Michel - Still some points to be checked in section 5.2

[LiCO.git] / PeCO-EO / articleeo.tex

diff --git a/PeCO-EO/articleeo.tex b/PeCO-EO/articleeo.tex
index 189f2363e4f773dcace0af8bd81a7c7213ff9c9e..b4925dbaa27323631f29c2c2f016d0f8f6db48a5 100644 (file)
--- a/PeCO-EO/articleeo.tex
+++ b/PeCO-EO/articleeo.tex
@@ -16,7 +16,7 @@
     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 $^{a}$
     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,
+$^{a}${\em{FEMTO-ST Institute, UMR 6174 CNRS, University of Franche-Comt\'e,

           Belfort, France}}}
 
 \maketitle
           Belfort, France}}}
 
 \maketitle


@@ -42,7 +42,6 @@ coverage for WSNs in comparison with some other protocols.
 
 \end{abstract}
 
 
 \end{abstract}
 

-

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


@@ -76,25 +75,25 @@ 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
+\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.
   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 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 deviations.
-\item We  have conducted  extensive simulation  experiments, 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:
+\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
+  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.
+\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.
   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.


@@ -112,9 +111,9 @@ Section~\ref{sec:Conclusion and Future Works}.
 \section{Related Literature}
 \label{sec:Literature Review}
 
 \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
 categories~\citep{li2013survey}   according  to   their  respective   monitoring
 
 The most  discussed coverage problems in  literature can be classified  in three
 categories~\citep{li2013survey}   according  to   their  respective   monitoring


@@ -137,47 +136,46 @@ sensor, and  $n$ is the  total number  of sensors in  the network. {\it  In PeCO
 
 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  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.}
+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
+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
 
 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{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
 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
 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
 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  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.}
+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
 
 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


@@ -308,7 +306,7 @@ 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*} 


@@ -350,7 +348,7 @@ 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
 \includegraphics[width=62.5mm]{figure3.eps}  
 \begin{figure}[h!]
 \centering
 \includegraphics[width=62.5mm]{figure3.eps}  


@@ -427,6 +425,7 @@ applied by a sensor node $s_k$ where $k$ is the node index in the WSN.
 %  \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} \;
 %  \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} \;

+  \label{alg:PeCO}

   \caption{PeCO pseudocode}
   \eIf{$RE_k \geq E_{th}$}{
     $s_k.status$ = COMMUNICATION\;
   \caption{PeCO pseudocode}
   \eIf{$RE_k \geq E_{th}$}{
     $s_k.status$ = COMMUNICATION\;


@@ -496,39 +495,48 @@ applied by a sensor node $s_k$ where $k$ is the node index in the WSN.
 %\label{alg:PeCO}
 %\end{algorithm}
 
 %\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.
-
-% TO BE CONTINUED
+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 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).
+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.

 
 

-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
 to the  method introduced in  subsection~\ref{CI}. For a coverage  interval $i$,
 \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
+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 \{ 


@@ -540,130 +548,136 @@ 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 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} 
-\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.
+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}
 

-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 : $M_{i}^{j}=l-l^{i}$, $V_{i}^{j}=0$. In the contrary, 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 : $M_{i}^{j}=0$, $V_{i}^{j}=l^{i}-l$.  
+%\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 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 variables $\alpha$ and $\beta$ should be made according to the needs of the application. $\alpha$ should be enough large to prevent undercoverage and so to reach the highest possible coverage ratio. $\beta$ should be enough large 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. 
+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
 be active during at most 20 periods.
 
 The values  of $\alpha^j_i$ and  $\beta^j_i$ have been  chosen to ensure  a good
 pre-sensing phases.  According  to the interval of initial energy,  a sensor may
 be active during at most 20 periods.
 
 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


@@ -676,42 +690,34 @@ 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


@@ -720,51 +726,81 @@ 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
+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}.
+
+% Questions on energy consumption calculation
+% 1 - How did you compute the value for COMPUTATION status ?
+% 2 - I have checked the paper of Chinh T. Vu (2006) and I wonder
+% why you completely deleted the energy due to the sensing range ?
+% => You should have use a fixed value for the sensing rangge Rs (5 meter)
+% => for all the nodes to compute f(Ri), which would have lead to energy values
+
+\begin{table}[h]
+\centering
+\caption{Energy consumption}
+\label{tab:EC}
+\begin{tabular}{|l||cccc|}
+  \hline
+  {\bf Sensor status} & MCU & Radio & Sensor & {\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.
+
+% No discussion about the execution of GLPK on a sensor ?
+
+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.   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 ($l=1$).
 
 sensor perimeter. In our experimentations, we chose a level of coverage equal to
 one ($l=1$).
 

-\subsubsection{\bf Coverage Ratio}
+\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 the DiLCO protocol  puts to  sleep status  more redundant  sensors (which
+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 PeCO  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
 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


@@ -780,20 +816,17 @@ 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


@@ -802,82 +835,92 @@ 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  over 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{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.

 
 

-\subsubsection{\bf Impact of $\alpha$ and $\beta$ on PeCO's performance}
-Table~\ref{my-labelx} shows network lifetime results for the different values of $\alpha$ and $\beta$, and for a network size equal to 200 sensor nodes. 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. With $\alpha \gg \beta$, we priviligie 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.     

 %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]
 %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]


@@ -905,16 +948,28 @@ $\alpha$ & $\beta$ & $Lifetime_{50}$ & $Lifetime_{95}$ \\ \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.

 
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