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

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.

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  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.}

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

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=127.5mm]{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} \;
  \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.  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

\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 WSN 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 $a^j_{ik}$ denotes  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} 
\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 : $M_{i}^{j}=l-l^{i}$, $V_{i}^{j}=0$. 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 : $M_{i}^{j}=0$, $V_{i}^{j}=l^{i}-l$.  

$\alpha^j_i$ 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. 

\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.

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:
  

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


  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.
\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:
 
\[
    \scriptsize
    \mbox{ASR}(\%) =  \frac{\sum\limits_{r=1}^R \mbox{$|A_r^p|$}}{\mbox{$|J|$}} \times 100
\]

  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.
\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:

\[  
  \scriptsize
    \mbox{EC} = \frac{\sum\limits_{p=1}^{P} \left( E^{\mbox{com}}_p+E^{\mbox{list}}_p+E^{\mbox{comp}}_p  
      + E^{a}_p+E^{s}_p \right)}{P},
\]
 
  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.  Finally, $E^a_{p}$ and $E^s_{p}$ indicate the energy
  consumed by the WSN during the sensing phase (active and 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.  Besides PeCO, two
other  protocols,  described  in  the  next paragraph,  will  be  evaluated  for
comparison purposes.   The simulations were run  on a DELL laptop  with an Intel
Core~i3~2370~M (1.8~GHz)  processor (2  cores) whose MIPS  (Million Instructions
Per Second) rate  is equal to 35330. To  be consistent with the use  of a sensor
node based on  Atmels AVR ATmega103L microcontroller (6~MHz) having  a MIPS rate
equal to 6,  the original execution time  on the laptop is  multiplied by 2944.2
$\left(\frac{35330}{2} \times  \frac{1}{6} \right)$.  The modeling  language for
Mathematical Programming (AMPL)~\citep{AMPL} is  employed to generate the integer
program instance  in a  standard format, which  is then read  and solved  by the
optimization solver  GLPK (GNU  linear Programming Kit  available in  the public
domain) \citep{glpk} through a Branch-and-Bound method.

As said previously, the PeCO is  compared to three other approaches. The first
one,  called  DESK,  is  a  fully distributed  coverage  algorithm  proposed  by
\citep{ChinhVu}. The second one,  called GAF~\citep{xu2001geography}, consists in
dividing  the monitoring  area into  fixed  squares. Then,  during the  decision
phase, in each square, one sensor is  chosen to remain active during the sensing
phase. The last  one, the DiLCO protocol~\citep{Idrees2}, is  an improved version
of a research work we presented in~\citep{idrees2014coverage}. Let us notice that
PeCO and  DiLCO protocols are  based on the  same framework. In  particular, the
choice for the simulations of a partitioning in 16~subregions was made because
it corresponds to the configuration producing  the best results for DiLCO. The
protocols are distinguished  from one another by the formulation  of the integer
program providing the set of sensors which  have to be activated in each sensing
phase. DiLCO protocol tries to satisfy the coverage of a set of primary points,
whereas the PeCO protocol objective is to reach a desired level of coverage for each
sensor perimeter. In our experimentations, we chose a level of coverage equal to
one ($l=1$).

\subsubsection{\bf Coverage Ratio}

Figure~\ref{figure5}  shows the  average coverage  ratio for  200 deployed  nodes
obtained with the  four protocols. DESK, GAF, and DiLCO  provide a slightly better
coverage ratio with respectively 99.99\%,  99.91\%, and 99.02\%, compared to the 98.76\%
produced by  PeCO for the  first periods. This  is due to  the fact that  at the
beginning the DiLCO protocol  puts to  sleep status  more redundant  sensors (which
slightly decreases the coverage ratio), while the three other protocols activate
more sensor  nodes. Later, when the  number of periods is  beyond~70, it clearly
appears that  PeCO provides a better  coverage ratio and keeps  a coverage ratio
greater  than 50\%  for  longer periods  (15  more compared  to  DiLCO, 40  more
compared to DESK). The energy saved by  PeCO in the early periods allows later a
substantial increase of the coverage performance.

\parskip 0pt    
\begin{figure}[h!]
\centering
 \includegraphics[scale=0.5] {figure5.eps} 
\caption{Coverage ratio for 200 deployed nodes.}
\label{figure5}
\end{figure} 




\subsubsection{\bf 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, 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{\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.

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



\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\%. 

\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)}
  \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  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.

\begin{figure}[h!]
\centering \includegraphics[scale=0.5]{figure9.eps}
\caption{Network lifetime for different coverage ratios.}
\label{figure9}
\end{figure} 


\subsubsection{\bf Impact of $\alpha$ and $\beta$ on PeCO's performance}
Table~\ref{my-labelx} 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]
\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-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.

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