized to a centralized algorithm. Indeed the exchange of messages in large networks may consume  a considerable amount of energy in a localized approach compared to a centralized one. 
 
 
 %In this paper we focus on centralized algorithms because distributed algorithms are outside the scope of our work. Note that centralized coverage algorithms have the advantage of requiring very low processing power from the sensor nodes which have usually limited processing capabilities. Moreover, a recent study conducted in \cite{pc10} concludes that there is a threshold in terms of network size to switch from a localized to a centralized algorithm. Indeed the exchange of messages in large networks may consume  a considerable amount of energy in a localized approach compared to a centralized one. 
 

-{\bf Centralized approaches}
+\subsection{Centralized approaches}
+%{\bf Centralized approaches}

 
 Power  efficient  centralized  schemes  differ  according  to  several
 criteria \cite{Cardei:2006:ECP:1646656.1646898},  such as the coverage
 
 Power  efficient  centralized  schemes  differ  according  to  several
 criteria \cite{Cardei:2006:ECP:1646656.1646898},  such as the coverage


@@ -264,24 +253,24 @@ to divide/organize  the sensors into  a suitable number of  set covers
 where each  set completely covers  an interest region and  to activate
 these set covers successively.
 
 where each  set completely covers  an interest region and  to activate
 these set covers successively.
 

-First algorithms  proposed in the  literature consider that  the cover
+The first algorithms  proposed in the  literature consider that  the cover

 sets  are  disjoint: a  sensor  node appears  in  exactly  one of  the
 generated  cover  sets.    For  instance,  Slijepcevic  and  Potkonjak
 sets  are  disjoint: a  sensor  node appears  in  exactly  one of  the
 generated  cover  sets.    For  instance,  Slijepcevic  and  Potkonjak

-\cite{Slijepcevic01powerefficient}   propose    an   algorithm   which
+\cite{Slijepcevic01powerefficient}   propose    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
 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,
+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 $n$
 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.  \cite{cardei02}~describes a graph coloring
-technique  to achieve energy  savings by  organizing the  sensor nodes
-into a maximum number of  disjoint dominating sets which are activated
+is the number of  sensors. In~\cite{cardei02}, a graph coloring
+technique is described to achieve energy  savings by  organizing the  sensor nodes
+into a maximum number of  disjoint dominating sets, which are activated

 successively. The dominating sets do not guarantee the coverage of the
 whole        region        of        interest.        Abrams        et
 al.~\cite{Abrams:2004:SKA:984622.984684}  design  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
 successively. The dominating sets do not guarantee the coverage of the
 whole        region        of        interest.        Abrams        et
 al.~\cite{Abrams:2004:SKA:984622.984684}  design  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.
+of covers that  includes 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.
 Their        work        builds        upon       previous        work
 in~\cite{Slijepcevic01powerefficient} and the  generated cover sets do
 not provide complete coverage of the monitoring zone.


@@ -301,9 +290,9 @@ integer      programming     solving.       %Cardei      and     Du
 \cite{Cardei:2005:IWS:1160086.1160098} propose 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
 \cite{Cardei:2005:IWS:1160086.1160098} propose 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
+maximum  flow   problem, which  is  formulated  as   a  mixed  integer

 programming (MIP). Then their heuristic  uses the output of the MIP to
 programming (MIP). Then their heuristic  uses the output of the MIP to

-compute  disjoint  set  covers.  Results  show  that  these  heuristic
+compute  disjoint  set  covers.  Results  show  that  this  heuristic

 provides  a   number  of  set  covers  slightly   larger  compared  to
 \cite{Slijepcevic01powerefficient}  but with  a larger  execution time
 due  to the complexity  of the  mixed integer  programming resolution.
 provides  a   number  of  set  covers  slightly   larger  compared  to
 \cite{Slijepcevic01powerefficient}  but with  a larger  execution time
 due  to the complexity  of the  mixed integer  programming resolution.


@@ -319,7 +308,7 @@ cover sets with a slight growth rate in execution time.
 %More recently Manju and Pujari\cite{Manju2011}
 
 In the  case of non-disjoint algorithms  \cite{Manju2011}, sensors may
 %More recently Manju and Pujari\cite{Manju2011}
 
 In the  case of non-disjoint algorithms  \cite{Manju2011}, sensors may

-participate  in more  than  one cover  set.   In some  cases this  may
+participate  in more  than  one cover  set.   In some  cases, this  may

 prolong  the lifetime  of the  network in  comparison to  the disjoint
 cover set algorithms, but  designing algorithms for non-disjoint cover
 sets  generally induces a  higher order  of complexity.   Moreover, in
 prolong  the lifetime  of the  network in  comparison to  the disjoint
 cover set algorithms, but  designing algorithms for non-disjoint cover
 sets  generally induces a  higher order  of complexity.   Moreover, in


@@ -333,13 +322,14 @@ that by allowing sensors to  participate in multiple sets, the network
 lifetime         increases        compared         with        related
 work~\cite{Cardei:2005:IWS:1160086.1160098}.   In~\cite{berman04}, the
 authors  have formulated  the lifetime  problem and  suggested another
 lifetime         increases        compared         with        related
 work~\cite{Cardei:2005:IWS:1160086.1160098}.   In~\cite{berman04}, the
 authors  have formulated  the lifetime  problem and  suggested another

-(LP)  technique to  solve this  problem. A  centralized  provably near
-optimal      solution     based      on      the     Garg-K\"{o}nemann
-algorithm~\cite{garg98} is also proposed.
+(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.

 
 

-{\bf Our contribution}
+\subsection{Our contribution}
+%{\bf Our contribution}

 
 

-There are  three main questions which  should be addressed  to build a
+There are  three main questions, which should be addressed  to build a

 scheduling strategy. We  give a brief answer to  these three questions
 to describe our  approach before going into details  in the subsequent
 sections.
 scheduling strategy. We  give a brief answer to  these three questions
 to describe our  approach before going into details  in the subsequent
 sections.


@@ -351,29 +341,29 @@ sections.
 
 \item {\bf What are the rules to decide which node has to be turned on
   or off?}  Our algorithm tends to limit the overcoverage of points of
 
 \item {\bf What are the rules to decide which node has to be turned on
   or off?}  Our algorithm tends to limit the overcoverage of points of

-  interest  to avoid  turning on  too much  sensors covering  the same
+  interest  to avoid  turning on  too many sensors covering  the same

   areas  at the  same time,  and tries  to prevent  undercoverage. The
   decision  is  a  good   compromise  between  these  two  conflicting
   objectives.
 
   areas  at the  same time,  and tries  to prevent  undercoverage. The
   decision  is  a  good   compromise  between  these  two  conflicting
   objectives.
 

-\item {\bf  Which node  should make such  decision?}  As  mentioned in
+\item {\bf Which  node should make such a  decision?}  As mentioned in

   \cite{pc10}, both centralized  and distributed algorithms have their
   own  advantages and  disadvantages. Centralized  coverage algorithms
   have the advantage  of requiring very low processing  power from the
   \cite{pc10}, both centralized  and distributed algorithms have their
   own  advantages and  disadvantages. Centralized  coverage algorithms
   have the advantage  of requiring very low processing  power from the

-  sensor  nodes which  have usually  limited  processing capabilities.
+  sensor  nodes, which  have usually  limited  processing capabilities.

   Distributed  algorithms  are  very  adaptable  to  the  dynamic  and
   scalable nature of sensors network.  Authors in \cite{pc10} conclude
   that there is a threshold in  terms of network size to switch from a
   Distributed  algorithms  are  very  adaptable  to  the  dynamic  and
   scalable nature of sensors network.  Authors in \cite{pc10} conclude
   that there is a threshold in  terms of network size to switch from a

-  localized  to  a  centralized  algorithm.  Indeed  the  exchange  of
+  localized  to  a  centralized  algorithm.  Indeed, the  exchange  of

   messages  in large  networks may  consume a  considerable  amount of
   messages  in large  networks may  consume a  considerable  amount of

-  energy in  a localized approach  compared to a centralized  one. Our
+  energy in a centralized approach  compared to a distributed one. Our

   work does not  consider only one leader to  compute and to broadcast
   work does not  consider only one leader to  compute and to broadcast

-  the  schedule decision  to all  the sensors.  When the  network size
-  increases,  the  network  is  divided  in many  subregions  and  the
+  the scheduling decision  to all the sensors.  When  the network size
+  increases,  the network  is  divided into  many  subregions and  the

   decision is made by a leader in each subregion.
 \end{itemize}
 
   decision is made by a leader in each subregion.
 \end{itemize}
 

-\section{Activity Scheduling}
+\section{Activity scheduling}

 \label{pd}
 
 We consider  a randomly and  uniformly deployed network  consisting of
 \label{pd}
 
 We consider  a randomly and  uniformly deployed network  consisting of


@@ -400,12 +390,12 @@ figure~\ref{fig1}.
 
 Each round  is divided  into 4 phases  : Information  (INFO) Exchange,
 Leader  Election, Decision,  and  Sensing.  For  each  round there  is
 
 Each round  is divided  into 4 phases  : Information  (INFO) Exchange,
 Leader  Election, Decision,  and  Sensing.  For  each  round there  is

-exactly one set cover responsible  for sensing task.  This protocol is
-more reliable  against the unexpectedly node failure  because it works
+exactly one set cover responsible  for the sensing task.  This protocol is
+more reliable  against an unexpected node failure  because it works

 in rounds.   On the  one hand,  if a node  failure is  detected before
 in rounds.   On the  one hand,  if a node  failure is  detected before

-taking the decision, the node will not participate to this phase, and,
+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
 on the other hand, if the  node failure occurs after the decision, the

-sensing task of the network  will be affected temporarily: only during
+sensing task of the network  will be temporarily affected: only during

 the period of sensing until a  new round starts, since a new set cover
 will take  charge of the  sensing task in  the next round.  The energy
 consumption  and  some other  constraints  can  easily  be taken  into
 the period of sensing until a  new round starts, since a new set cover
 will take  charge of the  sensing task in  the next round.  The energy
 consumption  and  some other  constraints  can  easily  be taken  into


@@ -414,12 +404,12 @@ information (including their residual energy) at the beginning of each
 round.  However,   the  pre-sensing  phases   (INFO  Exchange,  Leader
 Election,  Decision) are energy  consuming for  some nodes,  even when
 they do not  join the network to monitor the  area. Below, we describe
 round.  However,   the  pre-sensing  phases   (INFO  Exchange,  Leader
 Election,  Decision) are energy  consuming for  some nodes,  even when
 they do not  join the network to monitor the  area. Below, we describe

-each phase in more detail.
+each phase in more details.

 
 

-\subsection{INFOrmation Exchange Phase}
+\subsection{Information exchange phase}

 
 Each sensor node $j$ sends  its position, remaining energy $RE_j$, and
 
 Each sensor node $j$ sends  its position, remaining energy $RE_j$, and

-the number of local neighbors  $NBR_j$ to all wireless sensor nodes in
+the number of local neighbours  $NBR_j$ to all wireless sensor nodes in

 its subregion by using an INFO  packet and then listens to the packets
 sent from  other nodes.  After that, each  node will  have information
 about  all the  sensor  nodes in  the  subregion.  In  our model,  the
 its subregion by using an INFO  packet and then listens to the packets
 sent from  other nodes.  After that, each  node will  have information
 about  all the  sensor  nodes in  the  subregion.  In  our model,  the


@@ -430,32 +420,32 @@ active mode.
 
 %The working phase works in rounding fashion. Each round include 3 steps described as follow :
 
 
 %The working phase works in rounding fashion. Each round include 3 steps described as follow :
 

-\subsection{Leader Election Phase}
-This  step includes choosing  the Wireless  Sensor Node  Leader (WSNL)
-which  will  be  responsible  of executing  coverage  algorithm.  Each
+\subsection{Leader election phase}
+This  step includes 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  round.  All the  sensor  nodes cooperate  to
 select 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  in order  of priority  are: larger
 subregion  in  the   area  of  interest  will  select   its  own  WSNL
 independently  for each  round.  All the  sensor  nodes cooperate  to
 select 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  in order  of priority  are: larger

-number  of neighbors,  larger remaining  energy, and  then in  case of
+number  of neighbours,  larger remaining  energy, and  then in  case of

 equality, larger index.
 
 equality, larger index.
 

-\subsection{Decision Phase}
+\subsection{Decision phase}

 The  WSNL will  solve an  integer  program (see  section~\ref{cp})  to
 select which sensors will be  activated in the following sensing phase
 to cover  the subregion.  WSNL will send  Active-Sleep packet  to each
 The  WSNL will  solve an  integer  program (see  section~\ref{cp})  to
 select which sensors will be  activated in the following sensing phase
 to cover  the subregion.  WSNL will send  Active-Sleep packet  to each

-sensor in the subregion based on algorithm's results.
+sensor in the subregion based on the algorithm's results.

 %The main goal in this step after choosing the WSNL is to produce the best representative active nodes set that will take the responsibility of covering the whole region $A^k$ with minimum number of sensor nodes to prolong the lifetime in the wireless sensor network. For our problem, in each round we need to select the minimum set of sensor nodes to improve the lifetime of the network and in the same time taking into account covering the region $A^k$ . We need an optimal solution with tradeoff between our two conflicting objectives.
 %The above region coverage problem can be formulated as a Multi-objective optimization problem and we can use the Binary Particle Swarm Optimization technique to solve it. 
 
 %The main goal in this step after choosing the WSNL is to produce the best representative active nodes set that will take the responsibility of covering the whole region $A^k$ with minimum number of sensor nodes to prolong the lifetime in the wireless sensor network. For our problem, in each round we need to select the minimum set of sensor nodes to improve the lifetime of the network and in the same time taking into account covering the region $A^k$ . We need an optimal solution with tradeoff between our two conflicting objectives.
 %The above region coverage problem can be formulated as a Multi-objective optimization problem and we can use the Binary Particle Swarm Optimization technique to solve it. 
 

-\subsection{Sensing Phase}
+\subsection{Sensing phase}

 Active  sensors  in the  round  will  execute  their sensing  task  to
 preserve maximal  coverage in the  region of interest. We  will assume
 Active  sensors  in the  round  will  execute  their sensing  task  to
 preserve maximal  coverage in the  region of interest. We  will assume

-that the cost  of keeping a node awake (or sleep)  for sensing task is
+that the cost  of keeping a node awake (or asleep)  for sensing task is

 the same  for all wireless sensor  nodes in the  network.  Each sensor
 will receive  an Active-Sleep  packet from WSNL  informing it  to stay
 the same  for all wireless sensor  nodes in the  network.  Each sensor
 will receive  an Active-Sleep  packet from WSNL  informing it  to stay

-awake or  go sleep  for a time  equal to  the period of  sensing until
+awake or to go to sleep  for a time  equal to  the period of  sensing until

 starting a new round.
 
 %\subsection{Sensing coverage model}
 starting a new round.
 
 %\subsection{Sensing coverage model}


@@ -463,13 +453,13 @@ starting a new round.
 
 %\noindent We try to produce an adaptive scheduling which allows sensors to operate alternatively so as to prolong the network lifetime. For convenience, the notations and assumptions are described first.
 %The wireless sensor node use the  binary disk sensing model by which each sensor node will has a certain sensing range is reserved within a circular disk called radius $R_s$.
 
 %\noindent We try to produce an adaptive scheduling which allows sensors to operate alternatively so as to prolong the network lifetime. For convenience, the notations and assumptions are described first.
 %The wireless sensor node use the  binary disk sensing model by which each sensor node will has a certain sensing range is reserved within a circular disk called radius $R_s$.

-\noindent We consider a boolean  disk coverage model which is the most
+\indent We consider a boolean  disk coverage model which is the most

 widely used sensor coverage model in the literature. Each sensor has a
 constant sensing range $R_s$. All  space points within a disk centered
 at  the sensor with  the radius  of the  sensing range  is said  to be
 covered by this sensor. We also assume that the communication range is
 widely used sensor coverage model in the literature. Each sensor has a
 constant sensing range $R_s$. All  space points within a disk centered
 at  the sensor with  the radius  of the  sensing range  is said  to be
 covered by this sensor. We also assume that the communication range is

-at   least  twice   of  the   sensing  range.   In  fact,   Zhang  and
-Zhou~\cite{Zhang05} prove that if  the transmission range fulfills the
+at   least  twice    the size of the   sensing  range.   In  fact,   Zhang  and
+Zhou~\cite{Zhang05} proved that if  the transmission range fulfills the

 previous  hypothesis, a  complete coverage  of a  convex  area implies
 connectivity among the working nodes in the active mode.
 %To calculate the coverage ratio for the area of interest, we can propose the following coverage model which is called Wireless Sensor Node Area Coverage Model to ensure that all the area within each node sensing range are covered. We can calculate the positions of the points in the circle disc of the sensing range of wireless sensor node based on the Unit Circle in figure~\ref{fig:cluster1}:
 previous  hypothesis, a  complete coverage  of a  convex  area implies
 connectivity among the working nodes in the active mode.
 %To calculate the coverage ratio for the area of interest, we can propose the following coverage model which is called Wireless Sensor Node Area Coverage Model to ensure that all the area within each node sensing range are covered. We can calculate the positions of the points in the circle disc of the sensing range of wireless sensor node based on the Unit Circle in figure~\ref{fig:cluster1}:


@@ -488,9 +478,9 @@ connectivity among the working nodes in the active mode.
 %We choose to representEach wireless sensor node will be represented into a selected number of primary points by which we can know if the sensor node is covered or not.
 % Figure ~\ref{fig:cluster2} shows the selected primary points that represents the area of the sensor node and according to the sensing range of the wireless sensor node.
 
 %We choose to representEach wireless sensor node will be represented into a selected number of primary points by which we can know if the sensor node is covered or not.
 % Figure ~\ref{fig:cluster2} shows the selected primary points that represents the area of the sensor node and according to the sensing range of the wireless sensor node.
 

-\noindent Instead of working with  area coverage, we consider for each
+\indent Instead of working with the coverage area, we consider for each

 sensor a set of points called  primary points. We also assume that the
 sensor a set of points called  primary points. We also assume that the

-sensing disk defined  by a sensor is covered if  all primary points of
+sensing disk defined  by a sensor is covered if  all the primary points of

 this sensor are covered.
 %\begin{figure}[h!]
 %\centering
 this sensor are covered.
 %\begin{figure}[h!]
 %\centering


@@ -506,9 +496,9 @@ sensor node  and its $R_s$,  we calculate the primary  points directly
 based on the proposed model. We  use these primary points (that can be
 increased or decreased if necessary)  as references to ensure that the
 monitored  region  of interest  is  covered  by  the selected  set  of
 based on the proposed model. We  use these primary points (that can be
 increased or decreased if necessary)  as references to ensure that the
 monitored  region  of interest  is  covered  by  the selected  set  of

-sensors, instead of using all points in the area.
+sensors, instead of using all the points in the area.

 
 

-\noindent  We can  calculate  the positions  of  the selected  primary
+\indent  We can  calculate  the positions  of  the selected  primary

 points in  the circle disk of  the sensing range of  a wireless sensor
 node (see figure~\ref{fig2}) as follows:\\
 $(p_x,p_y)$ = point center of wireless sensor node\\  
 points in  the circle disk of  the sensing range of  a wireless sensor
 node (see figure~\ref{fig2}) as follows:\\
 $(p_x,p_y)$ = point center of wireless sensor node\\  


@@ -541,20 +531,20 @@ $X_{13}=( p_x + R_s * (0), p_y + R_s * (\frac{-\sqrt{2}}{2})) $.
 \label{fig2}
 \end{figure}
 
 \label{fig2}
 \end{figure}
 

-\section{Coverage Problem Formulation}
+\section{Coverage problem formulation}

 \label{cp}
 %We can formulate our optimization problem as energy cost minimization by minimize the number of active sensor nodes and maximizing the coverage rate at the same time in each $A^k$ . This optimization problem can be formulated as follow: Since that we use a homogeneous wireless sensor network, we will assume that the cost of keeping a node awake is the same for all wireless sensor nodes in the network. We can define the decision parameter  $X_j$ as in \eqref{eq11}:\\
 
 
 %To satisfy the coverage requirement, the set of the principal points that will represent all the sensor nodes in the monitored region as $PSET= \lbrace P_1,\ldots ,P_p, \ldots , P_{N_P^k} \rbrace $, where $N_P^k = N_{sp} * N^k $ and according to the proposed model in figure ~\ref{fig:cluster2}. These points can be used by the wireless sensor node leader which will be chosen in each region in A to build a new parameter $\alpha_{jp}$  that represents the coverage possibility for each principal point $P_p$ of each wireless sensor node $s_j$ in $A^k$ as in \eqref{eq12}:\\
 
 \label{cp}
 %We can formulate our optimization problem as energy cost minimization by minimize the number of active sensor nodes and maximizing the coverage rate at the same time in each $A^k$ . This optimization problem can be formulated as follow: Since that we use a homogeneous wireless sensor network, we will assume that the cost of keeping a node awake is the same for all wireless sensor nodes in the network. We can define the decision parameter  $X_j$ as in \eqref{eq11}:\\
 
 
 %To satisfy the coverage requirement, the set of the principal points that will represent all the sensor nodes in the monitored region as $PSET= \lbrace P_1,\ldots ,P_p, \ldots , P_{N_P^k} \rbrace $, where $N_P^k = N_{sp} * N^k $ and according to the proposed model in figure ~\ref{fig:cluster2}. These points can be used by the wireless sensor node leader which will be chosen in each region in A to build a new parameter $\alpha_{jp}$  that represents the coverage possibility for each principal point $P_p$ of each wireless sensor node $s_j$ in $A^k$ as in \eqref{eq12}:\\
 

-\noindent   Our   model   is   based   on  the   model   proposed   by
+\indent   Our   model   is   based   on  the   model   proposed   by

 \cite{pedraza2006} where the objective is  to find a maximum number of
 \cite{pedraza2006} where the objective is  to find a maximum number of

-disjoint  cover sets.   To accomplish  this goal,  authors  propose an
-integer program which forces undercoverage and overcoverage of targets
+disjoint  cover sets.   To accomplish  this goal,  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
 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 $X_{j}$  which  determine  the
+model,  we  consider  binary  variables $X_{j}$,  which  determine  the

 activation of  sensor $j$ in the  sensing phase of the  round. We also
 consider  primary points  as targets.   The set  of primary  points is
 denoted by $P$ and the set of sensors by $J$.
 activation of  sensor $j$ in the  sensing phase of the  round. We also
 consider  primary points  as targets.   The set  of primary  points is
 denoted by $P$ and the set of sensors by $J$.


@@ -584,7 +574,8 @@ We define the Overcoverage variable $\Theta_{p}$ as:
 \begin{equation}
  \Theta_{p} = \left \{ 
 \begin{array}{l l}
 \begin{equation}
  \Theta_{p} = \left \{ 
 \begin{array}{l l}

-  0 & \mbox{if point $p$ is not covered,}\\
+  0 & \mbox{if the primary point}\\
+    & \mbox{$p$ is not covered,}\\

   \left( \sum_{j \in J} \alpha_{jp} * X_{j} \right)- 1 & \mbox{otherwise.}\\
 \end{array} \right.
 \label{eq13} 
   \left( \sum_{j \in J} \alpha_{jp} * X_{j} \right)- 1 & \mbox{otherwise.}\\
 \end{array} \right.
 \label{eq13} 


@@ -596,7 +587,7 @@ by:
 \begin{equation}
 U_{p} = \left \{ 
 \begin{array}{l l}
 \begin{equation}
 U_{p} = \left \{ 
 \begin{array}{l l}

-  1 &\mbox{if point $p$ is not covered,} \\
+  1 &\mbox{if the primary point $p$ is not covered,} \\

   0 & \mbox{otherwise.}\\
 \end{array} \right.
 \label{eq14} 
   0 & \mbox{otherwise.}\\
 \end{array} \right.
 \label{eq14} 


@@ -623,17 +614,17 @@ X_{j} \in \{0,1\}, &\forall j \in J
   sensing in the round (1 if yes and 0 if not);
 \item $\Theta_{p}$  : {\it overcoverage}, the number  of sensors minus
   one that are covering the primary point $p$;
   sensing in the round (1 if yes and 0 if not);
 \item $\Theta_{p}$  : {\it overcoverage}, the number  of sensors minus
   one that are covering the primary point $p$;

-\item $U_{p}$  : {\it undercoverage},  indicates whether or  not point
+\item $U_{p}$  : {\it undercoverage},  indicates whether or  not the primary point

   $p$ is being covered (1 if not covered and 0 if covered).
 \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
   $p$ is being covered (1 if not covered and 0 if covered).
 \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  balance  the
-restriction  equation by taking  positive values.  There are  two main
-objectives.  First we limit overcoverage of primary points in order to
-activate a minimum number of sensors.  Second we prevent that parts of
-the  subregion are  not  monitored by  minimizing undercoverage.   The
+case,  overcoverage  and  undercoverage  variables  help  balancing  the
+restriction  equations by taking  positive values.  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.
 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.


@@ -663,25 +654,23 @@ round.
 
 %\end{itemize}
 
 
 %\end{itemize}
 

-\section{Simulation Results}
+\section{Simulation results}

 \label{exp}
 
 In this section, we conducted  a series of simulations to evaluate the
 \label{exp}
 
 In this section, we conducted  a series of simulations to evaluate the

-efficiency  and relevance of  our approach,  using the  discrete event
+efficiency  and the relevance of  our approach,  using the  discrete event

 simulator  OMNeT++  \cite{varga}. We  performed  simulations for  five
 different densities varying from 50 to 250~nodes. Experimental results
 were  obtained from  randomly generated  networks in  which  nodes are
 deployed over a  $(50 \times 25)~m^2 $ sensing  field. 
 simulator  OMNeT++  \cite{varga}. We  performed  simulations for  five
 different densities varying from 50 to 250~nodes. Experimental results
 were  obtained from  randomly generated  networks in  which  nodes are
 deployed over a  $(50 \times 25)~m^2 $ sensing  field. 

-% DEBUT MODIFICATION
-{\bf More precisely, the deployment is controlled at a coarse scale in
+More precisely, the deployment is controlled at a coarse scale in

   order to ensure that the  deployed nodes can fully cover the sensing
   order to ensure that the  deployed nodes can fully cover the sensing

-  field with the given sensing range.}
-% FIN MODIFICATION
+  field with the given sensing range.

 10~simulation  runs  are performed  with
 different  network  topologies for  each  node  density.  The  results
 presented hereafter  are the  average of these  10 runs.  A simulation
 ends  when  all the  nodes  are dead  or  the  sensor network  becomes
 10~simulation  runs  are performed  with
 different  network  topologies for  each  node  density.  The  results
 presented hereafter  are the  average of these  10 runs.  A simulation
 ends  when  all the  nodes  are dead  or  the  sensor network  becomes

-disconnected (some nodes may not be  able to sent to a base station an
+disconnected (some nodes may not be  able to send, to a base station, an

 event they sense).
 
 Our proposed coverage protocol uses the radio energy dissipation model
 event they sense).
 
 Our proposed coverage protocol uses the radio energy dissipation model


@@ -689,28 +678,28 @@ defined by~\cite{HeinzelmanCB02} as  energy consumption model for each
 wireless  sensor node  when  transmitting or  receiving packets.   The
 energy of  each node in a  network is initialized  randomly within the
 range 24-60~joules, and each sensor node will consume 0.2 watts during
 wireless  sensor node  when  transmitting or  receiving packets.   The
 energy of  each node in a  network is initialized  randomly within the
 range 24-60~joules, and each sensor node will consume 0.2 watts during

-the sensing period which will have  a duration of 60 seconds. Thus, an
-active  node will  consume  12~joules during  sensing  phase, while  a
+the sensing period, which will last 60 seconds. Thus, an
+active  node will  consume  12~joules during the sensing  phase, while  a

 sleeping  node will  use  0.002  joules.  Each  sensor  node will  not
 participate in the next round if its remaining energy is less than 12
 sleeping  node will  use  0.002  joules.  Each  sensor  node will  not
 participate in the next round if its remaining energy is less than 12

-joules.  In  all  experiments  the  parameters  are  set  as  follows:
-$R_s=5m$, $w_{\Theta}=1$, and $w_{U}=|P^2|$.
+joules.  In  all  experiments,  the  parameters  are  set  as  follows:
+$R_s=5~m$, $w_{\Theta}=1$, and $w_{U}=|P^2|$.

 
 

-We  evaluate the  efficiency of  our approach  using  some performance
+We  evaluate the  efficiency of  our approach by using  some performance

 metrics such as: coverage ratio,  number of active nodes ratio, energy
 saving  ratio, energy consumption,  network lifetime,  execution time,
 metrics such as: coverage ratio,  number of active nodes ratio, energy
 saving  ratio, energy consumption,  network lifetime,  execution time,

-and number of stopped simulation runs.  Our approach called Strategy~2
-(with Two Leaders)  works with two subregions, each  one having a size
+and number of stopped simulation runs.  Our approach called strategy~2
+(with two leaders)  works with two subregions, each  one having a size

 of $(25 \times 25)~m^2$.  Our strategy will be compared with two other
 of $(25 \times 25)~m^2$.  Our strategy will be compared with two other

-approaches. The first one,  called Strategy~1 (with One Leader), works
-as Strategy~2, but considers only one region of $(50 \times 25)$ $m^2$
+approaches. The first one,  called strategy~1 (with one leader), works
+as strategy~2, but considers only one region of $(50 \times 25)$ $m^2$

 with only  one leader.  The  other approach, called  Simple Heuristic,
 with only  one leader.  The  other approach, called  Simple Heuristic,

-consists in dividing  uniformly the region into squares  of $(5 \times
+consists in uniformly dividing   the region into squares  of $(5 \times

 5)~m^2$.   During the  decision phase,  in  each square,  a sensor  is
 randomly  chosen, it  will remain  turned  on for  the coming  sensing
 phase.
 
 5)~m^2$.   During the  decision phase,  in  each square,  a sensor  is
 randomly  chosen, it  will remain  turned  on for  the coming  sensing
 phase.
 

-\subsection{The impact of the Number of Rounds on Coverage Ratio} 
+\subsection{The impact of the number of rounds on the coverage ratio} 

 
 In this experiment, the coverage ratio measures how much the area of a
 sensor field is  covered. In our case, the  coverage ratio is regarded
 
 In this experiment, the coverage ratio measures how much the area of a
 sensor field is  covered. In our case, the  coverage ratio is regarded


@@ -725,26 +714,26 @@ $90\%$ for five more rounds.  Coverage ratio decreases when the number
 of rounds increases  due to dead nodes. Although  some nodes are dead,
 thanks to  strategy~1 or~2,  other nodes are  preserved to  ensure the
 coverage. Moreover, when  we have a dense sensor  network, it leads to
 of rounds increases  due to dead nodes. Although  some nodes are dead,
 thanks to  strategy~1 or~2,  other nodes are  preserved to  ensure the
 coverage. Moreover, when  we have a dense sensor  network, it leads to

-maintain the full coverage for  larger number of rounds. Strategy~2 is
-slightly more efficient that strategy 1, because strategy~2 subdivides
+maintain the full coverage for a larger number of rounds. Strategy~2 is
+slightly more efficient than strategy 1, because strategy~2 subdivides

 the region into 2~subregions and  if one of the two subregions becomes
 the region into 2~subregions and  if one of the two subregions becomes

-disconnected,  coverage   may  be  still  ensured   in  the  remaining
+disconnected,  the coverage   may  be  still  ensured   in  the  remaining

 subregion.
 
 \parskip 0pt 
 \begin{figure}[h!]
 \centering
 subregion.
 
 \parskip 0pt 
 \begin{figure}[h!]
 \centering

-\includegraphics[scale=0.55]{TheCoverageRatio150.eps} %\\~ ~ ~(a)
-\caption{The impact of the Number of Rounds on Coverage Ratio for 150 deployed nodes}
+\includegraphics[scale=0.5]{TheCoverageRatio150g.eps} %\\~ ~ ~(a)
+\caption{The impact of the number of rounds on the coverage ratio for 150 deployed nodes}

 \label{fig3}
 \end{figure} 
 
 \label{fig3}
 \end{figure} 
 

-\subsection{The impact of the Number of Rounds on Active Sensors Ratio} 
+\subsection{The impact of the number of rounds on the active sensors ratio} 

 
 It is important to have as few active nodes as possible in each round,
 in  order to  minimize  the communication  overhead  and maximize  the
 network lifetime.  This point is  assessed through the  Active Sensors
 
 It is important to have as few active nodes as possible in each round,
 in  order to  minimize  the communication  overhead  and maximize  the
 network lifetime.  This point is  assessed through the  Active Sensors

-Ratio, which is defined as follows:
+Ratio (ASR), which is defined as follows:

 \begin{equation*}
 \scriptsize
 \mbox{ASR}(\%) = \frac{\mbox{Number of active sensors 
 \begin{equation*}
 \scriptsize
 \mbox{ASR}(\%) = \frac{\mbox{Number of active sensors 


@@ -756,32 +745,32 @@ for 150 deployed nodes.
 
 \begin{figure}[h!]
 \centering
 
 \begin{figure}[h!]
 \centering

-\includegraphics[scale=0.55]{TheActiveSensorRatio150.eps} %\\~ ~ ~(a)
-\caption{The impact of the Number of Rounds on Active Sensors Ratio for 150 deployed nodes }
+\includegraphics[scale=0.5]{TheActiveSensorRatio150g.eps} %\\~ ~ ~(a)
+\caption{The impact of the number of rounds on the active sensors ratio for 150 deployed nodes }

 \label{fig4}
 \end{figure} 
 
 The  results presented  in figure~\ref{fig4}  show the  superiority of
 \label{fig4}
 \end{figure} 
 
 The  results presented  in figure~\ref{fig4}  show the  superiority of

-both proposed  strategies, the Strategy  with Two Leaders and  the one
-with a  single Leader,  in comparison with  the Simple  Heuristic. The
-Strategy with One Leader uses less active nodes than the Strategy with
-Two Leaders until the last  rounds, because it uses central control on
-the whole sensing field.  The  advantage of the Strategy~2 approach is
+both proposed  strategies, the strategy  with two leaders and  the one
+with a  single leader,  in comparison with  the simple  heuristic. The
+strategy with one leader uses less active nodes than the strategy with
+two leaders until the last  rounds, because it uses central control on
+the whole sensing field.  The  advantage of the strategy~2 approach is

 that even if a network is disconnected in one subregion, the other one
 usually  continues  the optimization  process,  and  this extends  the
 lifetime of the network.
 
 that even if a network is disconnected in one subregion, the other one
 usually  continues  the optimization  process,  and  this extends  the
 lifetime of the network.
 

-\subsection{The impact of the Number of Rounds on Energy Saving Ratio} 
+\subsection{The impact of the number of rounds on the energy saving ratio} 

 
 In this experiment, we consider a performance metric linked to energy.
 
 In this experiment, we consider a performance metric linked to energy.

-This metric, called Energy Saving Ratio, is defined by:
+This metric, called Energy Saving Ratio (ESR), is defined by:

 \begin{equation*}
 \scriptsize
 \mbox{ESR}(\%) = \frac{\mbox{Number of alive sensors during this round}}
 {\mbox{Total number of sensors in the network for the region}} \times 100.
 \end{equation*}
 \begin{equation*}
 \scriptsize
 \mbox{ESR}(\%) = \frac{\mbox{Number of alive sensors during this round}}
 {\mbox{Total number of sensors in the network for the region}} \times 100.
 \end{equation*}

-The  longer the ratio  is high,  the more  redundant sensor  nodes are
-switched off, and consequently  the longer the  network may  be alive.
+The  longer the ratio  is,  the more  redundant sensor  nodes are
+switched off, and consequently  the longer the  network may  live.

 Figure~\ref{fig5} shows the average  Energy Saving Ratio versus rounds
 for all three approaches and for 150 deployed nodes.
 
 Figure~\ref{fig5} shows the average  Energy Saving Ratio versus rounds
 for all three approaches and for 150 deployed nodes.
 


@@ -789,99 +778,96 @@ for all three approaches and for 150 deployed nodes.
 %\centering
 % \begin{multicols}{6}
 \centering
 %\centering
 % \begin{multicols}{6}
 \centering

-\includegraphics[scale=0.55]{TheEnergySavingRatio150.eps} %\\~ ~ ~(a)
-\caption{The impact of the Number of Rounds on Energy Saving Ratio for 150 deployed nodes}
+\includegraphics[scale=0.5]{TheEnergySavingRatio150g.eps} %\\~ ~ ~(a)
+\caption{The impact of the number of rounds on the energy saving ratio for 150 deployed nodes}

 \label{fig5}
 \end{figure} 
 
 The simulation  results show that our strategies  allow to efficiently
 save energy by  turning off some sensors during  the sensing phase. As
 \label{fig5}
 \end{figure} 
 
 The simulation  results show that our strategies  allow to efficiently
 save energy by  turning off some sensors during  the sensing phase. As

-expected, the Strategy with One Leader is usually slightly better than
-the second  strategy, because the  global optimization permit  to turn
+expected, the strategy with one leader is usually slightly better than
+the second  strategy, because the  global optimization permits  to turn

 off more  sensors. Indeed,  when there are  two subregions  more nodes
 remain awake  near the border shared  by them. Note that  again as the
 off more  sensors. Indeed,  when there are  two subregions  more nodes
 remain awake  near the border shared  by them. Note that  again as the

-number of  rounds increases  the two leader  strategy becomes  the most
-performing, since its takes longer  to have the two subregion networks
+number of  rounds increases  the two leaders'  strategy becomes  the most
+performing one, since it takes longer  to have the two subregion networks

 simultaneously disconnected.
 
 simultaneously disconnected.
 

-\subsection{The Number of Stopped Simulation Runs}
+\subsection{The percentage of stopped simulation runs}

 
 

-We  will now  study  the number  of  simulation which  stopped due  to
-network  disconnection, per round  for each  of the  three approaches.
-Figure~\ref{fig6} illustrates the average number of stopped simulation
+We  will now  study  the percentage  of  simulations, which  stopped due  to
+network  disconnections per round  for each  of the  three approaches.
+Figure~\ref{fig6} illustrates the percentage of stopped simulation

 runs per  round for 150 deployed  nodes.  It can be  observed that the
 runs per  round for 150 deployed  nodes.  It can be  observed that the

-heuristic is  the approach which  stops the earlier because  the nodes
-are  chosen   randomly.   Among  the  two   proposed  strategies,  the
-centralized  one  first  exhibits  network  disconnection.   Thus,  as
+simple heuristic is  the approach, which  stops first because  the nodes
+are   randomly chosen.   Among  the  two   proposed  strategies,  the
+centralized  one  first  exhibits  network  disconnections.   Thus,  as

 explained previously, in case  of the strategy with several subregions
 the  optimization effectively  continues as  long  as a  network in  a
 subregion   is  still   connected.   This   longer   partial  coverage
 explained previously, in case  of the strategy with several subregions
 the  optimization effectively  continues as  long  as a  network in  a
 subregion   is  still   connected.   This   longer   partial  coverage

-optimization participates in extending the lifetime.
+optimization participates in extending the network lifetime.

 
 \begin{figure}[h!]
 \centering
 
 \begin{figure}[h!]
 \centering

-\includegraphics[scale=0.55]{TheNumberofStoppedSimulationRuns150.eps} 
-\caption{The Number of Stopped Simulation Runs against Rounds for 150 deployed nodes }
+\includegraphics[scale=0.5]{TheNumberofStoppedSimulationRuns150g.eps} 
+\caption{The percentage of stopped simulation runs compared to the number of rounds for 150 deployed nodes }

 \label{fig6}
 \end{figure} 
 
 \label{fig6}
 \end{figure} 
 

-\subsection{The Energy Consumption}
+\subsection{The energy consumption}

 
 In this experiment, we study the effect of the multi-hop communication
 
 In this experiment, we study the effect of the multi-hop communication

-protocol  on the  performance of  the  Strategy with  Two Leaders  and
+protocol  on the  performance of  the  strategy with  two leaders  and

 compare  it  with  the  other  two  approaches.   The  average  energy
 consumption  resulting  from  wireless  communications  is  calculated
 compare  it  with  the  other  two  approaches.   The  average  energy
 consumption  resulting  from  wireless  communications  is  calculated

-considering the  energy spent by  all the nodes when  transmitting and
+by taking into account the  energy spent by  all the nodes when  transmitting and

 receiving  packets during  the network  lifetime. This  average value,
 which  is obtained  for 10~simulation  runs,  is then  divided by  the
 average number of rounds to define a metric allowing a fair comparison
 between networks having different densities.
 
 receiving  packets during  the network  lifetime. This  average value,
 which  is obtained  for 10~simulation  runs,  is then  divided by  the
 average number of rounds to define a metric allowing a fair comparison
 between networks having different densities.
 

-Figure~\ref{fig7} illustrates the Energy Consumption for the different
+Figure~\ref{fig7} illustrates the energy consumption for the different

 network  sizes and  the three  approaches. The  results show  that the
 network  sizes and  the three  approaches. The  results show  that the

-Strategy  with  Two  Leaders  is  the  most  competitive  from  energy
-consumption point  of view.  A  centralized method, like  the Strategy
-with  One  Leader, has  a  high energy  consumption  due  to the  many
+strategy  with  two  leaders  is  the  most  competitive  from  the energy
+consumption point  of view.  A  centralized method, like  the strategy
+with  one  leader, has  a  high energy  consumption  due  to  many

 communications.   In fact,  a distributed  method greatly  reduces the
 number  of communications thanks  to the  partitioning of  the initial
 network in several independent subnetworks. Let us notice that even if
 a  centralized  method  consumes  far  more  energy  than  the  simple
 heuristic, since the energy cost of communications during a round is a
 small  part   of  the   energy  spent  in   the  sensing   phase,  the
 communications.   In fact,  a distributed  method greatly  reduces the
 number  of communications thanks  to the  partitioning of  the initial
 network in several independent subnetworks. Let us notice that even if
 a  centralized  method  consumes  far  more  energy  than  the  simple
 heuristic, since the energy cost of communications during a round is a
 small  part   of  the   energy  spent  in   the  sensing   phase,  the

-communications have a small impact on the lifetime.
+communications have a small impact on the network lifetime.

 
 \begin{figure}[h!]
 \centering
 
 \begin{figure}[h!]
 \centering

-\includegraphics[scale=0.55]{TheEnergyConsumption.eps} 
-\caption{The Energy Consumption  }
+\includegraphics[scale=0.5]{TheEnergyConsumptiong.eps} 
+\caption{The energy consumption}

 \label{fig7}
 \end{figure} 
 
 \label{fig7}
 \end{figure} 
 

-\subsection{The impact of Number of Sensors on Execution Time}
+\subsection{The impact of the number of sensors on 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
 
 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.
-Table~\ref{table1} gives the average  execution times {\bf in seconds}
-on a laptop of the decision phase
-% DEBUT AJOUT
-{\bf (resolution of the optimization problem)}
-% FIN AJOUT
+used   for  the  sensing   phase,  not   for  the   pre-sensing  ones.   
+Table~\ref{table1} gives the average  execution times  in seconds
+on a laptop of the decision phase (solving of the optimization problem)

 during one  round.  They  are given for  the different  approaches and
 various numbers of sensors.  The lack of any optimization explains why
 during one  round.  They  are given for  the different  approaches and
 various numbers of sensors.  The lack of any optimization explains why

-the heuristic has very  low execution times.  Conversely, the Strategy
-with  One  Leader which  requires  to  solve  an optimization  problem
+the heuristic has very  low execution times.  Conversely, the strategy
+with  one  leader, which  requires  to  solve  an optimization  problem

 considering  all  the  nodes  presents  redhibitory  execution  times.
 considering  all  the  nodes  presents  redhibitory  execution  times.

-Moreover, increasing of 50~nodes  the network size multiplies the time
-by  almost a  factor of  10. The  Strategy with  Two Leaders  has more
+Moreover, increasing the network size by 50~nodes   multiplies the time
+by  almost a  factor of  10. The  strategy with  two leaders  has more

 suitable times.  We  think that in distributed fashion  the solving of
 the  optimization problem  in a  subregion  can be  tackled by  sensor
 suitable times.  We  think that in distributed fashion  the solving of
 the  optimization problem  in a  subregion  can be  tackled by  sensor

-nodes.   Overall,  to  be  able   deal  with  very  large  networks  a
+nodes.   Overall,  to  be  able to  deal  with  very  large  networks,  a

 distributed method is clearly required.
 
 \begin{table}[ht]
 distributed method is clearly required.
 
 \begin{table}[ht]

-\caption{The Execution Time(s) vs The Number of Sensors}
+\caption{THE EXECUTION TIME(S) VS THE NUMBER OF SENSORS}

 % title of Table
 \centering
 
 % title of Table
 \centering
 


@@ -890,8 +876,8 @@ distributed method is clearly required.
 % centered columns (4 columns)
       \hline
 %inserts double horizontal lines
 % centered columns (4 columns)
       \hline
 %inserts double horizontal lines

-Sensors Number & Strategy~1 & Strategy~2  & Simple Heuristic \\ [0.5ex]
- & (with Two Leaders) & (with One Leader) & \\ [0.5ex]
+Sensors number & Strategy~2 & Strategy~1  & Simple heuristic \\ [0.5ex]
+ & (with two leaders) & (with one leader) & \\ [0.5ex]

 %Case & Strategy (with Two Leaders) & Strategy (with One Leader) & Simple Heuristic \\ [0.5ex]
 % inserts table
 %heading
 %Case & Strategy (with Two Leaders) & Strategy (with One Leader) & Simple Heuristic \\ [0.5ex]
 % inserts table
 %heading


@@ -915,46 +901,44 @@ Sensors Number & Strategy~1 & Strategy~2  & Simple Heuristic \\ [0.5ex]
 % is used to refer this table in the text
 \end{table}
 
 % is used to refer this table in the text
 \end{table}
 

-\subsection{The Network Lifetime}
+\subsection{The network lifetime}

 
 Finally, we  have defined the network  lifetime as the  time until all
 nodes  have  been drained  of  their  energy  or each  sensor  network
 
 Finally, we  have defined the network  lifetime as the  time until all
 nodes  have  been drained  of  their  energy  or each  sensor  network

-monitoring  an area becomes  disconnected.  In  figure~\ref{fig8}, the
-network  lifetime for different  network sizes  and for  both Strategy
-with Two  Leaders and the Simple  Heuristic is illustrated. 
-% DEBUT MODIFICATION
-{\bf  We do  not consider  anymore the  centralized Strategy  with One
-  Leader, because, as shown above, this strategy results  in execution
-  times that quickly become unsuitable for a sensor network.}
-% FIN MODIFICATION
+monitoring  an area has become  disconnected.  In  figure~\ref{fig8}, the
+network  lifetime for different  network sizes  and for  both strategy
+with two  leaders and the simple  heuristic is illustrated. 
+  We do  not consider  anymore the  centralized strategy  with one
+  leader, because, as shown above, this strategy results  in execution
+  times that quickly become unsuitable for a sensor network.

 
 \begin{figure}[h!]
 %\centering
 % \begin{multicols}{6}
 \centering
 
 \begin{figure}[h!]
 %\centering
 % \begin{multicols}{6}
 \centering

-\includegraphics[scale=0.5]{TheNetworkLifetime.eps} %\\~ ~ ~(a)
-\caption{The Network Lifetime }
+\includegraphics[scale=0.5]{TheNetworkLifetimeg.eps} %\\~ ~ ~(a)
+\caption{The network lifetime }

 \label{fig8}
 \end{figure} 
 
 As  highlighted by figure~\ref{fig8},  the network  lifetime obviously
 \label{fig8}
 \end{figure} 
 
 As  highlighted by figure~\ref{fig8},  the network  lifetime obviously

-increases when  the size  of the network  increase, with  our approach
-that leads to  the larger lifetime improvement.  By  choosing for each
-round the  well suited nodes  to cover the  region of interest  and by
+increases when  the size  of the network  increases, with  our approach
+that leads to  the larger lifetime improvement.  By  choosing the  best 
+suited nodes, for each round,  to cover the  region of interest  and by

 letting the other ones sleep in order to be used later in next rounds,
 letting the other ones sleep in order to be used later in next rounds,

-our strategy efficiently prolongs the lifetime. Comparison shows that
+our strategy efficiently prolonges the network lifetime. Comparison shows that

 the larger  the sensor number  is, the more our  strategies outperform
 the larger  the sensor number  is, the more our  strategies outperform

-the Simple Heuristic.  Strategy~2, which uses two leaders, is the best
+the simple heuristic.  Strategy~2, which uses two leaders, is the best

 one because it is robust to network disconnection in one subregion. It
 also  means   that  distributing  the  algorithm  in   each  node  and
 subdividing the sensing field  into many subregions, which are managed
 independently and simultaneously, is the most relevant way to maximize
 the lifetime of a network.
 
 one because it is robust to network disconnection in one subregion. It
 also  means   that  distributing  the  algorithm  in   each  node  and
 subdividing the sensing field  into many subregions, which are managed
 independently and simultaneously, is the most relevant way to maximize
 the lifetime of a network.
 

-\section{Conclusions and Future Works}
+\section{Conclusion and future works}

 \label{sec:conclusion}
 
 \label{sec:conclusion}
 

-In this paper, we have  addressed the problem of coverage and lifetime
+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
 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


@@ -962,16 +946,16 @@ is   divided   into   smaller   subregions  using   the   concept   of
 divide-and-conquer method,  and then a  multi-rounds coverage protocol
 will optimize  coverage and  lifetime performances in  each subregion.
 The  proposed  protocol  combines  two efficient  techniques:  network
 divide-and-conquer method,  and then a  multi-rounds coverage protocol
 will optimize  coverage and  lifetime performances in  each subregion.
 The  proposed  protocol  combines  two efficient  techniques:  network

-Leader Election  and sensor activity scheduling,  where the challenges
+leader election  and sensor activity scheduling,  where the challenges

 include how to select the  most efficient leader in each subregion and
 include how to select the  most efficient leader in each subregion and

-the best  representative active nodes that will  optimize the lifetime
+the best  representative active nodes that will  optimize the network lifetime

 while  taking   the  responsibility  of   covering  the  corresponding
 subregion.   The network lifetime  in each  subregion is  divided into
 rounds, each round consists  of four phases: (i) Information Exchange,
 (ii) Leader Election, (iii) an optimization-based Decision in order to
 select  the  nodes remaining  active  for  the  last phase,  and  (iv)
 while  taking   the  responsibility  of   covering  the  corresponding
 subregion.   The network lifetime  in each  subregion is  divided into
 rounds, each round consists  of four phases: (i) Information Exchange,
 (ii) Leader Election, (iii) an optimization-based Decision in order to
 select  the  nodes remaining  active  for  the  last phase,  and  (iv)

-Sensing.  The  simulations results show the relevance  of the proposed
-protocol in  terms of lifetime, coverage ratio,  active sensors Ratio,
+Sensing.  The  simulations show the relevance  of the proposed
+protocol in  terms of lifetime, coverage ratio,  active sensors ratio,

 energy saving,  energy consumption, execution time, and  the number of
 stopped simulation  runs due  to network disconnection.   Indeed, when
 dealing with  large and dense wireless sensor  networks, a distributed
 energy saving,  energy consumption, execution time, and  the number of
 stopped simulation  runs due  to network disconnection.   Indeed, when
 dealing with  large and dense wireless sensor  networks, a distributed


@@ -979,18 +963,14 @@ 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.
 
 single global optimization problem  by partitioning it in many smaller
 problems, one per subregion, that can be solved more easily.
 

-In  future, we plan  to study  and propose  a coverage  protocol which
-computes  all  active  sensor  schedules  in  a  single  round,  using
+In  future work, we plan  to study  and propose  a coverage  protocol, which
+computes  all  active  sensor  schedules  in  one time,  using

 optimization  methods  such  as  swarms optimization  or  evolutionary
 optimization  methods  such  as  swarms optimization  or  evolutionary

-algorithms.
-% DEBUT AJOUT
-{\bf  This single  round  will still  consists  of 4  phases, but  the
-  decision phase will compute the schedules for several sensing phases
-  which aggregated together define a kind of meta-sensing phase.}
-% FIN AJOUT
-The computation of all cover sets in one round is far more
+algorithms.  The round  will still  consist of 4  phases, but  the
+  decision phase will compute the schedules for several sensing phases,
+  which aggregated together, define a kind of meta-sensing phase.
+The computation of all cover sets in one time is far more

 difficult, but will reduce the communication overhead.
 difficult, but will reduce the communication overhead.

-

 % use section* for acknowledgement
 %\section*{Acknowledgment}
 
 % use section* for acknowledgement
 %\section*{Acknowledgment}