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+++ b/PeCO-EO/articleeo.tex~
@@ -5,55 +5,59 @@
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 \usepackage{indentfirst}
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+\usepackage[algo2e,ruled,vlined]{algorithm2e}

 \begin{document}
 
 %\jvol{00} \jnum{00} \jyear{2013} \jmonth{April}
 
 %\articletype{GUIDE}
 
 \begin{document}
 
 %\jvol{00} \jnum{00} \jyear{2013} \jmonth{April}
 
 %\articletype{GUIDE}
 

-\title{{\itshape Perimeter-based Coverage Optimization to Improve Lifetime in Wireless Sensor Networks}}
-
-\author{Ali Kadhum Idrees$^{a}$, Karine Deschinkel$^{a}$$^{\ast}$\thanks{$^\ast$Corresponding author. Email: karine.deschinkel@univ-fcomte.fr}, Michel Salomon$^{a}$ and Rapha\"el Couturier $^{a}$
-$^{a}${\em{FEMTO-ST Institute, UMR 6174 CNRS, University of Franche-Comte,
-          Belfort, France}};}
-
+\title{{\itshape Perimeter-based Coverage Optimization \\
+  to Improve Lifetime in Wireless Sensor Networks}}

 
 

+\author{Ali Kadhum Idrees$^{a,b}$, Karine Deschinkel$^{a}$$^{\ast}$\thanks{$^\ast$Corresponding author. Email: karine.deschinkel@univ-fcomte.fr}, Michel Salomon$^{a}$ and Rapha\"el Couturier $^{a}$
+  $^{a}${\em{FEMTO-ST Institute, UMR 6174 CNRS, \\
+  University Bourgogne Franche-Comt\'e, Belfort, France}} \\ 
+  $^{b}${\em{Department of Computer Science, University of Babylon, Babylon, Iraq}}
+}         
+         

 \maketitle
 
 \begin{abstract}
 The most important problem in a Wireless Sensor Network (WSN) is to optimize the
 \maketitle
 
 \begin{abstract}
 The most important problem in a Wireless Sensor Network (WSN) is to optimize the

-use of its limited energy provision, so that it can fulfill its monitoring task
-as long as  possible. Among  known  available approaches  that can  be used  to
+use of its limited energy provision, so  that it can fulfill its monitoring task
+as  long as  possible. Among  known  available approaches  that can  be used  to

 improve  power  management,  lifetime coverage  optimization  provides  activity
 improve  power  management,  lifetime coverage  optimization  provides  activity

-scheduling which ensures sensing coverage while minimizing the energy cost. We propose such an approach called Perimeter-based Coverage Optimization
-protocol (PeCO). It is a  hybrid of centralized and distributed methods: the
-region of interest is first subdivided into subregions and the protocol is then
-distributed among sensor nodes in each  subregion.
-The novelty of our approach lies essentially in the formulation of a new
-mathematical optimization  model based on the  perimeter coverage level  to schedule
-sensors' activities.  Extensive simulation experiments demonstrate that PeCO  can
-offer longer lifetime coverage for WSNs in comparison with some other protocols.
-
-\begin{keywords}Wireless Sensor Networks, Area Coverage, Energy efficiency, Optimization, Scheduling.
+scheduling which ensures  sensing coverage while minimizing the  energy cost. We
+propose such  an approach called Perimeter-based  Coverage Optimization protocol
+(PeCO). It  is a hybrid  of centralized and  distributed methods: the  region of
+interest  is  first  subdivided  into   subregions  and  the  protocol  is  then
+distributed among sensor  nodes in each subregion.  The novelty  of our approach
+lies essentially  in the  formulation of a  new mathematical  optimization model
+based  on  the  perimeter  coverage   level  to  schedule  sensors'  activities.
+Extensive simulation experiments demonstrate that PeCO can offer longer lifetime
+coverage for WSNs in comparison with some other protocols.
+
+\begin{keywords}
+  Wireless Sensor Networks, Area Coverage, Energy efficiency, Optimization, Scheduling.

 \end{keywords}
 
 \end{abstract}
 
 \end{keywords}
 
 \end{abstract}
 

-

 \section{Introduction}
 \label{sec:introduction}
 
 \section{Introduction}
 \label{sec:introduction}
 

-The continuous progress in Micro Electro-Mechanical Systems (MEMS) and
-wireless communication hardware  has given rise to the opportunity  to use large
-networks    of     tiny    sensors,    called    Wireless     Sensor    Networks
-(WSN)~\citep{akyildiz2002wireless,puccinelli2005wireless}, to  fulfill monitoring
+The continuous progress in Micro  Electro-Mechanical Systems (MEMS) and wireless
+communication hardware has  given rise to the opportunity to  use large networks
+of      tiny       sensors,      called      Wireless       Sensor      Networks
+(WSN)~\citep{akyildiz2002wireless,puccinelli2005wireless}, to fulfill monitoring

 tasks.   A  WSN  consists  of  small low-powered  sensors  working  together  by
 communicating with one another through multi-hop radio communications. Each node
 can send the data  it collects in its environment, thanks to  its sensor, to the
 user by means of  sink nodes. The features of a WSN made  it suitable for a wide
 range of application  in areas such as business,  environment, health, industry,
 tasks.   A  WSN  consists  of  small low-powered  sensors  working  together  by
 communicating with one another through multi-hop radio communications. Each node
 can send the data  it collects in its environment, thanks to  its sensor, to the
 user by means of  sink nodes. The features of a WSN made  it suitable for a wide
 range of application  in areas such as business,  environment, health, industry,

-military, and so on~\citep{yick2008wireless}.   Typically, a sensor node contains
-three main components~\citep{anastasi2009energy}: a  sensing unit able to measure
+military, and so on~\citep{yick2008wireless}.  Typically, a sensor node contains
+three main components~\citep{anastasi2009energy}: a sensing unit able to measure

 physical,  chemical, or  biological  phenomena observed  in  the environment;  a
 processing unit which will process and store the collected measurements; a radio
 communication unit for data transmission and receiving.
 physical,  chemical, or  biological  phenomena observed  in  the environment;  a
 processing unit which will process and store the collected measurements; a radio
 communication unit for data transmission and receiving.


@@ -62,192 +66,201 @@ The energy needed  by an active sensor node to  perform sensing, processing, and
 communication is supplied by a power supply which is a battery. This battery has
 a limited energy provision and it may  be unsuitable or impossible to replace or
 recharge it in  most applications. Therefore it is necessary  to deploy WSN with
 communication is supplied by a power supply which is a battery. This battery has
 a limited energy provision and it may  be unsuitable or impossible to replace or
 recharge it in  most applications. Therefore it is necessary  to deploy WSN with

-high density in order to increase  reliability and to exploit node redundancy
+high density  in order to  increase reliability  and to exploit  node redundancy

 thanks to energy-efficient activity  scheduling approaches.  Indeed, the overlap
 of sensing  areas can be exploited  to schedule alternatively some  sensors in a
 low power sleep mode and thus save  energy. Overall, the main question that must
 be answered is: how to extend the lifetime coverage of a WSN as long as possible
 thanks to energy-efficient activity  scheduling approaches.  Indeed, the overlap
 of sensing  areas can be exploited  to schedule alternatively some  sensors in a
 low power sleep mode and thus save  energy. Overall, the main question that must
 be answered is: how to extend the lifetime coverage of a WSN as long as possible

-while  ensuring   a  high  level  of   coverage?   These past few years  many
+while  ensuring  a   high  level  of  coverage?   These  past   few  years  many

 energy-efficient mechanisms have been suggested  to retain energy and extend the
 energy-efficient mechanisms have been suggested  to retain energy and extend the

-lifetime of the WSNs~\citep{rault2014energy}.\\\\
+lifetime of the WSNs~\citep{rault2014energy}.
+

 This paper makes the following contributions.
 \begin{enumerate}
 This paper makes the following contributions.
 \begin{enumerate}

-\item We have devised a framework to schedule nodes to be activated alternatively such
-  that the network lifetime is prolonged  while ensuring that a certain level of
-  coverage is preserved.  A key idea in  our framework is to exploit spatial and
-  temporal subdivision.   On the one hand,  the area of interest  is divided into
-  several smaller subregions and, on the other hand, the time line is divided into
-  periods of equal length. In each subregion the sensor nodes will cooperatively
-  choose a  leader which will schedule  nodes' activities, and this  grouping of
-  sensors is similar to typical cluster architecture.
-\item We have proposed a new mathematical  optimization model.  Instead of  trying to
+\item A  framework is devised  to schedule  nodes to be  activated alternatively
+  such that  the network  lifetime is  prolonged while  ensuring that  a certain
+  level of coverage  is preserved.  A key  idea in the proposed  framework is to
+  exploit  spatial and  temporal  subdivision.  On  the one  hand,  the area  of
+  interest is  divided into several smaller  subregions and, on the  other hand,
+  the time line is divided into periods  of equal length.  In each subregion the
+  sensor nodes  will cooperatively  choose a leader  which will  schedule nodes'
+  activities,  and  this grouping  of  sensors  is  similar to  typical  cluster
+  architecture.
+\item A new  mathematical optimization model is proposed.  Instead  of trying to

   cover a set of specified points/targets as  in most of the methods proposed in
   the literature, we formulate an integer program based on perimeter coverage of
   each sensor.  The  model involves integer variables to  capture the deviations
   cover a set of specified points/targets as  in most of the methods proposed in
   the literature, we formulate an integer program based on perimeter coverage of
   each sensor.  The  model involves integer variables to  capture the deviations

-  between  the actual  level of  coverage and  the required  level.  Hence, an
-  optimal schedule  will be  obtained by  minimizing a  weighted sum  of these
+  between  the actual  level  of coverage  and the  required  level.  Hence,  an
+  optimal  schedule will  be  obtained by  minimizing a  weighted  sum of  these

   deviations.
   deviations.

-\item We have conducted extensive simulation  experiments, using the  discrete event
-  simulator OMNeT++, to demonstrate the  efficiency of our protocol. We have compared
-  our   PeCO   protocol   to   two   approaches   found   in   the   literature:
-  DESK~\citep{ChinhVu} and  GAF~\citep{xu2001geography}, and also to  our previous
-  work published in~\citep{Idrees2} which is  based on another optimization model
-  for sensor scheduling.
+\item Extensive  simulation experiments are  conducted using the  discrete event
+  simulator  OMNeT++, to  demonstrate the  efficiency of  our protocol.  We have
+  compared  the  PeCO  protocol  to  two approaches  found  in  the  literature:
+  DESK~\citep{ChinhVu} and GAF~\citep{xu2001geography}, and also to our previous
+  protocol DiLCO published in~\citep{Idrees2}. DiLCO  uses the same framework as
+  PeCO but is based on another optimization model for sensor scheduling.

 \end{enumerate}
 
 \end{enumerate}
 

-
-
-
-
-
-The rest  of the paper is  organized as follows.  In the next section
-some related work in the  field is reviewed. Section~\ref{sec:The PeCO Protocol Description}
+The rest of the paper is organized as follows.  In the next section some related
+work in the  field is reviewed. Section~\ref{sec:The  PeCO Protocol Description}

 is devoted to the PeCO protocol  description and Section~\ref{cp} focuses on the
 coverage model  formulation which is used  to schedule the activation  of sensor
 nodes.  Section~\ref{sec:Simulation  Results and Analysis}  presents simulations
 results and discusses the comparison  with other approaches. Finally, concluding
 is devoted to the PeCO protocol  description and Section~\ref{cp} focuses on the
 coverage model  formulation which is used  to schedule the activation  of sensor
 nodes.  Section~\ref{sec:Simulation  Results and Analysis}  presents simulations
 results and discusses the comparison  with other approaches. Finally, concluding

-remarks   are  drawn   and  some   suggestions are  given  for   future  works   in
+remarks  are  drawn  and  some  suggestions   are  given  for  future  works  in

 Section~\ref{sec:Conclusion and Future Works}.
 
 \section{Related Literature}
 \label{sec:Literature Review}
 
 Section~\ref{sec:Conclusion and Future Works}.
 
 \section{Related Literature}
 \label{sec:Literature Review}
 

-In  this section, some  related works  regarding  the
-coverage problem is summarized, and specific aspects of the PeCO protocol from the  works presented in
-the literature are presented.
+This section  summarizes some related  works regarding the coverage  problem and
+presents  specific aspects  of the  PeCO protocol  common with  other literature
+works.

 
 The most  discussed coverage problems in  literature can be classified  in three
 
 The most  discussed coverage problems in  literature can be classified  in three

-categories~\citep{li2013survey}   according   to  their   respective   monitoring
-objective.  Hence,  area coverage \citep{Misra}  means that every point  inside a
-fixed area  must be monitored, while  target coverage~\citep{yang2014novel} refers
+categories~\citep{li2013survey}   according  to   their  respective   monitoring
+objective.  Hence, area  coverage \citep{Misra} means that every  point inside a
+fixed area must be monitored, while target coverage~\citep{yang2014novel} refers

 to  the objective  of coverage  for a  finite number  of discrete  points called
 to  the objective  of coverage  for a  finite number  of discrete  points called

-targets,  and  barrier coverage~\citep{HeShibo,kim2013maximum}  focuses  on
+targets,   and   barrier  coverage~\citep{HeShibo,kim2013maximum}   focuses   on

 preventing  intruders   from  entering   into  the   region  of   interest.   In
 preventing  intruders   from  entering   into  the   region  of   interest.   In

-\citep{Deng2012}  authors  transform the  area  coverage  problem into  the  target
+\citep{Deng2012} authors  transform the  area coverage  problem into  the target

 coverage one taking into account the  intersection points among disks of sensors
 nodes    or   between    disk   of    sensor   nodes    and   boundaries.     In
 coverage one taking into account the  intersection points among disks of sensors
 nodes    or   between    disk   of    sensor   nodes    and   boundaries.     In

-\citep{Huang:2003:CPW:941350.941367}  authors prove  that  if  the perimeters  of
+\citep{Huang:2003:CPW:941350.941367}  authors prove  that if  the perimeters  of

 sensors are sufficiently  covered it will be  the case for the  whole area. They
 provide an algorithm in $O(nd~log~d)$  time to compute the perimeter-coverage of
 sensors are sufficiently  covered it will be  the case for the  whole area. They
 provide an algorithm in $O(nd~log~d)$  time to compute the perimeter-coverage of

-each  sensor. $d$  denotes  the  maximum  number  of  sensors  that  are
-neighbors  to  a  sensor, and  $n$  is  the  total  number of  sensors  in  the
-network. {\it In PeCO protocol, instead  of determining the level of coverage of
-  a set  of discrete  points, our  optimization model is  based on  checking the
-  perimeter-coverage of each sensor to activate a minimal number of sensors.}
+each sensor. $d$ denotes  the maximum number of sensors that  are neighbors to a
+sensor, and  $n$ is the  total number  of sensors in  the network. {\it  In PeCO
+  protocol, instead  of determining the level  of coverage of a  set of discrete
+  points, our optimization model is  based on checking the perimeter-coverage of
+  each sensor to activate a minimal number of sensors.}

 
 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
 
 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)\citep{wang2011coverage}, 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
-\citep{ling2009energy}.  Active node selection is determined based on the problem
-requirements (e.g.   area monitoring,  connectivity, or power  efficiency).  For
-instance, \citet{jaggi2006}  address the problem of maximizing
+non-disjoint)  \citep{wang2011coverage},  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 decided  at the
+beginning of each  period \citep{ling2009energy}. In fact,  many authors propose
+algorithms       working       in       such      a       periodic       fashion
+\citep{chin2007,yan2008design,pc10}.  Active node  selection is determined based
+on  the problem  requirements  (e.g.  area  monitoring,  connectivity, or  power
+efficiency).  For instance, \citet{jaggi2006}  address the problem of maximizing

 the 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
 the 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.   
-\citet{chin2007},  \citet{yan2008design}, \citet{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,  PeCO 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 centralized  and distributed approaches, or  even a mixing  of these two
-concepts, have  been proposed  to extend the  network lifetime \citep{zhou2009variable}.   In distributed algorithms~\citep{Tian02,yangnovel,ChinhVu,qu2013distributed} each sensor decides of its
-own activity scheduling  after an information exchange with  its neighbors.  The
-main interest of such an approach is to avoid long range communications and thus
-to reduce the energy dedicated to the communications.  Unfortunately, since each
-node has only information on  its immediate neighbors (usually the one-hop ones)
-it may make a bad decision leading to a global suboptimal solution.  Conversely,
-centralized
-algorithms~\citep{cardei2005improving,zorbas2010solving,pujari2011high}     always
+activated in succession to achieve  the desired network lifetime. {\it Motivated
+  by these works,  PeCO 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 centralized  and distributed approaches, or  even a mixing of  these two
+concepts,    have   been    proposed    to   extend    the   network    lifetime
+\citep{zhou2009variable}.                      In                    distributed
+algorithms~\citep{ChinhVu,qu2013distributed,yangnovel}  each  sensor decides  of
+its own  activity scheduling after  an information exchange with  its neighbors.
+The main interest of such an approach  is to avoid long range communications and
+thus to reduce the energy dedicated to the communications.  Unfortunately, since
+each node has  only information on its immediate neighbors  (usually the one-hop
+ones)  it may  make a  bad  decision leading  to a  global suboptimal  solution.
+Conversely,                                                          centralized
+algorithms~\citep{cardei2005improving,zorbas2010solving,pujari2011high}   always

 provide nearly  or close to  optimal solution since  the algorithm has  a global
 view of the whole network. The disadvantage of a centralized method is obviously
 provide nearly  or close to  optimal solution since  the algorithm has  a global
 view of the whole network. The disadvantage of a centralized method is obviously

-its high  cost in communications needed to  transmit to a single  node, the base
-station which will globally schedule  nodes' activities, and data from all the other
-sensor nodes  in the area.  The price  in communications can be  huge since
-long range  communications will be  needed. In fact  the larger the WNS  is, the
-higher the  communication and  thus the energy  cost are.   {\it In order  to be
-  suitable for large-scale  networks, in the PeCO protocol,  the area of interest
-  is divided into several smaller subregions, and in each one, a node called the
-  leader  is  in  charge  of  selecting  the active  sensors  for  the  current
-  period.  Thus our  protocol is  scalable  and is a  globally distributed  method,
-  whereas it is centralized in each subregion.}
-
-Various  coverage scheduling  algorithms have  been developed  these past few years.
+its high cost  in communications needed to  transmit to a single  node, the base
+station which will globally schedule nodes'  activities, data from all the other
+sensor nodes in  the area.  The price  in communications can be  huge since long
+range communications will be needed. In fact  the larger the WSN, the higher the
+communication  energy  cost.  {\it  In  order  to  be suitable  for  large-scale
+  networks,  in PeCO  protocol  the area  of interest  is  divided into  several
+  smaller subregions, and in each one, a  node called the leader is in charge of
+  selecting the  active sensors for the  current period.  Thus PeCO  protocol is
+  scalable and a globally distributed method,  whereas it is centralized in each
+  subregion.}
+
+Various coverage scheduling algorithms have been developed these past few years.

 Many of  them, 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
 Many of  them, 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

-\citep{berman04,zorbas2010solving}.  Other  approaches are based  on mathematical
-programming formulations~\citep{cardei2005energy,5714480,pujari2011high,Yang2014}
-and dedicated techniques (solving with a branch-and-bound algorithm available in
+\citep{berman04,zorbas2010solving}.  Other approaches  are based on mathematical
+programming
+formulations~\citep{cardei2005energy,5714480,pujari2011high,Yang2014}        and
+dedicated  techniques (solving  with a  branch-and-bound algorithm  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
 also                                                                        been
 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
 also                                                                        been

-used~\citep{castano2013column,doi:10.1080/0305215X.2012.687732,deschinkel2012column}. {\it  In the PeCO
-  protocol, each  leader, in charge  of a  subregion, solves an  integer program
-  which has a twofold objective: minimize the overcoverage and the undercoverage
-  of the perimeter of each sensor.}
-
-
-
-The authors in \citep{Idrees2} propose a Distributed Lifetime Coverage Optimization (DiLCO) protocol, maintains the coverage and improves the lifetime in WSNs. It is  an improved version
-of a research work they presented in~\citep{idrees2014coverage}.  First, they partition the area of interest into subregions using a divide-and-conquer method. DiLCO protocol is then distributed on the sensor nodes in each subregion in a second step. DiLCO protocol combines two techniques: a leader election in each subregion, followed by an optimization-based node activity scheduling performed by each elected leader. The proposed DiLCO protocol is a periodic protocol where each period is decomposed into 4 phases: information exchange, leader election, decision, and sensing. The simulations show that DiLCO is able to increase the WSN lifetime and provides improved coverage performance. {\it  In the PeCO
-  protocol, We have proposed a new mathematical optimization model. Instead of trying to
-cover a set of specified points/targets as in DiLCO protocol, we formulate an integer program based
-on perimeter coverage of each sensor. The model involves integer variables to capture the deviations between the actual level of coverage and the required level. The idea is that an optimal scheduling will be obtained by minimizing a weighted sum of these deviations.}
-  
-  
+used~\citep{castano2013column,doi:10.1080/0305215X.2012.687732,deschinkel2012column}.
+{\it In  the PeCO  protocol, each leader,  in charge of  a subregion,  solves an
+  integer program which  has a twofold objective: minimize  the overcoverage and
+  the undercoverage of the perimeter of each sensor.}
+
+The  authors   in  \citep{Idrees2}  propose  a   Distributed  Lifetime  Coverage
+Optimization (DiLCO)  protocol, which  maintains the  coverage and  improves the
+lifetime  in WSNs.   It is  an  improved version  of a  research work  presented
+in~\citep{idrees2014coverage}.  First, the area  of interest is partitioned into
+subregions using a divide-and-conquer method. DiLCO protocol is then distributed
+on the  sensor nodes  in each  subregion in  a second  step. Hence this protocol
+combines two  techniques: a leader  election in  each subregion, followed  by an
+optimization-based   node  activity   scheduling  performed   by  each   elected
+leader. The proposed DiLCO protocol is  a periodic protocol where each period is
+decomposed into 4  phases: information exchange, leader  election, decision, and
+sensing. The  simulations show that DiLCO  is able to increase  the WSN lifetime
+and provides  improved coverage performance.  {\it  In the PeCO protocol,  a new
+  mathematical optimization model is proposed. Instead  of trying to cover a set
+  of  specified points/targets  as in  DiLCO protocol,  we formulate  an integer
+  program based on perimeter coverage of each sensor. The model involves integer
+  variables to capture  the deviations between the actual level  of coverage and
+  the required level. The idea is that an optimal scheduling will be obtained by
+  minimizing a weighted sum of these deviations.}

   
   

-

 \section{ The P{\scshape e}CO Protocol Description}
 \label{sec:The PeCO Protocol Description}
 
 \section{ The P{\scshape e}CO Protocol Description}
 \label{sec:The PeCO Protocol Description}
 

-In  this  section,  the Perimeter-based  Coverage
-Optimization protocol is decribed in details.  First we present the  assumptions we made and the models
-we considered (in particular the perimeter coverage one), second we describe the
-background idea of our protocol, and third  we give the outline of the algorithm
-executed by each node.
+%In  this  section,  the Perimeter-based  Coverage
+%Optimization protocol is decribed in details.  First we present the  assumptions we made and the models
+%we considered (in particular the perimeter coverage one), second we describe the
+%background idea of our protocol, and third  we give the outline of the algorithm
+%executed by each node.

 
 
 \subsection{Assumptions and Models}
 \label{CI}
 
 
 
 \subsection{Assumptions and Models}
 \label{CI}
 

-A WSN consisting of $J$ stationary sensor nodes randomly and uniformly
+A  WSN  consisting  of  $J$  stationary  sensor  nodes  randomly  and  uniformly

 distributed in  a bounded sensor field  is considered. The wireless  sensors are
 deployed in high density  to ensure initially a high coverage  ratio of the area
 distributed in  a bounded sensor field  is considered. The wireless  sensors are
 deployed in high density  to ensure initially a high coverage  ratio of the area

-of interest.  We  assume that all the  sensor nodes are homogeneous  in terms of
-communication,  sensing,  and  processing capabilities  and  heterogeneous  from
-the energy provision  point of  view.  The  location information  is available  to a
+of interest.  All  the sensor nodes are  supposed to be homogeneous  in terms of
+communication, sensing,  and processing capabilities and  heterogeneous from the
+energy provision  point of  view.  The  location information  is available  to a

 sensor node either  through hardware such as embedded GPS  or location discovery
 sensor node either  through hardware such as embedded GPS  or location discovery

-algorithms. We consider a Boolean disk coverage model,
-which is the most  widely used sensor coverage model in  the literature, and all
-sensor nodes  have a constant sensing  range $R_s$.  Thus, all  the space points
-within a disk centered at a sensor with  a radius equal to the sensing range are
-said to be covered  by this sensor. We also assume  that the communication range
-$R_c$ satisfies $R_c  \geq 2 \cdot R_s$. In fact,  \citet{Zhang05}
-proved  that if  the  transmission  range fulfills  the  previous hypothesis,  the
-complete coverage of a convex area implies connectivity among active nodes.
-
-The PeCO protocol  uses the  same perimeter-coverage  model as \citet{huang2005coverage}. It  can be expressed as follows:  a sensor is
-said to be perimeter  covered if all the points on its  perimeter are covered by
-at least  one sensor  other than  itself. Authors \citet{huang2005coverage}  proved that  a network  area is
-$k$-covered (every point in the area covered by at least k sensors) if and only if each sensor in the network is $k$-perimeter-covered (perimeter covered by at least $k$ sensors). 
+algorithms. A Boolean disk coverage model,  which is the most widely used sensor
+coverage model  in the  literature, is  considered and all  sensor nodes  have a
+constant sensing range $R_s$.  Thus, all the space points within a disk centered
+at a sensor with  a radius equal to the sensing range are  said to be covered by
+this sensor.  We  also assume that the communication range  $R_c$ satisfies $R_c
+\geq 2  \cdot R_s$.  In  fact, \citet{Zhang05}  proved that if  the transmission
+range fulfills the  previous hypothesis, the complete coverage of  a convex area
+implies connectivity among active nodes.
+
+The    PeCO   protocol    uses    the   same    perimeter-coverage   model    as
+\citet{huang2005coverage}. It can  be expressed as follows: a sensor  is said to
+be perimeter covered if all the points  on its perimeter are covered by at least
+one sensor other  than itself.  Authors \citet{huang2005coverage}  proved that a
+network area  is $k$-covered  (every point in  the area is  covered by  at least
+$k$~sensors) if and only if each  sensor in the network is $k$-perimeter-covered
+(perimeter covered by at least $k$ sensors).

  
  

-Figure~\ref{figure1}(a)  shows  the coverage  of  sensor  node~$0$. On  this
-figure, sensor~$0$ has  nine neighbors and we  have reported on
-its  perimeter (the  perimeter  of the  disk  covered by  the  sensor) for  each
-neighbor  the  two  points  resulting  from the intersection  of  the  two  sensing
-areas. These points are denoted for  neighbor~$i$ by $iL$ and $iR$, respectively
-for  left and  right from  a neighboing  point of  view.  The  resulting couples  of
-intersection points subdivide  the perimeter of sensor~$0$  into portions called
-arcs.
+Figure~\ref{figure1}(a) shows the coverage of  sensor node~$0$.  On this figure,
+sensor~$0$  has nine  neighbors  and  we have  reported  on  its perimeter  (the
+perimeter of the  disk covered by the  sensor) for each neighbor  the two points
+resulting from  the intersection  of the  two sensing  areas.  These  points are
+denoted for neighbor~$i$ by $iL$ and  $iR$, respectively for left and right from
+a  neighboring point  of view.   The  resulting couples  of intersection  points
+subdivide the perimeter of sensor~$0$ into portions called arcs.

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


@@ -260,52 +273,48 @@ arcs.
   \label{figure1}
 \end{figure} 
 
   \label{figure1}
 \end{figure} 
 

-Figure~\ref{figure1}(b) describes the geometric information used to find the
+Figure~\ref{figure1}(b)  describes the  geometric information  used to  find the

 locations of the  left and right points of  an arc on the perimeter  of a sensor
 node~$u$ covered by a sensor node~$v$. Node~$v$ is supposed to be located on the
 west  side of  sensor~$u$,  with  the following  respective  coordinates in  the
 locations of the  left and right points of  an arc on the perimeter  of a sensor
 node~$u$ covered by a sensor node~$v$. Node~$v$ is supposed to be located on the
 west  side of  sensor~$u$,  with  the following  respective  coordinates in  the

-sensing area~: $(v_x,v_y)$ and $(u_x,u_y)$. From the previous coordinates 
-the euclidean distance between nodes~$u$ and $v$ is computed: $Dist(u,v)=\sqrt{\vert
-  u_x  - v_x  \vert^2 +  \vert u_y-v_y  \vert^2}$, while  the angle~$\alpha$  is
-obtained through  the formula:
+sensing area~:  $(v_x,v_y)$ and $(u_x,u_y)$.  From the previous  coordinates the
+euclidean distance between nodes~$u$ and $v$ is computed as follows:
+$$
+  Dist(u,v)=\sqrt{\vert u_x - v_x \vert^2 + \vert u_y-v_y \vert^2},
+$$
+while the angle~$\alpha$ is obtained through the formula:

  \[
  \[

-\alpha =  \arccos \left(\frac{Dist(u,v)}{2R_s}
-\right).
+\alpha = \arccos \left(\frac{Dist(u,v)}{2R_s} \right).

 \] 
 \] 

-The arc on the perimeter of~$u$ defined by the angular interval $[\pi
-  - \alpha,\pi + \alpha]$ is said to be perimeter-covered by sensor~$v$.
+The  arc  on the  perimeter  of~$u$  defined by  the  angular  interval $[\pi  -
+  \alpha,\pi + \alpha]$ is then said to be perimeter-covered by sensor~$v$.

 
 Every couple of intersection points is placed on the angular interval $[0,2\pi)$
 in  a  counterclockwise manner,  leading  to  a  partitioning of  the  interval.
 Figure~\ref{figure1}(a)  illustrates  the arcs  for  the  nine neighbors  of
 
 Every couple of intersection points is placed on the angular interval $[0,2\pi)$
 in  a  counterclockwise manner,  leading  to  a  partitioning of  the  interval.
 Figure~\ref{figure1}(a)  illustrates  the arcs  for  the  nine neighbors  of

-sensor $0$ and  Figure~\ref{figure2} gives the position of  the corresponding arcs
+sensor $0$ and  Table~\ref{my-label} gives the position of  the corresponding arcs

 in  the interval  $[0,2\pi)$. More  precisely, the  points are
 ordered according  to the  measures of  the angles  defined by  their respective
 positions. The intersection points are  then visited one after another, starting
 from the first  intersection point  after  point~zero,  and  the maximum  level  of
 coverage is determined  for each interval defined by two  successive points. The
 maximum  level of  coverage is  equal to  the number  of overlapping  arcs.  For
 in  the interval  $[0,2\pi)$. More  precisely, the  points are
 ordered according  to the  measures of  the angles  defined by  their respective
 positions. The intersection points are  then visited one after another, starting
 from the first  intersection point  after  point~zero,  and  the maximum  level  of
 coverage is determined  for each interval defined by two  successive points. The
 maximum  level of  coverage is  equal to  the number  of overlapping  arcs.  For

-example, 
-between~$5L$  and~$6L$ the maximum  level of  coverage is equal  to $3$
+example, between~$5L$  and~$6L$ the maximum  level of  coverage is equal  to $3$

 (the value is highlighted in yellow  at the bottom of Figure~\ref{figure2}), which
 means that at most 2~neighbors can cover  the perimeter in addition to node $0$. 
 Table~\ref{my-label} summarizes for each coverage  interval the maximum level of
 coverage and  the sensor  nodes covering the  perimeter.  The  example discussed
 above is thus given by the sixth line of the table.
 
 (the value is highlighted in yellow  at the bottom of Figure~\ref{figure2}), which
 means that at most 2~neighbors can cover  the perimeter in addition to node $0$. 
 Table~\ref{my-label} summarizes for each coverage  interval the maximum level of
 coverage and  the sensor  nodes covering the  perimeter.  The  example discussed
 above is thus given by the sixth line of the table.
 

-

 \begin{figure*}[t!]
 \centering
 \begin{figure*}[t!]
 \centering

-\includegraphics[width=127.5mm]{figure2.eps}  
+\includegraphics[width=0.95\linewidth]{figure2.eps}  

 \caption{Maximum coverage levels for perimeter of sensor node $0$.}
 \label{figure2}
 \end{figure*} 
 
 \caption{Maximum coverage levels for perimeter of sensor node $0$.}
 \label{figure2}
 \end{figure*} 
 

-
-
-
- \begin{table}
- \tbl{Coverage intervals and contributing sensors for sensor node 0 \label{my-label}}
+\begin{table}
+\tbl{Coverage intervals and contributing sensors for node 0 \label{my-label}}

 {\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
 \hline
 \begin{tabular}[c]{@{}c@{}}Left \\ point \\ angle~$\alpha$ \end{tabular} & \begin{tabular}[c]{@{}c@{}}Interval \\ left \\ point\end{tabular} & \begin{tabular}[c]{@{}c@{}}Interval \\ right \\ point\end{tabular} & \begin{tabular}[c]{@{}c@{}}Maximum \\ coverage\\  level\end{tabular} & \multicolumn{5}{c|}{\begin{tabular}[c]{@{}c@{}}Set of sensors\\ involved \\ in coverage interval\end{tabular}} \\ \hline
 {\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
 \hline
 \begin{tabular}[c]{@{}c@{}}Left \\ point \\ angle~$\alpha$ \end{tabular} & \begin{tabular}[c]{@{}c@{}}Interval \\ left \\ point\end{tabular} & \begin{tabular}[c]{@{}c@{}}Interval \\ right \\ point\end{tabular} & \begin{tabular}[c]{@{}c@{}}Maximum \\ coverage\\  level\end{tabular} & \multicolumn{5}{c|}{\begin{tabular}[c]{@{}c@{}}Set of sensors\\ involved \\ in coverage interval\end{tabular}} \\ \hline


@@ -332,41 +341,43 @@ above is thus given by the sixth line of the table.
 
 \end{table}
 
 
 \end{table}
 

-
-
-
-In the PeCO  protocol, the scheduling of the sensor  nodes' activities is formulated  with an
-integer program  based on  coverage intervals. The  formulation of  the coverage
+In  the  PeCO protocol,  the  scheduling  of  the  sensor nodes'  activities  is
+formulated    with    an    mixed-integer     program    based    on    coverage
+intervals~\citep{doi:10.1155/2010/926075}.  The  formulation   of  the  coverage

 optimization problem is  detailed in~Section~\ref{cp}.  Note that  when a sensor
 node  has a  part of  its sensing  range outside  the WSN  sensing field,  as in
 optimization problem is  detailed in~Section~\ref{cp}.  Note that  when a sensor
 node  has a  part of  its sensing  range outside  the WSN  sensing field,  as in

-Figure~\ref{figure3}, the maximum coverage level for  this arc is set to $\infty$
+Figure~\ref{figure3}, the maximum coverage level for this arc is set to $\infty$

 and  the  corresponding  interval  will  not   be  taken  into  account  by  the
 optimization algorithm.
 
 and  the  corresponding  interval  will  not   be  taken  into  account  by  the
 optimization algorithm.
 

- \newpage
+%\newpage

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

-\includegraphics[width=62.5mm]{figure3.eps}  
+\includegraphics[width=57.5mm]{figure3.eps}  

 \caption{Sensing range outside the WSN's area of interest.}
 \label{figure3}
 \caption{Sensing range outside the WSN's area of interest.}
 \label{figure3}

-\end{figure} 
-
-
-\subsection{The Main Idea}
-
-The  WSN area of  interest is, in a  first step, divided  into regular
-homogeneous subregions  using a divide-and-conquer  algorithm. In a  second step
-our  protocol  will  be  executed  in   a  distributed  way  in  each  subregion
-simultaneously to schedule nodes' activities for one sensing period.
-
-As  shown in  Figure~\ref{figure4}, node  activity  scheduling is  produced by  our
-protocol in a periodic manner. Each period is divided into 4 stages: Information
-(INFO)  Exchange,  Leader Election,  Decision  (the  result of  an  optimization
-problem),  and  Sensing.   For  each  period there  is  exactly  one  set  cover
-responsible for  the sensing task.  Protocols  based on a periodic  scheme, like
-PeCO, are more  robust against an unexpected  node failure. On the  one hand, if
-a node failure is discovered before  taking the decision, the corresponding sensor
-node will  not be considered  by the optimization  algorithm. On  the other
+\end{figure}
+
+\vspace{-0.25cm}
+
+\subsection{Main Idea}
+
+The WSN area of  interest is, in a first step,  divided into regular homogeneous
+subregions using a  divide-and-conquer algorithm. In a second  step our protocol
+will  be executed  in  a distributed  way in  each  subregion simultaneously  to
+schedule nodes' activities  for one sensing period. Sensor nodes  are assumed to
+be deployed  almost uniformly over the  region. The regular subdivision  is made
+such that the number of hops between  any pairs of sensors inside a subregion is
+less than or equal to 3.
+
+As shown  in Figure~\ref{figure4}, node  activity scheduling is produced  by the
+proposed protocol  in a periodic manner.  Each period is divided  into 4 stages:
+Information  (INFO)  Exchange,  Leader  Election, Decision  (the  result  of  an
+optimization problem),  and Sensing.  For each  period there is exactly  one set
+cover responsible for  the sensing task.  Protocols based on  a periodic scheme,
+like PeCO, are more robust against an  unexpected node failure. On the one hand,
+if a  node failure is discovered  before taking the decision,  the corresponding
+sensor node will  not be considered by the optimization  algorithm. On the other

 hand, if the sensor failure happens after  the decision, the sensing task of the
 network will be temporarily affected: only  during the period of sensing until a
 new period starts, since a new set cover will take charge of the sensing task in
 hand, if the sensor failure happens after  the decision, the sensing task of the
 network will be temporarily affected: only  during the period of sensing until a
 new period starts, since a new set cover will take charge of the sensing task in


@@ -375,7 +386,8 @@ taken  into  account since  the  sensors  can  update  and then  exchange  their
 information (including their  residual energy) at the beginning  of each period.
 However, the pre-sensing  phases (INFO Exchange, Leader  Election, and Decision)
 are energy consuming, even for nodes that will not join the set cover to monitor
 information (including their  residual energy) at the beginning  of each period.
 However, the pre-sensing  phases (INFO Exchange, Leader  Election, and Decision)
 are energy consuming, even for nodes that will not join the set cover to monitor

-the area.
+the area. Sensing  period duration is adapted according to  the QoS requirements
+of the application.

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


@@ -385,7 +397,6 @@ the area.
 \end{figure} 
 
 We define two types of packets to be used by PeCO protocol:
 \end{figure} 
 
 We define two types of packets to be used by PeCO protocol:

-

 \begin{itemize} 
 \item INFO  packet: sent  by each  sensor node to  all the  nodes inside  a same
   subregion for information exchange.
 \begin{itemize} 
 \item INFO  packet: sent  by each  sensor node to  all the  nodes inside  a same
   subregion for information exchange.


@@ -394,9 +405,7 @@ We define two types of packets to be used by PeCO protocol:
   sensing phase.
 \end{itemize}
 
   sensing phase.
 \end{itemize}
 

-

 Five statuses are possible for a sensor node in the network:
 Five statuses are possible for a sensor node in the network:

-

 \begin{itemize} 
 \item LISTENING: waits for a decision (to be active or not);
 \item COMPUTATION: executes the optimization algorithm as leader to
 \begin{itemize} 
 \item LISTENING: waits for a decision (to be active or not);
 \item COMPUTATION: executes the optimization algorithm as leader to


@@ -406,89 +415,130 @@ Five statuses are possible for a sensor node in the network:
 \item COMMUNICATION: transmits or receives packets.
 \end{itemize}
 
 \item COMMUNICATION: transmits or receives packets.
 \end{itemize}
 

-

 \subsection{PeCO Protocol Algorithm}
 
 \subsection{PeCO Protocol Algorithm}
 

-The  pseudocode implementing the  protocol on  a node is  given below.
-More  precisely,  Algorithm~\ref{alg:PeCO}  gives  a brief  description  of  the
-protocol applied by a sensor node $s_k$ where $k$ is the node index in the WSN.
-
+The  pseudocode implementing  the  protocol  on a  node  is  given below.   More
+precisely, Algorithm~\ref{alg:PeCO}  gives a  brief description of  the protocol
+applied by a sensor node $s_k$ where $k$ is the node index in the WSN.

 
 
 
 

-\begin{algorithm}      
+\begin{algorithm2e}      

  % \KwIn{all the parameters related to information exchange}
 %  \KwOut{$winer-node$ (: the id of the winner sensor node, which is the leader of current round)}
 %  \BlankLine
  % \KwIn{all the parameters related to information exchange}
 %  \KwOut{$winer-node$ (: the id of the winner sensor node, which is the leader of current round)}
 %  \BlankLine

-  %\emph{Initialize the sensor node and determine it's position and subregion} \; 
-  
-\noindent{\bf If} $RE_k \geq E_{th}$ {\bf then}\\
-\hspace*{0.6cm} \emph{$s_k.status$ = COMMUNICATION;}\\
-\hspace*{0.6cm}  \emph{Send $INFO()$ packet to other nodes in subregion;}\\
-\hspace*{0.6cm}  \emph{Wait $INFO()$ packet from other nodes in subregion;}\\
-\hspace*{0.6cm} \emph{Update K.CurrentSize;}\\
-\hspace*{0.6cm}  \emph{LeaderID = Leader election;}\\
-\hspace*{0.6cm} {\bf If} $ s_k.ID = LeaderID $ {\bf then}\\
-\hspace*{1.2cm}   \emph{$s_k.status$ = COMPUTATION;}\\
-\hspace*{1.2cm}{\bf If} \emph{$ s_k.ID $ is Not previously selected as a Leader} {\bf then}\\
-\hspace*{1.8cm} \emph{ Execute the perimeter coverage model;}\\
-\hspace*{1.2cm} {\bf end}\\
-\hspace*{1.2cm}{\bf If} \emph{($s_k.ID $ is the same Previous Leader)~And~(K.CurrentSize = K.PreviousSize)}\\
-\hspace*{1.8cm} \emph{ Use the same previous cover set for current sensing stage;}\\
-\hspace*{1.2cm}  {\bf end}\\
-\hspace*{1.2cm}  {\bf else}\\
-\hspace*{1.8cm}\emph{Update $a^j_{ik}$; prepare data for IP~Algorithm;}\\
-\hspace*{1.8cm} \emph{$\left\{\left(X_{1},\dots,X_{l},\dots,X_{K}\right)\right\}$ = Execute Integer Program Algorithm($K$);}\\
-\hspace*{1.8cm} \emph{K.PreviousSize = K.CurrentSize;}\\
-\hspace*{1.2cm}  {\bf end}\\
-\hspace*{1.2cm}\emph{$s_k.status$ = COMMUNICATION;}\\
-\hspace*{1.2cm}\emph{Send $ActiveSleep()$ to each node $l$ in subregion;}\\
-\hspace*{1.2cm}\emph{Update $RE_k $;}\\
-\hspace*{0.6cm}  {\bf end}\\
-\hspace*{0.6cm}  {\bf else}\\
-\hspace*{1.2cm}\emph{$s_k.status$ = LISTENING;}\\
-\hspace*{1.2cm}\emph{Wait $ActiveSleep()$ packet from the Leader;}\\
-\hspace*{1.2cm}\emph{Update $RE_k $;}\\
-\hspace*{0.6cm}  {\bf end}\\
-{\bf end}\\
-{\bf else}\\
-\hspace*{0.6cm} \emph{Exclude $s_k$ from entering in the current sensing stage;}\\
-{\bf end}\\
-\label{alg:PeCO}
-\end{algorithm}
-
-
-
-In this  algorithm, K.CurrentSize and K.PreviousSize  respectively represent the
-current number and  the previous number of living nodes in  the subnetwork of the
-subregion.  Initially, the sensor node checks its remaining energy $RE_k$, which
-must be greater than a threshold $E_{th}$ in order to participate in the current
-period.  Each  sensor node  determines its position  and its subregion  using an
-embedded  GPS or a  location discovery  algorithm. After  that, all  the sensors
-collect position coordinates,  remaining energy, sensor node ID,  and the number
-of their  one-hop live  neighbors during the  information exchange.  The sensors
-inside a same region cooperate to elect a leader. The selection criteria for the
-leader, in order of priority,  are: larger numbers of neighbors, larger remaining
-energy, and  then in case  of equality, larger  index.  Once chosen,  the leader
-collects information to formulate and  solve the integer program which allows to
-construct the set of active sensors in the sensing stage.
-
+  %\emph{Initialize the sensor node and determine it's position and subregion} \;
+  \label{alg:PeCO}
+  \caption{PeCO pseudocode}
+  \eIf{$RE_k \geq E_{th}$}{
+    $s_k.status$ = COMMUNICATION\;
+    Send $INFO()$ packet to other nodes in subregion\;
+    Wait $INFO()$ packet from other nodes in subregion\;
+    Update K.CurrentSize\;
+    LeaderID = Leader election\;
+    \eIf{$s_k.ID = LeaderID$}{
+      $s_k.status$ = COMPUTATION\;
+      \If{$ s_k.ID $ is Not previously selected as a Leader}{
+        Execute the perimeter coverage model\;
+      }
+      \eIf{($s_k.ID $ is the same Previous Leader) {\bf and} \\
+        \indent (K.CurrentSize = K.PreviousSize)}{
+        Use the same previous cover set for current sensing stage\;
+      }{
+        Update $a^j_{ik}$; prepare data for IP~Algorithm\;
+        $\left\{\left(X_{1},\dots,X_{l},\dots,X_{K}\right)\right\}$ = Execute Integer Program Algorithm($K$)\;
+        K.PreviousSize = K.CurrentSize\;
+      }
+      $s_k.status$ = COMMUNICATION\;
+      Send $ActiveSleep()$ to each node $l$ in subregion\;
+      Update $RE_k $\;
+    }{
+      $s_k.status$ = LISTENING\;
+      Wait $ActiveSleep()$ packet from the Leader\;
+      Update $RE_k $\;
+    }
+  }{
+    Exclude $s_k$ from entering in the current sensing stage\;
+  }
+\end{algorithm2e}
+
+%\begin{algorithm}
+%\noindent{\bf If} $RE_k \geq E_{th}$ {\bf then}\\
+%\hspace*{0.6cm} \emph{$s_k.status$ = COMMUNICATION;}\\
+%\hspace*{0.6cm}  \emph{Send $INFO()$ packet to other nodes in subregion;}\\
+%\hspace*{0.6cm}  \emph{Wait $INFO()$ packet from other nodes in subregion;}\\
+%\hspace*{0.6cm} \emph{Update K.CurrentSize;}\\
+%\hspace*{0.6cm}  \emph{LeaderID = Leader election;}\\
+%\hspace*{0.6cm} {\bf If} $ s_k.ID = LeaderID $ {\bf then}\\
+%\hspace*{1.2cm}   \emph{$s_k.status$ = COMPUTATION;}\\
+%\hspace*{1.2cm}{\bf If} \emph{$ s_k.ID $ is Not previously selected as a Leader} {\bf then}\\
+%\hspace*{1.8cm} \emph{ Execute the perimeter coverage model;}\\
+%\hspace*{1.2cm} {\bf end}\\
+%\hspace*{1.2cm}{\bf If} \emph{($s_k.ID $ is the same Previous Leader)~And~(K.CurrentSize = K.PreviousSize)}\\
+%\hspace*{1.8cm} \emph{ Use the same previous cover set for current sensing stage;}\\
+%\hspace*{1.2cm}  {\bf end}\\
+%\hspace*{1.2cm}  {\bf else}\\
+%\hspace*{1.8cm}\emph{Update $a^j_{ik}$; prepare data for IP~Algorithm;}\\
+%\hspace*{1.8cm} \emph{$\left\{\left(X_{1},\dots,X_{l},\dots,X_{K}\right)\right\}$ = Execute Integer Program Algorithm($K$);}\\
+%\hspace*{1.8cm} \emph{K.PreviousSize = K.CurrentSize;}\\
+%\hspace*{1.2cm}  {\bf end}\\
+%\hspace*{1.2cm}\emph{$s_k.status$ = COMMUNICATION;}\\
+%\hspace*{1.2cm}\emph{Send $ActiveSleep()$ to each node $l$ in subregion;}\\
+%\hspace*{1.2cm}\emph{Update $RE_k $;}\\
+%\hspace*{0.6cm}  {\bf end}\\
+%\hspace*{0.6cm}  {\bf else}\\
+%\hspace*{1.2cm}\emph{$s_k.status$ = LISTENING;}\\
+%\hspace*{1.2cm}\emph{Wait $ActiveSleep()$ packet from the Leader;}\\
+%\hspace*{1.2cm}\emph{Update $RE_k $;}\\
+%\hspace*{0.6cm}  {\bf end}\\
+%{\bf end}\\
+%{\bf else}\\
+%\hspace*{0.6cm} \emph{Exclude $s_k$ from entering in the current sensing stage;}\\
+%{\bf end}\\
+%\label{alg:PeCO}
+%\end{algorithm}
+
+In this  algorithm, $K.CurrentSize$ and $K.PreviousSize$  respectively represent
+the current number and the previous number  of living nodes in the subnetwork of
+the  subregion.   At the  beginning  of  the  first period  $K.PreviousSize$  is
+initialized to  zero.  Initially,  the sensor node  checks its  remaining energy
+$RE_k$, which must be greater than  a threshold $E_{th}$ in order to participate
+in  the current  period.   Each  sensor node  determines  its  position and  its
+subregion using an  embedded GPS or a location discovery  algorithm. After that,
+all the sensors collect position  coordinates, remaining energy, sensor node ID,
+and the number of their one-hop  live neighbors during the information exchange.
+The sensors  inside a same  region cooperate to  elect a leader.   The selection
+criteria for the leader are (in order  of priority):
+\begin{enumerate}
+\item larger number of neighbors;
+\item larger  remaining energy;
+\item and then  in case  of equality,  larger index.
+\end{enumerate}
+Once chosen, the leader collects information  to formulate and solve the integer
+program  which allows  to construct  the set  of active  sensors in  the sensing
+stage.

 
 \section{Perimeter-based Coverage Problem Formulation}
 \label{cp}
 
 
 \section{Perimeter-based Coverage Problem Formulation}
 \label{cp}
 

-In this  section, the coverage model is  mathematically formulated. The following
-notations are used  throughout the
-section.\\
+In  this  section,  the   perimeter-based  coverage  problem  is  mathematically
+formulated.    It    has    been    proved   to    be    a    NP-hard    problem
+by \citep{doi:10.1155/2010/926075}. Authors  study the coverage of  the perimeter
+of a  large object requiring  to be monitored.  For the proposed  formulation in
+this paper,  the large  object to  be monitored  is the  sensor itself  (or more
+precisely its sensing area).
+
+The following notations are used  throughout the section.
+

 First, the following sets:
 \begin{itemize}
 First, the following sets:
 \begin{itemize}

-\item $S$ represents the set of WSN sensor nodes;
+\item $S$ represents the set of sensor nodes;

 \item $A \subseteq S $ is the subset of alive sensors;
 \item  $I_j$  designates  the  set  of  coverage  intervals  (CI)  obtained  for
   sensor~$j$.
 \end{itemize}
 $I_j$ refers to the set of  coverage intervals which have been defined according
 \item $A \subseteq S $ is the subset of alive sensors;
 \item  $I_j$  designates  the  set  of  coverage  intervals  (CI)  obtained  for
   sensor~$j$.
 \end{itemize}
 $I_j$ refers to the set of  coverage intervals which have been defined according

-to the  method introduced in  subsection~\ref{CI}. For a coverage  interval $i$,
-let $a^j_{ik}$ denotes  the indicator function of whether  sensor~$k$ is involved
+to the  method introduced in  Subsection~\ref{CI}. For a coverage  interval $i$,
+let $a^j_{ik}$ denote  the indicator function of whether  sensor~$k$ is involved

 in coverage interval~$i$ of sensor~$j$, that is:
 \begin{equation}
 a^j_{ik} = \left \{ 
 in coverage interval~$i$ of sensor~$j$, that is:
 \begin{equation}
 a^j_{ik} = \left \{ 


@@ -500,126 +550,141 @@ a^j_{ik} = \left \{
 \end{equation}
 Note that $a^k_{ik}=1$ by definition of the interval.
 
 \end{equation}
 Note that $a^k_{ik}=1$ by definition of the interval.
 

-Second, several binary  and integer  variables are defined.  Hence,  each binary
-variable $X_{k}$  determines the activation of  sensor $k$ in the  sensing phase
-($X_k=1$ if  the sensor $k$  is active or 0  otherwise).  $M^j_i$ is  an integer
-variable  which  measures  the  undercoverage  for  the  coverage  interval  $i$
-corresponding to  sensor~$j$. In  the same  way, the  overcoverage for  the same
-coverage interval is given by the variable $V^j_i$.
-
-If we decide to sustain a level of coverage equal to $l$ all along the perimeter
-of sensor  $j$, we have  to ensure  that at least  $l$ sensors involved  in each
-coverage  interval $i  \in I_j$  of  sensor $j$  are active.   According to  the
-previous notations, the number of active sensors in the coverage interval $i$ of
-sensor $j$  is given by  $\sum_{k \in A} a^j_{ik}  X_k$.  To extend  the network
-lifetime,  the objective  is to  activate a  minimal number  of sensors  in each
-period to  ensure the  desired coverage  level. As the  number of  alive sensors
-decreases, it becomes impossible to reach  the desired level of coverage for all
-coverage intervals. Therefore  variables  $M^j_i$ and $V^j_i$ are introduced as a measure
-of the  deviation between  the desired  number of active  sensors in  a coverage
-interval and  the effective  number. And  we try  to minimize  these deviations,
-first to  force the  activation of  a minimal  number of  sensors to  ensure the
-desired coverage level, and if the desired level cannot be completely satisfied,
-to reach a coverage level as close as possible to the desired one.
-
-
-
-
-Our coverage optimization problem can then be mathematically expressed as follows: 
-
-\begin{equation} 
-\left \{
-\begin{array}{ll}
-\min \sum_{j \in S} \sum_{i \in I_j} (\alpha^j_i ~ M^j_i + \beta^j_i ~ V^j_i )&\\
-\textrm{subject to :}&\\
-\sum_{k \in A} ( a^j_{ik} ~ X_{k}) + M^j_i  \geq l \quad \forall i \in I_j, \forall j \in S\\
-\sum_{k \in A} ( a^j_{ik} ~ X_{k}) - V^j_i  \leq l \quad \forall i \in I_j, \forall j \in S\\
-X_{k} \in \{0,1\}, \forall k \in A
-\end{array}
-\right.
+Second,  several variables  are defined.   Hence, each  binary variable  $X_{k}$
+determines the  activation of sensor  $k$ in the  sensing phase ($X_k=1$  if the
+sensor $k$ is active or 0 otherwise).   $M^j_i$ is a variable which measures the
+undercoverage for the coverage interval  $i$ corresponding to sensor~$j$. In the
+same  way, the  overcoverage for  the  same coverage  interval is  given by  the
+variable $V^j_i$.
+
+To sustain a  level of coverage equal  to $l$ all along the  perimeter of sensor
+$j$, at  least $l$  sensors involved in  each coverage interval  $i \in  I_j$ of
+sensor $j$ have  to be active.  According to the  previous notations, the number
+of  active sensors  in the  coverage  interval $i$  of  sensor $j$  is given  by
+$\sum_{k \in A} a^j_{ik} X_k$.  To extend the network lifetime, the objective is
+to activate  a minimal number  of sensors in each  period to ensure  the desired
+coverage level. As the number of  alive sensors decreases, it becomes impossible
+to reach  the desired level  of coverage  for all coverage  intervals. Therefore
+variables  $M^j_i$ and  $V^j_i$ are  introduced as  a measure  of the  deviation
+between the  desired number  of active  sensors in a  coverage interval  and the
+effective number.  And we try to  minimize these deviations, first  to force the
+activation of a minimal number of  sensors to ensure the desired coverage level,
+and if  the desired level  cannot be completely  satisfied, to reach  a coverage
+level as close as possible to the desired one.
+
+The coverage optimization problem can then be mathematically expressed as follows:
+\begin{equation}
+  \begin{aligned}
+    \text{Minimize } & \sum_{j \in S} \sum_{i \in I_j} (\alpha^j_i ~ M^j_i + \beta^j_i ~ V^j_i ) \\
+    \text{Subject to:} & \\
+    & \sum_{k \in A} ( a^j_{ik} ~ X_{k}) + M^j_i  \geq l \quad \forall i \in I_j, \forall j \in S  \\
+    & \sum_{k \in A} ( a^j_{ik} ~ X_{k}) - V^j_i  \leq l \quad \forall i \in I_j, \forall j \in S \\
+    & X_{k} \in \{0,1\}, \forall k \in A \\
+    & M^j_i, V^j_i \in \mathbb{R}^{+} 
+  \end{aligned}

 \end{equation}
 
 \end{equation}
 

+%\begin{equation} 
+%\left \{
+%\begin{array}{ll}
+%\min \sum_{j \in S} \sum_{i \in I_j} (\alpha^j_i ~ M^j_i + \beta^j_i ~ V^j_i ) & \\
+%\textrm{subject to :} &\\
+%\sum_{k \in A} ( a^j_{ik} ~ X_{k}) + M^j_i  \geq l \quad \forall i \in I_j, \forall j \in S\\
+%\sum_{k \in A} ( a^j_{ik} ~ X_{k}) - V^j_i  \leq l \quad \forall i \in I_j, \forall j \in S\\
+%X_{k} \in \{0,1\}, \forall k \in A \\
+%M^j_i, V^j_i \in \mathbb{R}^{+} 
+%\end{array}
+%\right.
+%\end{equation}
+
+If a given level of coverage $l$ is  required for one sensor, the sensor is said
+to be undercovered (respectively overcovered) if the level of coverage of one of
+its  CI  is  less  (respectively  greater)  than $l$.   If  the  sensor  $j$  is
+undercovered, there exists at least one of its CI (say $i$) for which the number
+of active  sensors (denoted by $l^{i}$)  covering this part of  the perimeter is
+less than $l$ and in this case : $M_{i}^{j}=l-l^{i}$, $V_{i}^{j}=0$. Conversely,
+if the sensor $j$ is overcovered, there exists  at least one of its CI (say $i$)
+for which the  number of active sensors (denoted by  $l^{i}$) covering this part
+of  the  perimeter  is  greater  than  $l$  and  in  this  case:  $M_{i}^{j}=0$,
+$V_{i}^{j}=l^{i}-l$.
+

 $\alpha^j_i$ and $\beta^j_i$  are nonnegative weights selected  according to the
 relative importance of satisfying the associated level of coverage. For example,
 weights associated with  coverage intervals of a specified part  of a region may
 $\alpha^j_i$ and $\beta^j_i$  are nonnegative weights selected  according to the
 relative importance of satisfying the associated level of coverage. For example,
 weights associated with  coverage intervals of a specified part  of a region may

-be  given by a  relatively larger  magnitude than  weights associated  with another
-region. This  kind of integer program  is inspired from the  model developed for
-brachytherapy treatment planning  for optimizing dose  distribution
-\citep{0031-9155-44-1-012}. The integer  program must be solved by  the leader in
-each subregion at the beginning of  each sensing phase, whenever the environment
-has  changed (new  leader,  death of  some  sensors). Note  that  the number  of
-constraints in the model is constant  (constraints of coverage expressed for all
-sensors), whereas the number of variables $X_k$ decreases over periods, since 
-only alive  sensors (sensors with enough energy to  be alive during one
-sensing phase) are considered in the model.
+be given by  a relatively larger magnitude than weights  associated with another
+region. This kind of mixed-integer program  is inspired from the model developed
+for   brachytherapy  treatment   planning  for   optimizing  dose   distribution
+\citep{0031-9155-44-1-012}.  The choice of the values for variables $\alpha$ and
+$\beta$  should be  made according  to the  needs of  the application.  $\alpha$
+should be  large enough  to prevent  undercoverage and so  to reach  the highest
+possible coverage ratio. $\beta$ should  be large enough to prevent overcoverage
+and so to activate a minimum  number of sensors.  The mixed-integer program must
+be solved  by the  leader in  each subregion  at the  beginning of  each sensing
+phase, whenever the environment has changed (new leader, death of some sensors).
+Note that  the number of  constraints in the  model is constant  (constraints of
+coverage  expressed for  all sensors),  whereas  the number  of variables  $X_k$
+decreases over periods, since only alive  sensors (sensors with enough energy to
+be alive during one sensing phase) are considered in the model.

 
 \section{Performance Evaluation and Analysis}  
 \label{sec:Simulation Results and Analysis}
 
 
 \section{Performance Evaluation and Analysis}  
 \label{sec:Simulation Results and Analysis}
 

-

 \subsection{Simulation Settings}
 
 \subsection{Simulation Settings}
 

-

 The WSN  area of interest is  supposed to be divided  into 16~regular subregions
 The WSN  area of interest is  supposed to be divided  into 16~regular subregions

-and we use the same energy consumption model as in our previous work~\citep{Idrees2}.
-Table~\ref{table3} gives the chosen parameters settings.
+and   we  use   the  same   energy  consumption   model  as   in  our   previous
+work~\citep{Idrees2}.  Table~\ref{table3} gives the chosen parameters settings.

 
 \begin{table}[ht]
 \tbl{Relevant parameters for network initialization \label{table3}}{
 
 \begin{table}[ht]
 \tbl{Relevant parameters for network initialization \label{table3}}{

-

 \centering
 \centering

-

 \begin{tabular}{c|c}
 \begin{tabular}{c|c}

-

 \hline
 Parameter & Value  \\ [0.5ex]
 \hline
 Parameter & Value  \\ [0.5ex]

-   

 \hline
 % inserts single horizontal line
 \hline
 % inserts single horizontal line

-Sensing field & $(50 \times 25)~m^2 $   \\
-
-WSN size &  100, 150, 200, 250, and 300~nodes   \\
-
-Initial energy  & in range 500-700~Joules  \\  
-
+Sensing field & $(50 \times 25)~m^2 $ \\
+WSN size &  100, 150, 200, 250, and 300~nodes \\
+Initial energy  & in range 500-700~Joules \\  

 Sensing period & duration of 60 minutes \\
 Sensing period & duration of 60 minutes \\

-$E_{th}$ & 36~Joules\\
-$R_s$ & 5~m   \\     
-$R_c$ & 10~m   \\   
-$\alpha^j_i$ & 0.6   \\
-
+$E_{th}$ & 36~Joules \\
+$R_s$ & 5~m \\     
+$R_c$ & 10~m \\   
+$\alpha^j_i$ & 0.6 \\

 $\beta^j_i$ & 0.4
 $\beta^j_i$ & 0.4

-

 \end{tabular}}
 \end{tabular}}

-
-

 \end{table}
 \end{table}

+

 To  obtain  experimental  results  which are  relevant,  simulations  with  five
 different node densities going from  100 to 300~nodes were performed considering
 each time 25~randomly  generated networks. The nodes are deployed  on a field of
 interest of $(50 \times 25)~m^2 $ in such a way that they cover the field with a
 high coverage ratio. Each node has an  initial energy level, in Joules, which is
 To  obtain  experimental  results  which are  relevant,  simulations  with  five
 different node densities going from  100 to 300~nodes were performed considering
 each time 25~randomly  generated networks. The nodes are deployed  on a field of
 interest of $(50 \times 25)~m^2 $ in such a way that they cover the field with a
 high coverage ratio. Each node has an  initial energy level, in Joules, which is

-randomly drawn in the interval $[500-700]$.   If its energy provision reaches a
+randomly drawn in  the interval $[500-700]$.  If its energy  provision reaches a

 value below  the threshold $E_{th}=36$~Joules,  the minimum energy needed  for 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 longer  participate in  the
+node to  stay active  during one period,  it will no  longer participate  in the

 coverage task. This value corresponds to the energy needed by the sensing phase,
 coverage task. This value corresponds to the energy needed by the sensing phase,

-obtained by multiplying  the energy consumed in the active state  (9.72 mW) with the
-time in  seconds for one  period (3600 seconds), and  adding the energy  for the
+obtained by multiplying  the energy consumed in the active  state (9.72 mW) with
+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
 pre-sensing phases.  According  to the interval of initial energy,  a sensor may

-be active during at most 20 periods.
+be active during at most 20 periods. Information exchange to update the coverage
+is executed every  hour, but the length  of the sensing period  could be reduced
+and adapted dynamically. On  the one hand a small sensing  period would allow to
+be more  reliable but would  have result in  higher communication costs.  On the
+other hand  the choice of a  long duration may  cause problems in case  of nodes
+failure during the sensing period.

 
 The values  of $\alpha^j_i$ and  $\beta^j_i$ have been  chosen to ensure  a good
 
 The values  of $\alpha^j_i$ and  $\beta^j_i$ have been  chosen to ensure  a good

-network coverage and a longer WSN lifetime.  Higher priority is given to
-the  undercoverage  (by  setting  the  $\alpha^j_i$ with  a  larger  value  than
-$\beta^j_i$)  so as  to prevent  the non-coverage  for the  interval~$i$ of  the
-sensor~$j$.  On the  other hand,  
-$\beta^j_i$ is assigned to a value which is slightly lower so as to minimize the number of active sensor nodes which contribute
-in covering the interval.
+network coverage  and a longer  WSN lifetime.  Higher  priority is given  to the
+undercoverage (by setting the $\alpha^j_i$ with a larger value than $\beta^j_i$)
+so as  to prevent the non-coverage  for the interval~$i$ of  the sensor~$j$.  On
+the other hand, $\beta^j_i$ is assigned to a value which is slightly lower so as
+to minimize the  number of active sensor nodes which  contribute in covering the
+interval. Subsection~\ref{sec:Impact} investigates more deeply how the values of
+both parameters affect the performance of PeCO protocol.

 
 The following performance metrics are used to evaluate the efficiency of the
 approach.
 
 The following performance metrics are used to evaluate the efficiency of the
 approach.

-
-

 \begin{itemize}
 \item {\bf Network Lifetime}: the lifetime  is defined as the time elapsed until
   the  coverage  ratio  falls  below a  fixed  threshold.   $Lifetime_{95}$  and
 \begin{itemize}
 \item {\bf Network Lifetime}: the lifetime  is defined as the time elapsed until
   the  coverage  ratio  falls  below a  fixed  threshold.   $Lifetime_{95}$  and


@@ -632,42 +697,34 @@ approach.
 \item {\bf  Coverage Ratio (CR)} : it  measures how  well the  WSN is  able to
   observe the area of interest. In our  case, the sensor field is discretized as
   a regular grid, which yields the following equation:
 \item {\bf  Coverage Ratio (CR)} : it  measures how  well the  WSN is  able to
   observe the area of interest. In our  case, the sensor field is discretized as
   a regular grid, which yields the following equation:

-  
-
-\[
+  \begin{equation*}

     \scriptsize
     \mbox{CR}(\%) = \frac{\mbox{$n$}}{\mbox{$N$}} \times 100
     \scriptsize
     \mbox{CR}(\%) = \frac{\mbox{$n$}}{\mbox{$N$}} \times 100

-\]
-
-
+  \end{equation*}

   where $n$  is the  number of covered  grid points by  active sensors  of every
   subregions during  the current sensing phase  and $N$ is total  number of grid
   where $n$  is the  number of covered  grid points by  active sensors  of every
   subregions during  the current sensing phase  and $N$ is total  number of grid

-  points in  the sensing  field.  In  simulations  a  layout of
-  $N~=~51~\times~26~=~1326$~grid points is considered.
+  points in the sensing field. A layout of $N~=~51~\times~26~=~1326$~grid points
+  is considered in the simulations.

 \item {\bf Active Sensors Ratio (ASR)}: a  major objective of our protocol is to
 \item {\bf Active Sensors Ratio (ASR)}: a  major objective of our protocol is to

-  activate  as few nodes as possible,  in order  to minimize  the communication
+  activate as  few nodes  as possible,  in order  to minimize  the communication

   overhead and maximize the WSN lifetime. The active sensors ratio is defined as
   follows:
   overhead and maximize the WSN lifetime. The active sensors ratio is defined as
   follows:

- 
-\[
-    \scriptsize
-    \mbox{ASR}(\%) =  \frac{\sum\limits_{r=1}^R \mbox{$|A_r^p|$}}{\mbox{$|J|$}} \times 100
-\]
-
+  \begin{equation*}
+   \scriptsize
+   \mbox{ASR}(\%) =  \frac{\sum\limits_{r=1}^R \mbox{$|A_r^p|$}}{\mbox{$|J|$}} \times 100
+  \end{equation*}

   where $|A_r^p|$ is  the number of active  sensors in the subregion  $r$ in the
   where $|A_r^p|$ is  the number of active  sensors in the subregion  $r$ in the

-  current sensing period~$p$, $|J|$ is the number of sensors in the network, and
-  $R$ is the number of subregions.
+  sensing period~$p$, $R$  is the number of subregions, and  $|J|$ is the number
+  of sensors in the network.

 \item {\bf Energy Consumption (EC)}: energy consumption can be seen as the total
   energy  consumed by  the  sensors during  $Lifetime_{95}$ or  $Lifetime_{50}$,
   divided by  the number of  periods. The value of  EC is computed  according to
   this formula:
 \item {\bf Energy Consumption (EC)}: energy consumption can be seen as the total
   energy  consumed by  the  sensors during  $Lifetime_{95}$ or  $Lifetime_{50}$,
   divided by  the number of  periods. The value of  EC is computed  according to
   this formula:

-
-\[  
-  \scriptsize
+  \begin{equation*} 
+    \scriptsize

     \mbox{EC} = \frac{\sum\limits_{p=1}^{P} \left( E^{\mbox{com}}_p+E^{\mbox{list}}_p+E^{\mbox{comp}}_p  
       + E^{a}_p+E^{s}_p \right)}{P},
     \mbox{EC} = \frac{\sum\limits_{p=1}^{P} \left( E^{\mbox{com}}_p+E^{\mbox{list}}_p+E^{\mbox{comp}}_p  
       + E^{a}_p+E^{s}_p \right)}{P},

-\]
- 
+  \end{equation*}

   where $P$ corresponds  to the number of periods. The  total energy consumed by
   the  sensors  comes  through  taking   into  consideration  four  main  energy
   factors. The first one, denoted $E^{\scriptsize \mbox{com}}_p$, represents the
   where $P$ corresponds  to the number of periods. The  total energy consumed by
   the  sensors  comes  through  taking   into  consideration  four  main  energy
   factors. The first one, denoted $E^{\scriptsize \mbox{com}}_p$, represents the


@@ -676,57 +733,87 @@ approach.
   the energy  consumed by the sensors  in LISTENING status before  receiving the
   decision to go active or sleep in period $p$.  $E^{\scriptsize \mbox{comp}}_p$
   refers to  the energy  needed by  all the  leader nodes  to solve  the integer
   the energy  consumed by the sensors  in LISTENING status before  receiving the
   decision to go active or sleep in period $p$.  $E^{\scriptsize \mbox{comp}}_p$
   refers to  the energy  needed by  all the  leader nodes  to solve  the integer

-  program during a period.  Finally, $E^a_{p}$ and $E^s_{p}$ indicate the energy
-  consumed by the WSN during the sensing phase (active and sleeping nodes).
+  program  during  a  period   (COMPUTATION  status).   Finally,  $E^a_{p}$  and
+  $E^s_{p}$ indicate  the energy consumed  by the  WSN during the  sensing phase
+  ({\it active} and {\it sleeping} nodes).

 \end{itemize}
 
 \end{itemize}
 

-

 \subsection{Simulation Results}
 
 In  order  to  assess and  analyze  the  performance  of  our protocol  we  have
 \subsection{Simulation Results}
 
 In  order  to  assess and  analyze  the  performance  of  our protocol  we  have

-implemented PeCO protocol in  OMNeT++~\citep{varga} simulator.  Besides PeCO, two
-other  protocols,  described  in  the  next paragraph,  will  be  evaluated  for
-comparison purposes.   The simulations were run  on a DELL laptop  with an Intel
-Core~i3~2370~M (1.8~GHz)  processor (2  cores) whose MIPS  (Million Instructions
-Per Second) rate  is equal to 35330. To  be consistent with the use  of a sensor
-node based on  Atmels AVR ATmega103L microcontroller (6~MHz) having  a MIPS rate
-equal to 6,  the original execution time  on the laptop is  multiplied by 2944.2
-$\left(\frac{35330}{2} \times  \frac{1}{6} \right)$.  The modeling  language for
-Mathematical Programming (AMPL)~\citep{AMPL} is  employed to generate the integer
-program instance  in a  standard format, which  is then read  and solved  by the
-optimization solver  GLPK (GNU  linear Programming Kit  available in  the public
-domain) \citep{glpk} through a Branch-and-Bound method.
-
-As said previously, the PeCO is  compared to three other approaches. The first
-one,  called  DESK,  is  a  fully distributed  coverage  algorithm  proposed  by
-\citep{ChinhVu}. The second one,  called GAF~\citep{xu2001geography}, consists in
-dividing  the monitoring  area into  fixed  squares. Then,  during the  decision
-phase, in each square, one sensor is  chosen to remain active during the sensing
-phase. The last  one, the DiLCO protocol~\citep{Idrees2}, is  an improved version
-of a research work we presented in~\citep{idrees2014coverage}. Let us notice that
-PeCO and  DiLCO protocols are  based on the  same framework. In  particular, the
-choice for the simulations of a partitioning in 16~subregions was made because
-it corresponds to the configuration producing  the best results for DiLCO. The
-protocols are distinguished  from one another by the formulation  of the integer
-program providing the set of sensors which  have to be activated in each sensing
-phase. DiLCO protocol tries to satisfy the coverage of a set of primary points,
-whereas the PeCO protocol objective is to reach a desired level of coverage for each
-sensor perimeter. In our experimentations, we chose a level of coverage equal to
-one ($l=1$).
-
-\subsubsection{\bf Coverage Ratio}
-
-Figure~\ref{figure5}  shows the  average coverage  ratio for  200 deployed  nodes
-obtained with the  four protocols. DESK, GAF, and DiLCO  provide a slightly better
-coverage ratio with respectively 99.99\%,  99.91\%, and 99.02\%, compared to the 98.76\%
-produced by  PeCO for the  first periods. This  is due to  the fact that  at the
-beginning the DiLCO protocol  puts to  sleep status  more redundant  sensors (which
-slightly decreases the coverage ratio), while the three other protocols activate
-more sensor  nodes. Later, when the  number of periods is  beyond~70, it clearly
-appears that  PeCO provides a better  coverage ratio and keeps  a coverage ratio
-greater  than 50\%  for  longer periods  (15  more compared  to  DiLCO, 40  more
-compared to DESK). The energy saved by  PeCO in the early periods allows later a
-substantial increase of the coverage performance.
+implemented PeCO  protocol in OMNeT++~\citep{varga} simulator.   The simulations
+were run  on a DELL laptop  with an Intel Core~i3~2370~M  (1.8~GHz) processor (2
+cores) whose MIPS  (Million Instructions Per Second) rate is  equal to 35330. To
+be consistent  with the  use of  a sensor  node based  on Atmels  AVR ATmega103L
+microcontroller (6~MHz)  having a MIPS rate  equal to 6, the  original execution
+time  on  the  laptop  is multiplied  by  2944.2  $\left(\frac{35330}{2}  \times
+\frac{1}{6} \right)$.  Energy  consumption is calculated according  to the power
+consumption  values,  in  milliWatt  per  second,  given  in  Table~\ref{tab:EC}.
+based on the energy model proposed in \citep{ChinhVu}.
+
+\begin{table}[h]
+\centering
+\caption{Power consumption values}
+\label{tab:EC}
+\begin{tabular}{|l||cccc|}
+  \hline
+  {\bf Sensor status} & MCU & Radio & Sensing & {\it Power (mW)} \\
+  \hline
+  LISTENING & On & On & On & 20.05 \\
+  ACTIVE & On & Off & On & 9.72 \\
+  SLEEP & Off & Off & Off & 0.02 \\
+  COMPUTATION & On & On & On & 26.83 \\
+  \hline
+  \multicolumn{4}{|l}{Energy needed to send or receive a 2-bit content message} & 0.515 \\
+  \hline
+\end{tabular}
+\end{table}
+
+The modeling  language for Mathematical Programming  (AMPL)~\citep{AMPL} is used
+to generate  the integer program  instance in a  standard format, which  is then
+read and  solved by  the optimization  solver GLPK  (GNU linear  Programming Kit
+available in the public domain)  \citep{glpk} through a Branch-and-Bound method.
+In practice, executing GLPK on a sensor node is obviously intractable due to the
+huge memory  use. Fortunately, to  solve the  optimization problem we  could use
+commercial  solvers  like  CPLEX  \citep{iamigo:cplex}  which  are  less  memory
+consuming and more efficient, or implement a lightweight heuristic. For example,
+for  a WSN  of 200  sensor nodes,  a leader  node has  to deal  with constraints
+induced  by about  12 sensor  nodes.  In  that case,  to solve  the optimization
+problem  a memory  consumption of  more  than 1~MB  can be  observed with  GLPK,
+whereas less than 300~kB would be needed with CPLEX.
+
+Besides  PeCO,   three  other  protocols   will  be  evaluated   for  comparison
+purposes. The first one, called DESK,  is a fully distributed coverage algorithm
+proposed      by     \citep{ChinhVu}.       The      second     one,      called
+GAF~\citep{xu2001geography}, consists in dividing the monitoring area into fixed
+squares. Then, during  the decision phase, in each square,  one sensor is chosen
+to  remain  active   during  the  sensing  phase.   The  last   one,  the  DiLCO
+protocol~\citep{Idrees2}, is an improved version of a research work we presented
+in~\citep{idrees2014coverage}. Let us  notice that PeCO and  DiLCO protocols are
+based on the same framework. In particular,  the choice for the simulations of a
+partitioning  in   16~subregions  was  made   because  it  corresponds   to  the
+configuration producing  the best results for  DiLCO. Of course, this  number of
+subregions should be adapted according to the  size of the area  of interest and
+the number of sensors.  The protocols  are distinguished from one another by the
+formulation of the integer program providing the set of sensors which have to be
+activated in each sensing phase. DiLCO protocol tries to satisfy the coverage of
+a set of primary  points, whereas PeCO protocol objective is  to reach a desired
+level of coverage for each sensor perimeter. In our experimentations, we chose a
+level of coverage equal to one ($l=1$).
+
+\subsubsection{Coverage Ratio}
+
+Figure~\ref{figure5} shows  the average  coverage ratio  for 200  deployed nodes
+obtained with the four protocols. DESK, GAF, and DiLCO provide a slightly better
+coverage ratio with respectively 99.99\%,  99.91\%, and 99.02\%, compared to the
+98.76\% produced by PeCO for the first periods.  This is due to the fact that at
+the  beginning DiLCO  and  PeCO protocols  put to  sleep  status more  redundant
+sensors  (which slightly  decreases the  coverage  ratio), while  the two  other
+protocols  activate more  sensor nodes.  Later, when  the number  of periods  is
+beyond~70, it  clearly appears that  PeCO provides  a better coverage  ratio and
+keeps a coverage ratio greater than 50\% for longer periods (15 more compared to
+DiLCO, 40 more compared to DESK). The  energy saved by PeCO in the early periods
+allows later a substantial increase of the coverage performance.

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


@@ -736,20 +823,17 @@ substantial increase of the coverage performance.
 \label{figure5}
 \end{figure} 
 
 \label{figure5}
 \end{figure} 
 

-
-
-
-\subsubsection{\bf Active Sensors Ratio}
+\subsubsection{Active Sensors Ratio}

 
 Having the less active sensor nodes in  each period is essential to minimize the
 
 Having the less active sensor nodes in  each period is essential to minimize the

-energy consumption  and thus to  maximize the network  lifetime.  Figure~\ref{figure6}
-shows the  average active nodes ratio  for 200 deployed nodes.   We observe that
-DESK and  GAF have 30.36  \% and  34.96 \% active  nodes for the  first fourteen
-rounds and  DiLCO and PeCO  protocols compete perfectly  with only 17.92~\% and
-20.16~\% active  nodes during the same  time interval. As the  number of periods
-increases, PeCO protocol  has a lower number of active  nodes in comparison with
-the three other approaches, while keeping a greater coverage ratio as shown in
-Figure \ref{figure5}.
+energy   consumption    and   thus    to   maximize   the    network   lifetime.
+Figure~\ref{figure6}  shows the  average  active nodes  ratio  for 200  deployed
+nodes.  We observe that DESK and GAF have 30.36~\% and 34.96~\% active nodes for
+the first fourteen  rounds, and DiLCO and PeCO protocols  compete perfectly with
+only 17.92~\%  and 20.16~\% active nodes  during the same time  interval. As the
+number of periods increases, PeCO protocol has a lower number of active nodes in
+comparison with the  three other approaches and exhibits a  slow decrease, while
+keeping a greater coverage ratio as shown in Figure \ref{figure5}.

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


@@ -758,82 +842,93 @@ Figure \ref{figure5}.
 \label{figure6}
 \end{figure} 
 
 \label{figure6}
 \end{figure} 
 

-\subsubsection{\bf Energy Consumption}
-
-We studied the effect of the energy  consumed by the WSN during the communication,
-computation, listening, active, and sleep status for different network densities
-and  compared  it for  the  four  approaches.  Figures~\ref{figure7}(a)  and  (b)
-illustrate  the  energy   consumption  for  different  network   sizes  and  for
-$Lifetime95$ and  $Lifetime50$. The results show  that our PeCO protocol  is the
-most competitive  from the energy  consumption point of  view. As shown  in both
-figures, PeCO consumes much less energy than the three other methods.  One might
-think that the  resolution of the integer  program is too costly  in energy, but
-the  results show  that it  is very  beneficial to  lose a  bit of  time in  the
-selection of  sensors to  activate.  Indeed the  optimization program  allows to
-reduce significantly the number of active  sensors and so the energy consumption
-while keeping a good coverage level.
+\subsubsection{Energy Consumption}
+
+The  effect  of  the  energy  consumed by  the  WSN  during  the  communication,
+computation,  listening,  active, and  sleep  status  is studied  for  different
+network densities  and the  four approaches  compared.  Figures~\ref{figure7}(a)
+and (b)  illustrate the energy consumption  for different network sizes  and for
+$Lifetime95$ and $Lifetime50$.  The results show  that PeCO protocol is the most
+competitive from the energy consumption point of view. As shown by both figures,
+PeCO consumes much less energy than the  other methods. One might think that the
+resolution of the integer program is too  costly in energy, but the results show
+that it is very beneficial to lose a  bit of time in the selection of sensors to
+activate.  Indeed  the optimization program  allows to reduce  significantly the
+number of  active sensors  and so  the energy consumption  while keeping  a good
+coverage level. Let  us notice that the energy overhead  when increasing network
+size is the lowest with PeCO.

 
 \begin{figure}[h!]
   \centering
   \begin{tabular}{@{}cr@{}}
 
 \begin{figure}[h!]
   \centering
   \begin{tabular}{@{}cr@{}}

-    \includegraphics[scale=0.475]{figure7a.eps} & \raisebox{2.75cm}{(a)} \\
-    \includegraphics[scale=0.475]{figure7b.eps} & \raisebox{2.75cm}{(b)}
+    \includegraphics[scale=0.5]{figure7a.eps} & \raisebox{2.75cm}{(a)} \\
+    \includegraphics[scale=0.5]{figure7b.eps} & \raisebox{2.75cm}{(b)}

   \end{tabular}
   \caption{Energy consumption per period for (a)~$Lifetime_{95}$ and (b)~$Lifetime_{50}$.}
   \label{figure7}
 \end{figure} 
 
   \end{tabular}
   \caption{Energy consumption per period for (a)~$Lifetime_{95}$ and (b)~$Lifetime_{50}$.}
   \label{figure7}
 \end{figure} 
 

+\subsubsection{Network Lifetime}

 
 

-
-\subsubsection{\bf Network Lifetime}
-
-We observe the superiority of PeCO and DiLCO protocols in comparison with the
-two    other   approaches    in    prolonging   the    network   lifetime.    In
-Figures~\ref{figure8}(a)  and (b),  $Lifetime95$ and  $Lifetime50$ are  shown for
-different  network  sizes.   As  highlighted  by  these  figures,  the  lifetime
-increases with the size  of the network, and it is clearly   largest for DiLCO
-and PeCO  protocols.  For instance,  for a  network of 300~sensors  and coverage
-ratio greater than 50\%, we can  see on Figure~\ref{figure8}(b) that the lifetime
-is about twice longer with  PeCO compared to DESK protocol.  The performance
-difference    is    more    obvious   in    Figure~\ref{figure8}(b)    than    in
-Figure~\ref{figure8}(a) because the gain induced  by our protocols increases with
- time, and the lifetime with a coverage  of 50\% is far  longer than with
-95\%.
+We observe the  superiority of both PeCO and DiLCO  protocols in comparison with
+the   two   other  approaches   in   prolonging   the  network   lifetime.    In
+Figures~\ref{figure8}(a) and  (b), $Lifetime95$  and $Lifetime50$ are  shown for
+different  network  sizes.  As  can  be  seen  in  these figures,  the  lifetime
+increases with the size of the network,  and it is clearly largest for DiLCO and
+PeCO protocols.  For  instance, for a network of 300~sensors  and coverage ratio
+greater than  50\%, we can see  on Figure~\ref{figure8}(b) that the  lifetime is
+about  twice  longer with  PeCO  compared  to  DESK protocol.   The  performance
+difference    is   more    obvious    in    Figure~\ref{figure8}(b)   than    in
+Figure~\ref{figure8}(a) because the gain induced by our protocols increases with
+time, and the lifetime with a coverage over 50\% is far longer than with 95\%.

 
 \begin{figure}[h!]
   \centering
   \begin{tabular}{@{}cr@{}}
 
 \begin{figure}[h!]
   \centering
   \begin{tabular}{@{}cr@{}}

-    \includegraphics[scale=0.475]{figure8a.eps} & \raisebox{2.75cm}{(a)} \\  
-    \includegraphics[scale=0.475]{figure8b.eps} & \raisebox{2.75cm}{(b)}
+    \includegraphics[scale=0.5]{figure8a.eps} & \raisebox{2.75cm}{(a)} \\  
+    \includegraphics[scale=0.5]{figure8b.eps} & \raisebox{2.75cm}{(b)}

   \end{tabular}
   \end{tabular}

-  \caption{Network Lifetime for (a)~$Lifetime_{95}$ \\
-    and (b)~$Lifetime_{50}$.}
+  \caption{Network Lifetime for (a)~$Lifetime_{95}$ and (b)~$Lifetime_{50}$.}

   \label{figure8}
 \end{figure} 
 
   \label{figure8}
 \end{figure} 
 

-
-
-Figure~\ref{figure9}  compares  the  lifetime  coverage of  our  protocols  for
-different coverage  ratios. We denote by  Protocol/50, Protocol/80, Protocol/85,
-Protocol/90, and  Protocol/95 the amount  of time  during which the  network can
-satisfy an area coverage greater than $50\%$, $80\%$, $85\%$, $90\%$, and $95\%$
-respectively, where the term Protocol refers to  DiLCO  or PeCO.  Indeed there  are applications
-that do not require a 100\% coverage of  the area to be monitored. PeCO might be
-an interesting  method since  it achieves  a good balance  between a  high level
-coverage ratio and network lifetime. PeCO always outperforms DiLCO for the three
-lower  coverage  ratios,  moreover  the   improvements  grow  with  the  network
-size. DiLCO is better  for coverage ratios near 100\%, but in  that case PeCO is
-not ineffective for the smallest network sizes.
+Figure~\ref{figure9} compares the lifetime coverage  of DiLCO and PeCO protocols
+for  different   coverage  ratios.   We  denote  by   Protocol/50,  Protocol/80,
+Protocol/85, Protocol/90,  and Protocol/95 the  amount of time during  which the
+network  can satisfy  an  area  coverage greater  than  $50\%$, $80\%$,  $85\%$,
+$90\%$, and  $95\%$ respectively,  where the  term Protocol  refers to  DiLCO or
+PeCO.  Indeed there are applications that do not require a 100\% coverage of the
+area to be  monitored. PeCO might be  an interesting method since  it achieves a
+good balance  between a  high level  coverage ratio  and network  lifetime. PeCO
+always  outperforms DiLCO  for the  three  lower coverage  ratios, moreover  the
+improvements grow  with the network  size. DiLCO  is better for  coverage ratios
+near 100\%, but  in that case PeCO  is not ineffective for  the smallest network
+sizes.

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

-\centering \includegraphics[scale=0.5]{figure9.eps}
+\centering \includegraphics[scale=0.55]{figure9.eps}

 \caption{Network lifetime for different coverage ratios.}
 \label{figure9}
 \end{figure} 
 
 \caption{Network lifetime for different coverage ratios.}
 \label{figure9}
 \end{figure} 
 

-
-\subsubsection{\bf Impact of $\alpha$ and $\beta$ on PeCO's performance}
-Table~\ref{my-labelx} explains all possible network lifetime result of the relation between the different values of $\alpha$ and $\beta$, and for a network size equal to 200 sensor nodes. As can be seen in Table~\ref{my-labelx},  it is obvious and clear that when $\alpha$ decreased and $\beta$ increased by any step, the network lifetime for $Lifetime_{50}$ increased and the $Lifetime_{95}$ decreased. Therefore, selecting the values of $\alpha$ and $\beta$ depend on the application type used in the sensor nework. In PeCO protocol, $\alpha$ and $\beta$ are chosen based on the largest value of network lifetime for $Lifetime_{95}$.
+\subsubsection{Impact of $\alpha$ and $\beta$ on PeCO's performance}
+\label{sec:Impact}
+
+Table~\ref{my-labelx}  shows network  lifetime results  for different  values of
+$\alpha$ and $\beta$, and  a network size equal to 200 sensor  nodes. On the one
+hand, the choice  of $\beta \gg \alpha$ prevents the  overcoverage, and so limit
+the activation of a large number of  sensors, but as $\alpha$ is low, some areas
+may be poorly covered.  This explains  the results obtained for {\it Lifetime50}
+with $\beta \gg \alpha$: a large number  of periods with low coverage ratio.  On
+the other hand, when we choose $\alpha \gg \beta$, we favor the coverage even if
+some areas may  be overcovered, so high  coverage ratio is reached,  but a large
+number  of  sensors are  activated  to  achieve  this goal.   Therefore  network
+lifetime is reduced.   The choice $\alpha=0.6$ and $\beta=0.4$  seems to achieve
+the best compromise  between lifetime and coverage ratio.  That  explains why we
+have  chosen  this  setting  for  the  experiments  presented  in  the  previous
+subsections.
+
+%As can be seen in Table~\ref{my-labelx},  it is obvious and clear that when $\alpha$ decreased and $\beta$ increased by any step, the network lifetime for $Lifetime_{50}$ increased and the $Lifetime_{95}$ decreased. Therefore, selecting the values of $\alpha$ and $\beta$ depend on the application type used in the sensor nework. In PeCO protocol, $\alpha$ and $\beta$ are chosen based on the largest value of network lifetime for $Lifetime_{95}$.

 
 \begin{table}[h]
 \centering
 
 \begin{table}[h]
 \centering


@@ -848,7 +943,7 @@ $\alpha$ & $\beta$ & $Lifetime_{50}$ & $Lifetime_{95}$ \\ \hline
 0.3 & 0.7 & 134 & 0 \\ \hline
 0.4 & 0.6 & 125 & 0 \\ \hline
 0.5 & 0.5 & 118 & 30 \\ \hline
 0.3 & 0.7 & 134 & 0 \\ \hline
 0.4 & 0.6 & 125 & 0 \\ \hline
 0.5 & 0.5 & 118 & 30 \\ \hline

-0.6 & 0.4 & 94 & 57 \\ \hline
+{\bf 0.6} & {\bf 0.4} & {\bf 94} & {\bf 57} \\ \hline

 0.7 & 0.3 & 97 & 49 \\ \hline
 0.8 & 0.2 & 90 & 52 \\ \hline
 0.9 & 0.1 & 77 & 50 \\ \hline
 0.7 & 0.3 & 97 & 49 \\ \hline
 0.8 & 0.2 & 90 & 52 \\ \hline
 0.9 & 0.1 & 77 & 50 \\ \hline


@@ -860,16 +955,37 @@ $\alpha$ & $\beta$ & $Lifetime_{50}$ & $Lifetime_{95}$ \\ \hline
 \section{Conclusion and Future Works}
 \label{sec:Conclusion and Future Works}
 
 \section{Conclusion and Future Works}
 \label{sec:Conclusion and Future Works}
 

-In this paper  we have studied the problem of  Perimeter-based Coverage Optimization in WSNs. We have designed  a new protocol, called Perimeter-based  Coverage Optimization, which schedules nodes'  activities (wake up  and sleep  stages) with the  objective of maintaining a  good coverage ratio  while maximizing the network  lifetime. This protocol is  applied in a distributed  way in regular subregions  obtained after partitioning the area of interest in a preliminary step. It works in periods and
-is based on the resolution of an integer program to select the subset of sensors operating in active status for each period. Our work is original in so far as it proposes for  the first  time an  integer program  scheduling the  activation of sensors  based on  their perimeter  coverage level,  instead of  using a  set of targets/points to be covered.   
-
-
-We have carried out  several simulations  to  evaluate the  proposed protocol. The simulation  results  show   that  PeCO  is  more   energy-efficient  than  other approaches, with respect to lifetime,  coverage ratio, active sensors ratio, and energy consumption. 
-
-We plan to extend our framework so that the schedules are planned for multiple sensing periods. We also want to improve our integer program to  take into account heterogeneous sensors  from both  energy and node characteristics point of views. Finally,  it would  be interesting to implement our protocol using  a sensor-testbed to evaluate it in real world applications.
+In this paper we have studied  the problem of perimeter coverage optimization in
+WSNs.   We  have  designed  a  new  protocol,  called  Perimeter-based  Coverage
+Optimization, which schedules nodes' activities  (wake up and sleep stages) with
+the objective of maintaining a good  coverage ratio while maximizing the network
+lifetime.  This protocol  is applied in a distributed way  in regular subregions
+obtained after partitioning the area of interest in a preliminary step. It works
+in periods and  is based on the  resolution of an integer program  to select the
+subset  of sensors  operating in  active status  for each  period.  Our  work is
+original  in so  far  as it  proposes  for  the first  time  an integer  program
+scheduling the  activation of sensors  based on their perimeter  coverage level,
+instead of using a set of targets/points to be covered. Several simulations have
+been carried out to evaluate the  proposed protocol. The simulation results show
+that  PeCO is  more  energy-efficient  than other  approaches,  with respect  to
+lifetime, coverage ratio, active sensors ratio, and energy consumption.
+
+We plan to extend  our framework so that the schedules  are planned for multiple
+sensing  periods. We  also want  to  improve the  integer program  to take  into
+account heterogeneous sensors from both energy and node characteristics point of
+views.  Finally,  it would  be interesting  to implement  PeCO protocol  using a
+sensor-testbed to evaluate it in real world applications.
+
+
+\subsection*{Acknowledgements}
+The  authors  are   deeply  grateful  to  the  anonymous   reviewers  for  their
+constructive advice,  which improved the  technical quality  of the paper.  As a
+Ph.D.   student, Ali  Kadhum IDREES  would  like to  gratefully acknowledge  the
+University of  Babylon - Iraq  for financial support  and Campus France  for the
+received support. This work is also partially funded by the Labex ACTION program
+(contract ANR-11-LABX-01-01).

 
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