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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
  our   PeCO   protocol   to   two   approaches   found   in   the   literature:
  DESK~\citep{ChinhVu} and  GAF~\citep{xu2001geography}, and also to  our previous
  work published in~\citep{Idrees2} which is  based on another optimization model
  for sensor scheduling.
\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, and data from all the other
sensor nodes  in the area.  The price  in communications can be  huge since
long range  communications will be  needed. In fact  the larger the WNS  is, the
higher the  communication and  thus the energy  cost are.   {\it In order  to be
  suitable for large-scale  networks, in the PeCO protocol,  the area of interest
  is divided into several smaller subregions, and in each one, a node called the
  leader  is  in  charge  of  selecting  the active  sensors  for  the  current
  period.  Thus our  protocol is  scalable  and is a  globally distributed  method,
  whereas it is centralized in each subregion.}

Various  coverage scheduling  algorithms have  been developed  these past few years.
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, maintains the coverage and improves the lifetime in WSNs. It is  an improved version
of a research work they presented in~\citep{idrees2014coverage}.  First, they partition the area of interest into subregions using a divide-and-conquer method. DiLCO protocol is then distributed on the sensor nodes in each subregion in a second step. DiLCO protocol combines two techniques: a leader election in each subregion, followed by an optimization-based node activity scheduling performed by each elected leader. The proposed DiLCO protocol is a periodic protocol where each period is decomposed into 4 phases: information exchange, leader election, decision, and sensing. The simulations show that DiLCO is able to increase the WSN lifetime and provides improved coverage performance. {\it  In the PeCO
  protocol, We have proposed a new mathematical optimization model. Instead of trying to
cover a set of specified points/targets as in DiLCO protocol, we formulate an integer program based
on perimeter coverage of each sensor. The model involves integer variables to capture the deviations between the actual level of coverage and the required level. The idea is that an optimal scheduling will be obtained by minimizing a weighted sum of these deviations.}
  
  
  

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

\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: $Dist(u,v)=\sqrt{\vert
  u_x  - v_x  \vert^2 +  \vert u_y-v_y  \vert^2}$, 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 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  Figure~\ref{figure2} 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 sensor 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
integer program  based on  coverage intervals. 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} 


\subsection{The Main Idea}

The  WSN area of  interest is, in a  first step, divided  into regular
homogeneous subregions  using a divide-and-conquer  algorithm. In a  second step
our  protocol  will  be  executed  in   a  distributed  way  in  each  subregion
simultaneously to schedule nodes' activities for one sensing period.

As  shown in  Figure~\ref{figure4}, node  activity  scheduling is  produced by  our
protocol in a periodic manner. Each period is divided into 4 stages: Information
(INFO)  Exchange,  Leader Election,  Decision  (the  result of  an  optimization
problem),  and  Sensing.   For  each  period there  is  exactly  one  set  cover
responsible for  the sensing task.  Protocols  based on a periodic  scheme, like
PeCO, are more  robust against an unexpected  node failure. On the  one hand, if
a node failure is discovered before  taking the decision, the corresponding sensor
node will  not be considered  by the optimization  algorithm. On  the other
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.

\begin{figure}[t!]
\centering
\includegraphics[width=80mm]{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{algorithm}      
 % \KwIn{all the parameters related to information exchange}
%  \KwOut{$winer-node$ (: the id of the winner sensor node, which is the leader of current round)}
%  \BlankLine
  %\emph{Initialize the sensor node and determine it's position and subregion} \; 
  
\noindent{\bf If} $RE_k \geq E_{th}$ {\bf then}\\
\hspace*{0.6cm} \emph{$s_k.status$ = COMMUNICATION;}\\
\hspace*{0.6cm}  \emph{Send $INFO()$ packet to other nodes in subregion;}\\
\hspace*{0.6cm}  \emph{Wait $INFO()$ packet from other nodes in subregion;}\\
\hspace*{0.6cm} \emph{Update K.CurrentSize;}\\
\hspace*{0.6cm}  \emph{LeaderID = Leader election;}\\
\hspace*{0.6cm} {\bf If} $ s_k.ID = LeaderID $ {\bf then}\\
\hspace*{1.2cm}   \emph{$s_k.status$ = COMPUTATION;}\\
\hspace*{1.2cm}{\bf If} \emph{$ s_k.ID $ is Not previously selected as a Leader} {\bf then}\\
\hspace*{1.8cm} \emph{ Execute the perimeter coverage model;}\\
\hspace*{1.2cm} {\bf end}\\
\hspace*{1.2cm}{\bf If} \emph{($s_k.ID $ is the same Previous Leader)~And~(K.CurrentSize = K.PreviousSize)}\\
\hspace*{1.8cm} \emph{ Use the same previous cover set for current sensing stage;}\\
\hspace*{1.2cm}  {\bf end}\\
\hspace*{1.2cm}  {\bf else}\\
\hspace*{1.8cm}\emph{Update $a^j_{ik}$; prepare data for IP~Algorithm;}\\
\hspace*{1.8cm} \emph{$\left\{\left(X_{1},\dots,X_{l},\dots,X_{K}\right)\right\}$ = Execute Integer Program Algorithm($K$);}\\
\hspace*{1.8cm} \emph{K.PreviousSize = K.CurrentSize;}\\
\hspace*{1.2cm}  {\bf end}\\
\hspace*{1.2cm}\emph{$s_k.status$ = COMMUNICATION;}\\
\hspace*{1.2cm}\emph{Send $ActiveSleep()$ to each node $l$ in subregion;}\\
\hspace*{1.2cm}\emph{Update $RE_k $;}\\
\hspace*{0.6cm}  {\bf end}\\
\hspace*{0.6cm}  {\bf else}\\
\hspace*{1.2cm}\emph{$s_k.status$ = LISTENING;}\\
\hspace*{1.2cm}\emph{Wait $ActiveSleep()$ packet from the Leader;}\\
\hspace*{1.2cm}\emph{Update $RE_k $;}\\
\hspace*{0.6cm}  {\bf end}\\
{\bf end}\\
{\bf else}\\
\hspace*{0.6cm} \emph{Exclude $s_k$ from entering in the current sensing stage;}\\
{\bf end}\\
\label{alg:PeCO}
\end{algorithm}



In this  algorithm, K.CurrentSize and K.PreviousSize  respectively represent the
current number and  the previous number of living nodes in  the subnetwork of the
subregion.  Initially, the sensor node checks its remaining energy $RE_k$, which
must be greater than a threshold $E_{th}$ in order to participate in the current
period.  Each  sensor node  determines its position  and its subregion  using an
embedded  GPS or a  location discovery  algorithm. After  that, all  the sensors
collect position coordinates,  remaining energy, sensor node ID,  and the number
of their  one-hop live  neighbors during the  information exchange.  The sensors
inside a same region cooperate to elect a leader. The selection criteria for the
leader, in order of priority,  are: larger numbers of neighbors, larger remaining
energy, and  then in case  of equality, larger  index.  Once chosen,  the leader
collects information to formulate and  solve the integer program which allows to
construct the set of active sensors in the sensing stage.


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

In this  section, the coverage model is  mathematically formulated. The following
notations are used  throughout the
section.\\
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 binary  and integer  variables are defined.  Hence,  each binary
variable $X_{k}$  determines the activation of  sensor $k$ in the  sensing phase
($X_k=1$ if  the sensor $k$  is active or 0  otherwise).  $M^j_i$ is  an integer
variable  which  measures  the  undercoverage  for  the  coverage  interval  $i$
corresponding to  sensor~$j$. In  the same  way, the  overcoverage for  the same
coverage interval is given by the variable $V^j_i$.

If we decide to sustain a level of coverage equal to $l$ all along the perimeter
of sensor  $j$, we have  to ensure  that at least  $l$ sensors involved  in each
coverage  interval $i  \in I_j$  of  sensor $j$  are active.   According to  the
previous notations, the number of active sensors in the coverage interval $i$ of
sensor $j$  is given by  $\sum_{k \in A} a^j_{ik}  X_k$.  To extend  the network
lifetime,  the objective  is to  activate a  minimal number  of sensors  in each
period to  ensure the  desired coverage  level. As the  number of  alive sensors
decreases, it becomes impossible to reach  the desired level of coverage for all
coverage intervals. Therefore  variables  $M^j_i$ and $V^j_i$ are introduced as a measure
of the  deviation between  the desired  number of active  sensors in  a coverage
interval and  the effective  number. And  we try  to minimize  these deviations,
first to  force the  activation of  a minimal  number of  sensors to  ensure the
desired coverage level, and if the desired level cannot be completely satisfied,
to reach a coverage level as close as possible to the desired one.




Our coverage optimization problem can then be mathematically expressed as follows: 

\begin{equation} 
\left \{
\begin{array}{ll}
\min \sum_{j \in S} \sum_{i \in I_j} (\alpha^j_i ~ M^j_i + \beta^j_i ~ V^j_i )&\\
\textrm{subject to :}&\\
\sum_{k \in A} ( a^j_{ik} ~ X_{k}) + M^j_i  \geq l \quad \forall i \in I_j, \forall j \in S\\
\sum_{k \in A} ( a^j_{ik} ~ X_{k}) - V^j_i  \leq l \quad \forall i \in I_j, \forall j \in S\\
X_{k} \in \{0,1\}, \forall k \in A
M^j_i, V^j_i \in  \mathbb{R}^{+}
\end{array}
\right.
\end{equation}

$\alpha^j_i$ and $\beta^j_i$  are nonnegative weights selected  according to the
relative importance of satisfying the associated level of coverage. For example,
weights associated with  coverage intervals of a specified part  of a region may
be  given by a  relatively larger  magnitude than  weights associated  with another
region. This  kind of integer program  is inspired from the  model developed for
brachytherapy treatment planning  for optimizing dose  distribution
\citep{0031-9155-44-1-012}. The integer  program must be solved by  the leader in
each subregion at the beginning of  each sensing phase, whenever the environment
has  changed (new  leader,  death of  some  sensors). Note  that  the number  of
constraints in the model is constant  (constraints of coverage expressed for all
sensors), whereas the number of variables $X_k$ decreases over periods, since 
only alive  sensors (sensors with enough energy to  be alive during one
sensing phase) are considered in the model.

\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  of 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} explains all possible network lifetime result of the relation between the different values of $\alpha$ and $\beta$, and for a network size equal to 200 sensor nodes. As can be seen in Table~\ref{my-labelx},  it is obvious and clear that when $\alpha$ decreased and $\beta$ increased by any step, the network lifetime for $Lifetime_{50}$ increased and the $Lifetime_{95}$ decreased. Therefore, selecting the values of $\alpha$ and $\beta$ depend on the application type used in the sensor nework. In PeCO protocol, $\alpha$ and $\beta$ are chosen based on the largest value of network lifetime for $Lifetime_{95}$.

\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
0.6 & 0.4 & 94 & 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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