Michel - Still some points to be checked in section 5.2

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

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\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-Comt\'e,
          Belfort, France}}}

\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
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.
\end{keywords}

\end{abstract}

\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
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
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.

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
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
while  ensuring  a   high  level  of  coverage?   These  past   few  years  many
energy-efficient mechanisms have been suggested  to retain energy and extend the
lifetime of the WSNs~\citep{rault2014energy}.

This paper makes the following contributions.
\begin{enumerate}
\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
  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.
\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}
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
Section~\ref{sec:Conclusion and Future Works}.

\section{Related Literature}
\label{sec:Literature Review}

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
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
targets,   and   barrier  coverage~\citep{HeShibo,kim2013maximum}   focuses   on
preventing  intruders   from  entering   into  the   region  of   interest.   In
\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
\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
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
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
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
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
\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
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}

%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}

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
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
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 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  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{tabular}{@{}cr@{}}
    \includegraphics[width=75mm]{figure1a.eps} & \raisebox{3.25cm}{(a)} \\
    \includegraphics[width=75mm]{figure1b.eps} & \raisebox{2.75cm}{(b)}
  \end{tabular}
  \caption{(a) Perimeter  coverage of sensor node  0 and (b) finding  the arc of
    $u$'s perimeter covered by $v$.}
  \label{figure1}
\end{figure} 

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
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:
 \[
\alpha = \arccos \left(\frac{Dist(u,v)}{2R_s} \right).
\] 
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
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
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.

\begin{figure*}[t!]
\centering
\includegraphics[width=0.95\linewidth]{figure2.eps}  
\caption{Maximum coverage levels for perimeter of sensor node $0$.}
\label{figure2}
\end{figure*} 

\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
0.0291    & 1L                                                                        & 2L                                                        & 4                                                                     & 0                     & 1                     & 3                    & 4                    &                      \\ \hline
0.104     & 2L                                                                        & 3R                                                        & 5                                                                     & 0                     & 1                     & 3                    & 4                    & 2                    \\ \hline
0.3168    & 3R                                                                        & 4R                                                        & 4                                                                     & 0                     & 1                     & 4                    & 2                    &                      \\ \hline
0.6752    & 4R                                                                        & 1R                                                        & 3                                                                     & 0                     & 1                     & 2                    &                      &                      \\ \hline
1.8127    & 1R                                                                        & 5L                                                        & 2                                                                     & 0                     & 2                     &                      &                      &                      \\ \hline
1.9228    & 5L                                                                        & 6L                                                        & 3                                                                     & 0                     & 2                     & 5                    &                      &                      \\ \hline
2.3959    & 6L                                                                        & 2R                                                        & 4                                                                     & 0                     & 2                     & 5                    & 6                    &                      \\ \hline
2.4258    & 2R                                                                        & 7L                                                        & 3                                                                     & 0                     & 5                     & 6                    &                      &                      \\ \hline
2.7868    & 7L                                                                        & 8L                                                        & 4                                                                     & 0                     & 5                     & 6                    & 7                    &                      \\ \hline
2.8358    & 8L                                                                        & 5R                                                        & 5                                                                     & 0                     & 5                     & 6                    & 7                    & 8                    \\ \hline
2.9184    & 5R                                                                        & 7R                                                        & 4                                                                     & 0                     & 6                     & 7                    & 8                    &                      \\ \hline
3.3301    & 7R                                                                        & 9R                                                        & 3                                                                     & 0                     & 6                     & 8                    &                      &                      \\ \hline
3.9464    & 9R                                                                        & 6R                                                        & 4                                                                     & 0                     & 6                     & 8                    & 9                    &                      \\ \hline
4.767     & 6R                                                                        & 3L                                                        & 3                                                                     & 0                     & 8                     & 9                    &                      &                      \\ \hline
4.8425    & 3L                                                                        & 8R                                                        & 4                                                                     & 0                     & 3                     & 8                    & 9                    &                      \\ \hline
4.9072    & 8R                                                                        & 4L                                                        & 3                                                                     & 0                     & 3                     & 9                    &                      &                      \\ \hline
5.3804    & 4L                                                                        & 9R                                                        & 4                                                                     & 0                     & 3                     & 4                    & 9                    &                      \\ \hline
5.9157    & 9R                                                                        & 1L                                                        & 3                                                                     & 0                     & 3                     & 4                    &                      &                      \\ \hline
\end{tabular}}


\end{table}

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
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.

%\newpage
\begin{figure}[h!]
\centering
\includegraphics[width=62.5mm]{figure3.eps}  
\caption{Sensing range outside the WSN's area of interest.}
\label{figure3}
\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. Node Sensors  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
the next period. The energy consumption and some other constraints can easily be
taken  into  account since  the  sensors  can  update  and then  exchange  their
information (including their  residual energy) at the beginning  of each period.
However, the pre-sensing  phases (INFO Exchange, Leader  Election, and Decision)
are energy consuming, even for nodes that will not join the set cover to monitor
the area. Sensing  period duration is adapted according to  the QoS requirements
of the application.

\begin{figure}[t!]
\centering
\includegraphics[width=85mm]{figure4.eps}  
\caption{PeCO protocol.}
\label{figure4}
\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.
\item ActiveSleep packet: sent  by the leader to all the  nodes in its subregion
  to transmit to  them their respective status (stay Active  or go Sleep) during
  sensing phase.
\end{itemize}

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
  determine the activities scheduling;
\item ACTIVE: node is sensing;
\item SLEEP: node is turned off;
\item COMMUNICATION: transmits or receives packets.
\end{itemize}

\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.


\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
  %\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}

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}
\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 $i$,
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 \{ 
\begin{array}{lll}
  1 & \mbox{if sensor $k$ is involved in the } \\
        &       \mbox{coverage interval $i$ of sensor $j$}, \\
  0 & \mbox{otherwise.}\\
\end{array} \right.
\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}
  \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}

%\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
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}

\subsection{Simulation Settings}

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.

\begin{table}[ht]
\tbl{Relevant parameters for network initialization \label{table3}}{
\centering
\begin{tabular}{c|c}
\hline
Parameter & Value  \\ [0.5ex]
\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 period & duration of 60 minutes \\
$E_{th}$ & 36~Joules \\
$R_s$ & 5~m \\     
$R_c$ & 10~m \\   
$\alpha^j_i$ & 0.6 \\
$\beta^j_i$ & 0.4
\end{tabular}}
\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
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
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,
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
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.
\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
  $Lifetime_{50}$  denote, respectively,  the  amount of  time  during which  is
  guaranteed a  level of coverage  greater than $95\%$  and $50\%$. The  WSN can
  fulfill the expected  monitoring task until all its nodes  have depleted their
  energy or if the network is no  more connected. This last condition is crucial
  because without  network connectivity a  sensor may not be  able to send  to a
  base station an event it has sensed.
\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
  \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
  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
  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:
  \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
  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:
  \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},
  \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
  energy consumption spent  by all the nodes for  wireless communications during
  period $p$.  $E^{\scriptsize \mbox{list}}_p$,  the next factor, corresponds to
  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   (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}

\subsection{Simulation Results}

In  order  to  assess and  analyze  the  performance  of  our protocol  we  have
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 ($l=1$).

\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 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
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!]
\centering
 \includegraphics[scale=0.5] {figure5.eps} 
\caption{Coverage ratio for 200 deployed nodes.}
\label{figure5}
\end{figure} 

\subsubsection{Active Sensors Ratio}

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 and exhibits a  slow decrease, while
keeping a greater coverage ratio as shown in Figure \ref{figure5}.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{figure6.eps}  
\caption{Active sensors ratio for 200 deployed nodes.}
\label{figure6}
\end{figure} 

\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@{}}
    \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} 

\subsubsection{Network Lifetime}

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@{}}
    \includegraphics[scale=0.5]{figure8a.eps} & \raisebox{2.75cm}{(a)} \\  
    \includegraphics[scale=0.5]{figure8b.eps} & \raisebox{2.75cm}{(b)}
  \end{tabular}
  \caption{Network Lifetime for (a)~$Lifetime_{95}$ and (b)~$Lifetime_{50}$.}
  \label{figure8}
\end{figure} 

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!]
\centering \includegraphics[scale=0.55]{figure9.eps}
\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.

%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
\caption{The impact of $\alpha$ and $\beta$ on PeCO's performance}
\label{my-labelx}
\begin{tabular}{|c|c|c|c|}
\hline
$\alpha$ & $\beta$ & $Lifetime_{50}$ & $Lifetime_{95}$ \\ \hline
0.0 & 1.0 & 151 & 0 \\ \hline
0.1 & 0.9 & 145 & 0 \\ \hline
0.2 & 0.8 & 140 & 0 \\ \hline
0.3 & 0.7 & 134 & 0 \\ \hline
0.4 & 0.6 & 125 & 0 \\ \hline
0.5 & 0.5 & 118 & 30 \\ \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
1.0 & 0.0 & 60 & 44 \\ \hline
\end{tabular}
\end{table}


\section{Conclusion and Future Works}
\label{sec:Conclusion and Future Works}

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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