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%\title{Authors' Instructions  \subtitle{Preparation of Camera-Ready Contributions to SCITEPRESS Proceedings} }

\title{Distributed Lifetime Coverage Optimization Protocol \\
  in Wireless Sensor Networks}

\author{Ali Kadhum Idrees$^{a,b}$, Karine Deschinkel$^{a}$,\\ Michel Salomon$^{a}$, and Rapha\"el Couturier$^{a}$\\
$^{a}$FEMTO-ST Institute, UMR 6174 CNRS, \\ University  Bourgogne  Franche-Comt\'e, Belfort, France\\
$^{b}${\em{Department of Computer Science, University of Babylon, Babylon, Iraq}}\\
email: ali.idness@edu.univ-fcomte.fr,\\ $\lbrace$karine.deschinkel, michel.salomon, raphael.couturier$\rbrace$@univ-fcomte.fr}

%\author{Ali   Kadhum   Idrees$^{a,b}$,   Karine  Deschinkel$^{a}$,\\  Michel Salomon$^{a}$,   and  Rapha\"el   Couturier   $^{a}$  \\   
%$^{a}${\em{FEMTO-ST Institute,  UMR  6174  CNRS,   University  Bourgogne  Franche-Comt\'e,\\ Belfort, France}} \\ 
%$^{b}${\em{Department of Computer Science, University of Babylon, Babylon, Iraq}} }

\begin{document}
 \maketitle 
%\keywords{Wireless   Sensor   Networks,   Area   Coverage,   Network   lifetime,Optimization, Scheduling.}

\abstract{ One of the main research challenges faced in Wireless Sensor Networks
  (WSNs) is to preserve continuously and effectively the coverage of an area (or
  region) of interest  to be monitored, while simultaneously  preventing as much
  as possible a network failure due to battery-depleted nodes.  In this paper we
  propose a protocol, called Distributed Lifetime Coverage Optimization protocol
  (DiLCO), which maintains the coverage and  improves the lifetime of a wireless
  sensor network. First, we partition the area of interest into subregions using
  a classical divide-and-conquer method.  Our DiLCO protocol is then distributed
  on  the sensor  nodes in  each subregion  in a  second step.   To fulfill  our
  objective, the proposed  protocol combines two effective  techniques: a leader
  election in  each subregion, followed  by an optimization-based  node activity
  scheduling  performed by  each elected  leader.  This  two-step process  takes
  place periodically, in  order to choose a small set  of nodes remaining active
  for sensing during a time slot.  Each set is built to ensure coverage at a low
  energy cost,  allowing to optimize  the network lifetime. Simulations are conducted using the discrete event simulator OMNET++. We refer to the characterictics  of a Medusa II  sensor for the energy consumption  and the computation time.   In comparison with two other existing  methods, our approach is able to  increase the WSN lifetime and provides improved coverage performances. }


%\onecolumn


%\normalsize \vfill

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

\noindent 
Energy efficiency is  a crucial issue in wireless sensor  networks since sensory
consumption, in  order to  maximize the network  lifetime, represents  the major
difficulty when designing WSNs. As a consequence, one of the scientific research
challenges in  WSNs, which has  been addressed by  a large amount  of literature
during the  last few  years, is  the design of  energy efficient  approaches for
coverage and connectivity~\cite{conti2014mobile}.  Coverage  reflects how well a
sensor  field is  monitored. On  the one  hand we  want to  monitor the  area of
interest in the most  efficient way~\cite{Nayak04}, which means
  that we want to maintain the best coverage as long as possible.  On the other
hand  we  want  to  use  as   little  energy  as  possible.   Sensor  nodes  are
battery-powered  with  no means  of  recharging  or  replacing, usually  due  to
environmental (hostile or unpractical environments) or cost reasons.  Therefore,
it is desired  that the WSNs are  deployed with high densities so  as to exploit
the overlapping sensing  regions of some sensor nodes to  save energy by turning
off  some   of  them   during  the   sensing  phase   to  prolong   the  network
lifetime.  A WSN  can  use  various types  of  sensors such  as
  \cite{ref17,ref19}:  thermal, seismic,  magnetic, visual,  infrared, acoustic,
  and  radar.  These  sensors  are   capable  of  observing  different  physical
  conditions  such  as:  temperature,   humidity,  pressure,  speed,  direction,
  movement, light,  soil makeup,  noise levels, presence  or absence  of certain
  kinds  of  objects,   and  mechanical  stress  levels   on  attached  objects.
  Consequently, there is a wide  range of WSN applications such as~\cite{ref22}:
  health-care, environment, agriculture, public safety, military, transportation
  systems, and industry applications.

In this  paper we design  a protocol that focuses  on the area  coverage problem
with  the objective  of maximizing  the network  lifetime. Our  proposition, the
Distributed  Lifetime  Coverage  Optimization (DiLCO)  protocol,  maintains  the
coverage  and improves  the lifetime  in  WSNs. The  area of  interest is  first
divided into  subregions using  a divide-and-conquer  algorithm and  an activity
scheduling  for sensor  nodes is  then  planned by  the elected  leader in  each
subregion. In fact, the nodes in a subregion can be seen as a cluster where each
node sends sensing data to the cluster  head or the sink node.  Furthermore, the
activities in a subregion/cluster can continue even if another cluster stops due
to too many node failures.  Our DiLCO protocol considers periods, where a period
starts with  a discovery phase  to exchange  information between sensors  of the
same subregion,  in order  to choose  in a  suitable manner  a sensor  node (the
leader) to carry out the coverage  strategy. In each subregion the activation of
the sensors for the  sensing phase of the current period  is obtained by solving
an integer program.  The resulting activation vector is broadcast by a leader to
every node of its subregion.

% MODIF - BEGIN
Our previous  paper ~\cite{idrees2014coverage} relies almost  exclusively on the
framework of the  DiLCO approach and the coverage problem  formulation.  In this
paper  we   made  more  realistic   simulations  by  taking  into   account  the
characteristics of  a Medusa II sensor  ~\cite{raghunathan2002energy} to measure
the energy consumption and the computation  time.  We have implemented two other
existing and distributed approaches (DESK ~\cite{ChinhVu}, and
GAF ~\cite{xu2001geography})  in order  to compare  their performances  with our
approach. We focused  on DESK and GAF protocols  for two reasons.
  First our protocol is inspired by both  of them: DiLCO uses a regular division
  of the  area of interest  as in GAF  and a temporal  division in rounds  as in
  DESK.  Second, DESK  and GAF are well-known protocols, easy  to implement, and
  often  used  as references  for  comparison.  We  also focus  on  performance
analysis based on the number of subregions.
% MODIF - END

The remainder  of the  paper continues with  Section~\ref{sec:Literature Review}
where a  review of some related  works is presented. The  next section describes
the  DiLCO  protocol,  followed  in   Section~\ref{cp}  by  the  coverage  model
formulation    which    is    used     to    schedule    the    activation    of
sensors. Section~\ref{sec:Simulation Results and  Analysis} shows the simulation
results. The paper ends with a  conclusion and some suggestions for further work
in Section~\ref{sec:Conclusion and Future Works}.

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

\noindent  In  this section,  we  summarize  some  related works  regarding  the
coverage problem and distinguish our  DiLCO protocol from the works presented in
the literature.

The most discussed coverage problems  in literature can be classified into three
types \cite{li2013survey}:  area coverage \cite{Misra} where  every point inside
an area is to be  monitored, target coverage \cite{yang2014novel} where the main
objective is  to cover only a  finite number of discrete  points called targets,
and barrier coverage \cite{Kumar:2005}\cite{kim2013maximum} to prevent intruders
from entering into the region  of interest. In \cite{Deng2012} authors transform
the area coverage problem to the target coverage problem taking into account the
intersection points among disks of sensors nodes or between disk of sensor nodes
and boundaries.  {\it In DiLCO protocol, the area coverage, i.e. the coverage of
  every  point in  the  sensing region,  is  transformed to  the  coverage of  a
  fraction of points called primary points. }

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), 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
\cite{ling2009energy}.  Active node selection is determined based on the problem
requirements  (e.g.  area   monitoring,  connectivity,  power  efficiency).  For
instance,  Jaggi  et al.  \cite{jaggi2006}  address  the  problem of  maximizing
network lifetime by dividing sensors into the maximum number of disjoint subsets
so 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.   Vu
\cite{chin2007}, Padmatvathy et al. \cite{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,  DiLCO 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 approaches, including centralized,  or distributed algorithms, have been
proposed     to    extend    the     network    lifetime.      In    distributed
algorithms~\cite{yangnovel,ChinhVu,qu2013distributed},       information      is
disseminated  throughout  the  network   and  sensors  decide  cooperatively  by
communicating with their neighbors which of them will remain in sleep mode for a
certain         period         of         time.          The         centralized
algorithms~\cite{cardei2005improving,zorbas2010solving,pujari2011high}     always
provide nearly or close to optimal  solution since the algorithm has global view
of the whole  network. But such a method has the  disadvantage of requiring high
communication costs,  since the  node (located at  the base station)  making the
decision needs information from all the  sensor nodes in the area and the amount
of  information can  be huge.   {\it  In order  to be  suitable for  large-scale
  network,  in the DiLCO  protocol, the  area coverage  is divided  into several
  smaller subregions, and in each one, a node called the leader is in charge for
  selecting the active sensors for the current period.}

% MODIF - BEGIN
 Our approach to select the leader node in a subregion is quite
  different   from    cluster   head    selection   methods   used    in   LEACH
  \cite{DBLP:conf/hicss/HeinzelmanCB00}         or          its         variants
  \cite{ijcses11}. Contrary  to LEACH, the division  of the area of  interest is
  supposed to be performed before the  leader election. Moreover, we assume that
  the sensors are deployed almost uniformly  and with high density over the area
  of interest, so  that the division is  fixed and regular.  As  in LEACH, our
  protocol works in round fashion.  In each round, during the pre-sensing phase,
  nodes make autonomous  decisions. In LEACH, each sensor elects  itself to be a
  cluster head,  and each non-cluster  head will  determine its cluster  for the
  round. In  our protocol, nodes in  the same subregion select  their leader. In
  both protocols,  the amount  of remaining  energy in each  node is  taken into
  account  to promote  the nodes  that have  the most  energy to  become leader.
  Contrary to  the LEACH protocol  where all sensors  will be active  during the
  sensing-phase, our protocol  allows to deactivate a subset  of sensors through
  an optimization process which significantly reduces  the energy consumption.
% MODIF - END

A large  variety of coverage scheduling  algorithms has been  developed. Many of
the existing  algorithms, 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
\cite{berman04,zorbas2010solving}.  Other  approaches are based  on mathematical
programming formulations~\cite{cardei2005energy,5714480,pujari2011high,Yang2014}
and dedicated  techniques (solving with a  branch-and-bound algorithms 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~\cite{castano2013column,rossi2012exact,deschinkel2012column}. {\it In DiLCO
  protocol, each  leader, in  each subregion, solves  an integer program  with a
  double objective  consisting in minimizing  the overcoverage and  limiting the
  undercoverage.  This  program is inspired from the  work of \cite{pedraza2006}
  where the objective is to maximize the number of cover sets.}

\section{\uppercase{Description of the DiLCO protocol}}
\label{sec:The DiLCO Protocol Description}

\noindent In this section, we  introduce the DiLCO protocol which is distributed
on  each subregion  in  the area  of interest.   It  is based  on two  efficient
techniques: network leader election  and sensor activity scheduling for coverage
preservation  and  energy  conservation,  applied  periodically  to  efficiently
maximize the lifetime in the network.

\subsection{Assumptions and models}

\noindent  We consider  a sensor  network composed  of static  nodes distributed
independently and uniformly at random.  A high density deployment ensures a high
coverage ratio of the interested area at the start. The nodes are supposed to
have homogeneous characteristics from a  communication and a processing point of
view, whereas they  have heterogeneous energy provisions.  Each  node has access
to its location thanks,  either to a hardware component (like a  GPS unit), or a
location discovery algorithm. 

\indent We consider a boolean disk  coverage model which is the most widely used
sensor coverage  model in the  literature. Thus, since  a sensor has  a constant
sensing range $R_s$, every space points  within a disk centered at a sensor with
the radius of  the sensing range is said  to be covered by this  sensor. We also
assume  that  the communication  range  $R_c \geq  2R_s$.   In  fact, Zhang  and
Hou~\cite{Zhang05} proved  that if the transmission range  fulfills the previous
hypothesis, a complete coverage of  a convex area implies connectivity among the
working nodes in the active mode.

\indent  For  each  sensor  we  also  define a  set  of  points  called  primary
points~\cite{idrees2014coverage} to  approximate the area  coverage it provides,
rather  than  working  with  a   continuous  coverage.   Thus,  a  sensing  disk
corresponding to  a sensor node is covered  by its neighboring nodes  if all its
primary points are covered. Obviously,  the approximation of coverage is more or
less accurate according to the number of primary points.

\subsection{Main idea}
\label{main_idea}
\noindent We start  by applying a divide-and-conquer algorithm  to partition the
area of interest  into smaller areas called subregions and  then our protocol is
executed  simultaneously in  each subregion. Sensor nodes  are
  assumed to be deployed almost uniformly over the region and the subdivision of
  the area of interest is regular.

\begin{figure}[ht!]
\centering
\includegraphics[width=75mm]{FirstModel.pdf} % 70mm
\caption{DiLCO protocol}
\label{fig2}
\end{figure} 

As  shown  in Figure~\ref{fig2},  the  proposed  DiLCO  protocol is  a  periodic
protocol where  each period is  decomposed into 4~phases:  Information Exchange,
Leader Election,  Decision, and Sensing. For  each period there  will be exactly
one  cover  set  in charge  of  the  sensing  task.   A periodic  scheduling  is
interesting  because it  enhances the  robustness  of the  network against  node failures.
% \textcolor{blue}{Many WSN applications have communication requirements that are periodic and known previously such as collecting temperature statistics at regular intervals. This periodic nature can be used to provide a regular schedule to sensor nodes and thus avoid a sensor failure. If the period time increases, the reliability and energy consumption are decreased and vice versa}. 
First,  a node  that has not  enough energy  to complete a  period, or
which fails before  the decision is taken, will be  excluded from the scheduling
process. Second,  if a node  fails later, whereas  it was supposed to  sense the
region of  interest, it will only affect  the quality of the  coverage until the
definition of  a new  cover set  in the next  period.  Constraints,  like energy
consumption, can be easily taken into consideration since the sensors can update
and exchange their  information during the first phase.  Let  us notice that the
phases  before  the sensing  one  (Information  Exchange,  Leader Election,  and
Decision) are  energy consuming for all the  nodes, even nodes that  will not be
retained by the leader to keep watch over the corresponding area.

During the execution of the DiLCO protocol, two kinds of packet will be used:
%\begin{enumerate}[(a)]
\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 inform them to stay Active or to go Sleep during the sensing phase.
\end{itemize}
%\end{enumerate}
and each sensor node will have five possible status in the network:
%\begin{enumerate}[(a)] 
\begin{itemize} 
\item LISTENING: sensor is waiting for a decision (to be active or not);
\item COMPUTATION: sensor applies the optimization process as leader;
\item ACTIVE: sensor is active;
\item SLEEP: sensor is turned off;
\item COMMUNICATION: sensor is transmitting or receiving packet.
\end{itemize}
%\end{enumerate}

An outline of the protocol  implementation is given by Algorithm~\ref{alg:DiLCO}
which describes  the execution of  a period  by a node  (denoted by $s_j$  for a
sensor node  indexed by  $j$). At  the beginning  a node  checks whether  it has
enough  energy (its energy  should  be  greater than  a  fixed
  treshold $E_{th}$) to  stay active during the next sensing  phase. If yes, it
exchanges information with all the other  nodes belonging to the same subregion:
it collects from each node  its position coordinates, remaining energy ($RE_j$),
ID,  and the  number of  one-hop neighbors  still alive.  INFO
  packet contains two parts: header and payload data. The sensor ID is included
  in  the header,  where the  header  size is  8  bits. The  data part  includes
  position coordinates (64 bits), remaining energy  (32 bits), and the number of
  one-hop live neighbors (8 bits). Therefore the  size of the INFO packet is 112
  bits. Once the  first phase is completed,  the nodes of a  subregion choose a
leader to  take the  decision based  on the  following criteria  with decreasing
importance: larger  number of  neighbors, larger remaining  energy, and  then in
case of equality,  larger index.  After that,  if the sensor node  is leader, it
will  solve an  integer program  (see Section~\ref{cp}).  This
  integer program  contains boolean variables  $X_j$ where ($X_j=1$)  means that
  sensor $j$ will be active in the  next sensing phase. Only sensors with enough
  remaining energy are  involved in the integer  program ($J$ is the  set of all
  sensors  involved).  As  the  leader consumes  energy  (computation energy  is
  denoted by $E^{comp}$) to solve the  optimization problem, it will be included
  in the integer program only if it has enough energy to achieve the computation
  and to  stay alive during the  next sensing phase, that  is to say if  $RE_j >
  E^{comp}+E_{th}$. Once  the optimization problem  is solved, each  leader will
  send an ActiveSleep packet to each sensor in the same subregion to indicate it
  if it has to be active or not.  Otherwise, if the sensor is not the leader, it
  will wait for the ActiveSleep packet to  know its state for the coming sensing
  phase.
%which provides a set of  sensors planned to be
%active in the next sensing phase.

\begin{algorithm}[h!]                

  \BlankLine
  %\emph{Initialize the sensor node and determine it's position and subregion} \; 
  
  \If{ $RE_j \geq E_{th}$ }{
      \emph{$s_j.status$ = COMMUNICATION}\;
      \emph{Send $INFO()$ packet to other nodes in the subregion}\;
      \emph{Wait $INFO()$ packet from other nodes in the subregion}\; 
      %\emph{UPDATE $RE_j$ for every sent or received INFO Packet}\;
      %\emph{ Collect information and construct the list L for all nodes in the subregion}\;
      
      %\If{ the received INFO Packet = No. of nodes in it's subregion -1  }{
      \emph{LeaderID = Leader election}\;
      \If{$ s_j.ID = LeaderID $}{
        \emph{$s_j.status$ = COMPUTATION}\;
        \emph{$\left\{\left(X_{1},\dots,X_{k},\dots,X_{J}\right)\right\}$ =
          Execute Integer Program Algorithm($J$)}\;
        \emph{$s_j.status$ = COMMUNICATION}\;
        \emph{Send $ActiveSleep()$ to each node $k$ in subregion} \;
        \emph{Update $RE_j $}\;
      }   
      \Else{
        \emph{$s_j.status$ = LISTENING}\;
        \emph{Wait $ActiveSleep()$ packet from the Leader}\;

        \emph{Update $RE_j $}\;
      }  
      %  }
  }
  \Else { Exclude $s_j$ from entering in the current sensing phase}
  
 %   \emph{return X} \;
\caption{DiLCO($s_j$)}
\label{alg:DiLCO}

\end{algorithm}

\section{\uppercase{Coverage problem formulation}}
\label{cp}

% MODIF - BEGIN
We formulate the coverage optimization problem with an integer program.
The objective function consists in minimizing the undercoverage and the overcoverage of the area as suggested in \cite{pedraza2006}. 
The area coverage problem is expressed as the coverage of a fraction of points called primary points. 
Details on the choice and the number of primary points can be found in \cite{idrees2014coverage}. The set of primary points is denoted by $P$
and the set of alive sensors by $J$. As we consider a boolean disk coverage model, we use the boolean indicator $\alpha_{jp}$ which is equal to 1 if the primary point $p$ is in the sensing range of the sensor $j$. The binary variable $X_j$ represents the activation or not of the sensor $j$. So we can express the number of  active sensors  that cover  the primary  point $p$ by $\sum_{j \in J} \alpha_{jp} * X_{j}$. We deduce the overcoverage denoted by $\Theta_p$ of the primary point $p$ :
\begin{equation}
 \Theta_{p} = \left \{ 
\begin{array}{l l}
  0 & \mbox{if the primary point}\\
    & \mbox{$p$ is not covered,}\\
  \left( \sum_{j \in J} \alpha_{jp} * X_{j} \right)- 1 & \mbox{otherwise.}\\
\end{array} \right.
\label{eq13} 
\end{equation}
More  precisely, $\Theta_{p}$ represents  the number of  active sensor
nodes minus  one that  cover the primary  point~$p$.
In the same way, we define the  undercoverage variable
$U_{p}$ of the primary point $p$ as:
\begin{equation}
U_{p} = \left \{ 
\begin{array}{l l}
  1 &\mbox{if the primary point $p$ is not covered,} \\
  0 & \mbox{otherwise.}\\
\end{array} \right.
\label{eq14} 
\end{equation}
There is, of course, a relationship between the three variables $X_j$, $\Theta_p$, and $U_p$ which can be formulated as follows :
\begin{equation}
\sum_{j \in J}  \alpha_{jp} X_{j} - \Theta_{p}+ U_{p} =1, \forall p \in P
\end{equation}
If the point $p$ is not covered, $U_p=1$,  $\sum_{j \in J}  \alpha_{jp} X_{j}=0$ and $\Theta_{p}=0$ by definition, so the equality is satisfied.
On the contrary, if the point $p$ is covered, $U_p=0$, and $\Theta_{p}=\left( \sum_{j \in J} \alpha_{jp}  X_{j} \right)- 1$. 
\noindent Our coverage optimization problem can then be formulated as follows:
\begin{equation} \label{eq:ip2r}
\left \{
\begin{array}{ll}
\min \sum_{p \in P} (w_{\theta} \Theta_{p} + w_{U} U_{p})&\\
\textrm{subject to :}&\\
\sum_{j \in J}  \alpha_{jp} X_{j} - \Theta_{p}+ U_{p} =1, &\forall p \in P\\
%\label{c1} 
%\sum_{t \in T} X_{j,t} \leq \frac{RE_j}{e_t} &\forall j \in J \\
%\label{c2}
\Theta_{p}\in \mathbb{N}, &\forall p \in P\\
U_{p} \in \{0,1\}, &\forall p \in P \\
X_{j} \in \{0,1\}, &\forall j \in J
\end{array}
\right.
\end{equation}
The objective function is a weighted sum of overcoverage and undercoverage. The goal is to limit the overcoverage in order to activate a minimal number of sensors while simultaneously preventing undercoverage.  By
    choosing  $w_{U}$ much  larger than $w_{\theta}$,  the coverage  of a
    maximum of  primary points  is ensured.  Then for the  same number  of covered
    primary points,  the solution  with a  minimal number  of active  sensors is
    preferred. 
%Both  weights $w_\theta$  and $w_U$ must  be carefully  chosen in
%order to  guarantee that the  maximum number of  points are covered  during each
%period.
% MODIF - END

\iffalse 

\indent Our model is based on the model proposed by \cite{pedraza2006} where the
objective is  to find a  maximum number of  disjoint cover sets.   To accomplish
this goal,  the authors proposed  an integer program which  forces undercoverage
and overcoverage of targets to become minimal at the same time.  They use binary
variables $x_{jl}$ to  indicate if sensor $j$ belongs to cover  set $l$.  In our
model, we consider that the binary variable $X_{j}$ determines the activation of
sensor $j$  in the sensing  phase. We also  consider primary points  as targets.
The set of primary points is denoted by $P$ and the set of sensors by $J$.

\noindent Let $\alpha_{jp}$ denote the indicator function of whether the primary
point $p$ is covered, that is:
\begin{equation}
\alpha_{jp} = \left \{ 
\begin{array}{l l}
  1 & \mbox{if the primary point $p$ is covered} \\
 & \mbox{by sensor node $j$}, \\
  0 & \mbox{otherwise.}\\
\end{array} \right.
%\label{eq12} 
\end{equation}
The  number of  active sensors  that cover  the primary  point $p$  can  then be
computed by $\sum_{j \in J} \alpha_{jp} * X_{j}$ where:
\begin{equation}
X_{j} = \left \{ 
\begin{array}{l l}
  1& \mbox{if sensor $j$  is active,} \\
  0 &  \mbox{otherwise.}\\
\end{array} \right.
%\label{eq11} 
\end{equation}
We define the Overcoverage variable $\Theta_{p}$ as:
\begin{equation}
 \Theta_{p} = \left \{ 
\begin{array}{l l}
  0 & \mbox{if the primary point}\\
    & \mbox{$p$ is not covered,}\\
  \left( \sum_{j \in J} \alpha_{jp} * X_{j} \right)- 1 & \mbox{otherwise.}\\
\end{array} \right.
\label{eq13} 
\end{equation}
\noindent More  precisely, $\Theta_{p}$ represents  the number of  active sensor
nodes minus  one that  cover the primary  point~$p$. The  Undercoverage variable
$U_{p}$ of the primary point $p$ is defined by:
\begin{equation}
U_{p} = \left \{ 
\begin{array}{l l}
  1 &\mbox{if the primary point $p$ is not covered,} \\
  0 & \mbox{otherwise.}\\
\end{array} \right.
\label{eq14} 
\end{equation}

\noindent Our coverage optimization problem can then be formulated as follows:
\begin{equation} \label{eq:ip2r}
\left \{
\begin{array}{ll}
\min \sum_{p \in P} (w_{\theta} \Theta_{p} + w_{U} U_{p})&\\
\textrm{subject to :}&\\
\sum_{j \in J}  \alpha_{jp} X_{j} - \Theta_{p}+ U_{p} =1, &\forall p \in P\\
%\label{c1} 
%\sum_{t \in T} X_{j,t} \leq \frac{RE_j}{e_t} &\forall j \in J \\
%\label{c2}
\Theta_{p}\in \mathbb{N}, &\forall p \in P\\
U_{p} \in \{0,1\}, &\forall p \in P \\
X_{j} \in \{0,1\}, &\forall j \in J
\end{array}
\right.
\end{equation}

\begin{itemize}
\item $X_{j}$ :  indicates whether or not the sensor $j$  is actively sensing (1
  if yes and 0 if not);
\item $\Theta_{p}$  : {\it overcoverage}, the  number of sensors  minus one that
  are covering the primary point $p$;
\item $U_{p}$ : {\it undercoverage},  indicates whether or not the primary point
  $p$ is being covered (1 if not covered and 0 if covered).
\end{itemize}

The first group  of constraints indicates that some primary  point $p$ should be
covered by at least  one sensor and, if it is not  always the case, overcoverage
and undercoverage  variables help balancing the restriction  equations by taking
positive values. Two objectives can be noticed in our model. First, we limit the
overcoverage of primary  points to activate as few  sensors as possible. Second,
to  avoid   a  lack  of  area   monitoring  in  a  subregion   we  minimize  the
undercoverage. Both  weights $w_\theta$  and $w_U$ must  be carefully  chosen in
order to  guarantee that the  maximum number of  points are covered  during each
period.

\fi

\section{\uppercase{Protocol evaluation}}  
\label{sec:Simulation Results and Analysis}
\noindent \subsection{Simulation framework}

To assess the performance of our DiLCO protocol, we have used the discrete
event simulator OMNeT++ \cite{varga} to run different series of simulations.
Table~\ref{table3} gives the chosen parameters setting.

\begin{table}[ht]
\caption{Relevant parameters for network initializing.}
% title of Table
\centering
% used for centering table
\begin{tabular}{c|c}
% centered columns (4 columns)
      \hline
%inserts double horizontal lines
Parameter & Value  \\ [0.5ex]
   
%Case & Strategy (with Two Leaders) & Strategy (with One Leader) & Simple Heuristic \\ [0.5ex]
% inserts table
%heading
\hline
% inserts single horizontal line
Sensing  Field  & $(50 \times 25)~m^2 $   \\
% inserting body of the table
%\hline
Nodes Number &  50, 100, 150, 200 and 250~nodes   \\
%\hline
Initial Energy  & 500-700~joules  \\  
%\hline
Sensing Period & 60 Minutes \\
$E_{th}$ & 36 Joules\\
$R_s$ & 5~m   \\     
%\hline
$w_{\Theta}$ & 1   \\
% [1ex] adds vertical space
%\hline
$w_{U}$ & $|P|^2$
%inserts single line
\end{tabular}
\label{table3}
% is used to refer this table in the text
\end{table}

Simulations with five  different node densities going from  50 to 250~nodes were
performed  considering  each  time  25~randomly generated  networks,  to  obtain
experimental results  which are relevant. 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.

We chose as energy consumption model the one proposed proposed by~\cite{ChinhVu}
and based on ~\cite{raghunathan2002energy} with slight modifications. The energy
consumed by  the communications  is added  and the part  relative to  a variable
sensing range is removed. We also assume that the nodes have the characteristics
of the  Medusa II sensor  node platform \cite{raghunathan2002energy}.   A sensor
node typically  consists of  four units: a  MicroController Unit, an  Atmels AVR
ATmega103L in  case of Medusa II,  to perform the  computations; a communication
(radio) unit able to send and  receive messages; a sensing unit to collect data;
a power supply  which provides the energy consumed by  node. Except the battery,
all the other unit  can be switched off to save  energy according to the node
status.   Table~\ref{table4} summarizes  the energy  consumed (in  milliWatt per
second) by a node for each of its possible status.

\begin{table}[ht]
\caption{Energy consumption model}
% title of Table
\centering
% used for centering table
{\scriptsize
\begin{tabular}{|c|c|c|c|c|}
% centered columns (4 columns)
      \hline
%inserts double horizontal lines
Sensor status & MCU   & Radio & Sensing & Power (mW) \\ [0.5ex]
\hline
% inserts single horizontal line
Listening & ON & ON & ON & 20.05 \\
% inserting body of the table
\hline
Active & ON & OFF & ON & 9.72 \\
\hline
Sleep & OFF & OFF & OFF & 0.02 \\
\hline
Computation & ON & ON & ON & 26.83 \\
%\hline
%\multicolumn{4}{|c|}{Energy needed to send/receive a 1-bit} & 0.2575\\
 \hline
\end{tabular}
}

\label{table4}
% is used to refer this table in the text
\end{table}

Less  influent  energy consumption  sources  like  when  turning on  the  radio,
starting the sensor node, changing the status of a node, etc., will be neglected
for the  sake of simplicity. Each node  saves energy by switching  off its radio
once it has  received its decision status from the  corresponding leader (it can
be itself).  As explained previously in subsection~\ref{main_idea}, two kinds of
packets  for communication  are  considered  in our  protocol:  INFO packet  and
ActiveSleep  packet. To  compute the  energy  needed by  a node  to transmit  or
receive such  packets, we  use the equation  giving the  energy spent to  send a
1-bit-content   message  defined   in~\cite{raghunathan2002energy}   (we  assume
symmetric  communication costs), and  we set  their respective  size to  112 and
24~bits. The energy required to send  or receive a 1-bit-content message is thus
 equal to 0.2575~mW.

Each node  has an initial  energy level, in  Joules, which is randomly  drawn in
$[500-700]$.   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  take part  in  the coverage  task. This  value
corresponds to the  energy needed by the sensing  phase, obtained by multiplying
the energy  consumed in active state  (9.72 mW) by  the time in seconds  for one
period  (3,600 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.

In the simulations,  we introduce the following performance  metrics to evaluate
the efficiency of our approach:

%\begin{enumerate}[i)]
\begin{itemize}
\item {{\bf Network Lifetime}:} we define the network lifetime as the time until
  the  coverage  ratio  drops  below  a  predefined  threshold.   We  denote  by
  $Lifetime_{95}$ (respectively $Lifetime_{50}$) the amount of time during which
  the  network can  satisfy an  area coverage  greater than  $95\%$ (respectively
  $50\%$). We assume that the sensor  network can fulfill its task until all its
  nodes have  been drained of their  energy or it  becomes disconnected. Network
  connectivity  is crucial because  an active  sensor node  without connectivity
  towards a base  station cannot transmit any information  regarding an observed
  event in the area that it monitors.
     
\item {{\bf Coverage Ratio (CR)}:} it measures how well the WSN is able to 
  observe the area of interest. In our case, we discretized the sensor field
  as a regular grid, which yields the following equation to compute the
  coverage ratio: 
\begin{equation*}
\scriptsize
\mbox{CR}(\%) = \frac{\mbox{$n$}}{\mbox{$N$}} \times 100.
\end{equation*}
where  $n$ is  the number  of covered  grid points  by active  sensors  of every
subregions during  the current  sensing phase  and $N$ is the total number  of grid
points in  the sensing field. In  our simulations, we have  a layout of  $N = 51
\times 26 = 1326$ grid points.

\item {{\bf  Energy Consumption}:}  energy consumption (EC)  can be seen  as the
  total amount of  energy   consumed   by   the   sensors   during   $Lifetime_{95}$   
  or $Lifetime_{50}$, divided  by the number of periods.  Formally, the computation
  of EC can be expressed as follows:
  \begin{equation*}
    \scriptsize
    \mbox{EC} = \frac{\sum\limits_{m=1}^{M} \left( E^{\mbox{com}}_m+E^{\mbox{list}}_m+E^{\mbox{comp}}_m  
      + E^{a}_m+E^{s}_m \right)}{M},
  \end{equation*}

where $M$  corresponds to  the number  of periods.  The  total amount  of energy
consumed by the  sensors (EC) comes through taking  into consideration four main
energy   factors.  The  first   one,  denoted   $E^{\scriptsize  \mbox{com}}_m$,
represents  the  energy  consumption  spent   by  all  the  nodes  for  wireless
communications  during period  $m$.  $E^{\scriptsize  \mbox{list}}_m$,  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  $m$.
$E^{\scriptsize \mbox{comp}}_m$  refers to the  energy needed by all  the leader
nodes  to solve the  integer program  during a  period.  Finally,  $E^a_{m}$ and
$E^s_{m}$ indicate the energy consumed by the whole network in the sensing phase
(active and sleeping nodes).

\end{itemize}
%\end{enumerate}

%\subsection{Performance Analysis for different subregions}
\subsection{Performance analysis}
\label{sub1}

In this subsection, we first focus  on the performance of our DiLCO protocol for
different numbers  of subregions.  We consider partitions  of the WSN  area into
$2$, $4$, $8$, $16$, and $32$ subregions. Thus the DiLCO protocol is declined in
five versions:  DiLCO-2, DiLCO-4,  DiLCO-8, DiLCO-16, and  DiLCO-32. Simulations
without  partitioning  the  area  of  interest,  cases  which  correspond  to  a
centralized  approach, are  not presented  because they  require  high execution
times to solve the integer program and therefore consume too much energy.

We compare our protocol to two  other approaches. The first one, called DESK and
proposed  by ~\cite{ChinhVu}  is a  fully distributed  coverage  algorithm.  The
second one, called GAF  ~\cite{xu2001geography}, consists in dividing the region
into fixed  squares.  During the decision  phase, in each square,  one sensor is
chosen to remain active during the sensing phase.

\subsubsection{Coverage ratio} 

Figure~\ref{fig3} shows  the average coverage  ratio for 150 deployed  nodes. It
can be seen  that both DESK and  GAF provide a coverage ratio  which is slightly
better  compared to  DiLCO  in the  first  thirty periods.  This  can be  easily
explained  by  the number  of  active nodes:  the  optimization  process of  our
protocol activates less  nodes than DESK or GAF, resulting  in a slight decrease
of the coverage  ratio. In case of DiLCO-2  (respectively DiLCO-4), the coverage
ratio exhibits a fast decrease with the number of periods and reaches zero value
in period~18 (respectively  46), whereas the other versions  of DiLCO, DESK, and
GAF ensure a coverage ratio above  50\% for subsequent periods.  We believe that
the  results  obtained  with these  two  methods  can  be  explained by  a  high
consumption of energy and we will check this assumption in the next subsection.

Concerning  DiLCO-8, DiLCO-16,  and  DiLCO-32,  these methods  seem  to be  more
efficient than DESK  and GAF, since they can provide the  same level of coverage
(except in the first periods where  DESK and GAF slightly outperform them) for a
greater number  of periods. In fact, when  our protocol is applied  with a large
number of subregions (from 8 to 32~regions), it activates a restricted number of
nodes, and thus enables the extension of the network lifetime.

\parskip 0pt    
\begin{figure}[t!]
\centering
 \includegraphics[scale=0.475] {CR.pdf} 
\caption{Coverage ratio}
\label{fig3}
\end{figure} 


\subsubsection{Energy consumption}

Based on  the results shown in  Figure~\ref{fig3}, we focus on  the DiLCO-16 and
DiLCO-32 versions of our protocol,  and we compare their energy consumption with
the DESK and GAF approaches. For each sensor node we measure the energy consumed
according to its successive status,  for different network densities.  We denote
by $\mbox{\it  Protocol}/50$ (respectively $\mbox{\it  Protocol}/95$) the amount
of energy consumed  while the area coverage is  greater than $50\%$ (repectively
$95\%$),  where  {\it  Protocol}  is  one  of the  four  protocols  we  compare.
Figure~\ref{fig95} presents  the energy consumptions observed  for network sizes
going from 50  to 250~nodes. Let us  notice that the same network  sizes will be
used for the different performance metrics.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.475]{EC.pdf} 
\caption{Energy consumption per period}
\label{fig95}
\end{figure} 

The  results  depict the  good  performance of  the  different  versions of  our
protocol.   Indeed,  the protocols  DiLCO-16/50,  DiLCO-32/50, DiLCO-16/95,  and
DiLCO-32/95  consume less  energy than  their DESK  and GAF  counterparts  for a
similar level of area coverage.   This observation reflects the larger number of
nodes set active  by DESK and GAF. 

Now, if we consider a same  protocol, we can notice that the average consumption
per  period increases slightly  for our  protocol when  increasing the  level of
coverage and the number of node,  whereas it increases more largely for DESK and
GAF.  In case of DiLCO, it means that even if a larger network allows to improve
the number of periods with a  minimum coverage level value, this improvement has
a  higher energy  cost  per period  due  to communication  overhead  and a  more
difficult optimization problem.   However, in comparison with DESK  and GAF, our
approach has a reasonable energy overcost.

\subsubsection{Execution time}

Another interesting point to investigate  is the evolution of the execution time
with the size of the WSN and  the number of subregions. Therefore, we report for
every version of  our protocol the average execution times  in seconds needed to
solve the optimization problem for  different WSN sizes. The execution times are
obtained on a laptop DELL  which has an Intel Core~i3~2370~M~(2.4~GHz) dual core
processor and a MIPS rating equal to 35330. The corresponding execution times on
a MEDUSA II sensor node are then  extrapolated according to the MIPS rate of the
Atmels  AVR  ATmega103L  microcontroller  (6~MHz),  which  is  equal  to  6,  by
multiplying    the    laptop     times    by    $\left(\frac{35330}{2}    \times
\frac{1}{6}\right)$.  The  expected times  on  a  sensor  node are  reported  on
Figure~\ref{fig8}.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.475]{T.pdf}  
\caption{Execution time in seconds}
\label{fig8}
\end{figure} 

Figure~\ref{fig8} shows that DiLCO-32 has very low execution times in comparison
with  other DiLCO  versions, because  the activity  scheduling is  tackled by  a
larger  number of  leaders and  each  leader solves  an integer  problem with  a
limited number  of variables and  constraints.  Conversely, DiLCO-2  requires to
solve an optimization problem with half of the network nodes and thus presents a
high execution time.  Nevertheless if  we refer to Figure~\ref{fig3}, we observe
that DiLCO-32  is slightly less efficient  than DilCO-16 to maintain  as long as
possible high coverage. In fact an excessive subdivision of the area of interest
prevents  it  to  ensure a  good  coverage  especially  on  the borders  of  the
subregions. Thus,  the optimal number of  subregions can be seen  as a trade-off
between execution time and coverage performance.

\subsubsection{Network lifetime}

In the next figure, the network lifetime is illustrated. Obviously, the lifetime
increases with  the network  size, whatever the  considered protocol,  since the
correlated node  density also  increases.  A high  network density means  a high
node redundancy  which allows  to turn-off  many nodes and  thus to  prolong the
network lifetime.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.475]{LT.pdf}  
\caption{Network lifetime}
\label{figLT95}
\end{figure} 

As  highlighted by  Figure~\ref{figLT95},  when the  coverage  level is  relaxed
($50\%$) the network lifetime also  improves. This observation reflects the fact
that  the higher  the coverage  performance, the  more nodes  must be  active to
ensure the wider monitoring.  For a similar level of coverage, DiLCO outperforms
DESK and GAF for the lifetime of  the network. More specifically, if we focus on
the  larger  level  of coverage  ($95\%$)  in  the  case  of our  protocol,  the
subdivision in $16$~subregions seems to be the most appropriate.


\section{\uppercase{Conclusion and future work}}
\label{sec:Conclusion and Future Works} 

A crucial problem in WSN is to  schedule the sensing activities of the different
nodes  in order  to ensure  both coverage  of the  area of  interest and  longer
network lifetime. The inherent limitations of sensor nodes, in energy provision,
communication and computing capacities, require  protocols that optimize the use
of  the available  resources  to  fulfill the  sensing  task.   To address  this
problem, this paper proposes a two-step  approach. Firstly, the field of sensing
is  divided into  smaller  subregions using  the  concept of  divide-and-conquer
method. Secondly,  a distributed  protocol called Distributed  Lifetime Coverage
Optimization is applied in each subregion  to optimize the coverage and lifetime
performances.  In a  subregion, our protocol consists in electing  a leader node
which will then perform a sensor activity scheduling. The challenges include how
to  select   the  most  efficient  leader   in  each  subregion  and   the  best
representative set of active nodes to ensure a high level of coverage. To assess
the performance of our approach, we  compared it with two other approaches using
many performance metrics  like coverage ratio or network lifetime.  We have also
studied the impact of  the number of subregions chosen to  subdivide the area of
interest,  considering  different  network  sizes.  The  experiments  show  that
increasing the number  of subregions improves the lifetime.  The more subregions
there are, the more robust the network is against random disconnection resulting
from dead nodes.  However,  for a given sensing field and  network size there is
an optimal number of subregions.  Therefore, in case of our simulation context a
subdivision in $16$~subregions seems to be the most relevant. The optimal number
of subregions will be investigated in the future.

\section*{\uppercase{Acknowledgements}}

\noindent  As a  Ph.D.   student, Ali  Kadhum  IDREES would  like to  gratefully
acknowledge  the University  of Babylon  - IRAQ  for the  financial  support and
Campus France for  the received support. This paper is  also partially funded by
the Labex ACTION program (contract ANR-11-LABX-01-01).

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