update figure of primary points model by Ali

[JournalMultiPeriods.git] / article.tex

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\journal{Ad Hoc Networks}

\begin{document}

\begin{frontmatter}

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

%% use optional labels to link authors explicitly to addresses:
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%\author{Ali Kadhum Idrees, Karine Deschinkel, \\
%Michel Salomon, and Rapha\"el Couturier}

%\thanks{are members in the AND team - DISC department - FEMTO-ST Institute, University of Franche-Comt\'e, Belfort, France.
% e-mail: ali.idness@edu.univ-fcomte.fr, $\lbrace$karine.deschinkel, michel.salomon, raphael.couturier$\rbrace$@univ-fcomte.fr.}% <-this % stops a space
%\thanks{}% <-this % stops a space
 
%\address{FEMTO-ST Institute, University of Franche-Comt\'e, Belfort, France. \\ 
%e-mail: 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{abstract}
%One of  the fundamental challenges in Wireless Sensor Networks (WSNs)
%is the coverage preservation and the extension of the network lifetime
%continuously  and  effectively  when  monitoring a  certain  area  (or
%region) of  interest. 
Coverage and  lifetime are  two paramount problems  in Wireless  Sensor Networks
(WSNs). In this paper, a  method called Multiround Distributed Lifetime Coverage
Optimization  protocol (MuDiLCO)  is proposed  to maintain  the coverage  and to
improve the lifetime in wireless sensor  networks. The area of interest is first
divided into  subregions and  then the  MuDiLCO protocol  is distributed  on the
sensor nodes in  each subregion. The proposed MuDiLCO protocol  works in periods
during which sets of sensor nodes are  scheduled, with one set for each round of
a period, to remain active during the  sensing phase and thus ensure coverage so
as  to maximize  the  WSN lifetime.   \textcolor{blue}{The  decision process  is
  carried out by a leader node,  which solves an optimization problem to produce
  the  best representative  sets to  be used  during the  rounds of  the sensing
  phase. The optimization problem formulated as  an integer program is solved to
  optimality through a Branch-and-Bound method  for small instances.  For larger
  instances, the best  feasible solution found by the solver  after a given time
  limit threshold is considered.}
%The decision process is  carried out by a  leader node, which
%solves an  integer program to  produce the best  representative sets to  be used
%during the rounds  of the sensing phase. 
%\textcolor{red}{The integer program is solved by either GLPK solver or Genetic Algorithm (GA)}. 
Compared  with some  existing protocols,  simulation results  based on  multiple
criteria (energy consumption, coverage ratio, and  so on) show that the proposed
protocol can prolong  efficiently the network lifetime and  improve the coverage
performance.
\end{abstract}

\begin{keyword}
Wireless   Sensor   Networks,   Area   Coverage,   Network   Lifetime,
Optimization, Scheduling, Distributed Computation.
\end{keyword}

\end{frontmatter}

\section{Introduction}
 
\indent  The   fast  developments  of  low-cost  sensor   devices  and  wireless
communications have allowed the emergence of WSNs. A WSN includes a large number
of small, limited-power sensors that  can sense, process, and transmit data over
a wireless  communication. They communicate  with each other by  using multi-hop
wireless communications and cooperate together  to monitor the area of interest,
so that  each measured data can be  reported to a monitoring  center called sink
for further  analysis~\cite{Sudip03}.  There  are several fields  of application
covering  a wide  spectrum for  a  WSN, including  health, home,  environmental,
military, and industrial applications~\cite{Akyildiz02}.

On the one hand sensor nodes run on batteries with limited capacities, and it is
often  costly  or  simply  impossible  to  replace  and/or  recharge  batteries,
especially in remote and hostile environments. Obviously, to achieve a long life
of the  network it is important  to conserve battery  power. Therefore, lifetime
optimization is one of the most  critical issues in wireless sensor networks. On
the other hand we must guarantee  coverage over the area of interest. To fulfill
these two objectives, the main idea  is to take advantage of overlapping sensing
regions to turn-off redundant sensor nodes  and thus save energy. In this paper,
we concentrate  on the area coverage  problem, with the  objective of maximizing
the network lifetime by using an optimized multiround scheduling.

% One of the major scientific research challenges in WSNs, which are addressed by a large number of literature during the last few years is to design energy efficient approaches for coverage and connectivity in WSNs~\cite{conti2014mobile}. The coverage problem is one  of the
%fundamental challenges in WSNs~\cite{Nayak04} that consists in monitoring efficiently and continuously
%the area of interest. The limited energy of sensors represents the main challenge in the WSNs
%design~\cite{Sudip03}, where it is difficult to replace and/or recharge their batteries because the the area of interest nature (such as hostile environments) and the cost. So, it is necessary that a WSN
%deployed  with high  density because  spatial redundancy  can  then be exploited to increase  the lifetime of the network. However, turn on all the sensor nodes, which monitor the same region at the same time
%leads to decrease the lifetime of the network. To extend the lifetime of the network, the main idea is to take advantage of the overlapping sensing regions  of some  sensor nodes to  save energy by  turning off
%some  of them  during the  sensing phase~\cite{Misra05}. WSNs require energy-efficient solutions to improve the network lifetime that is constrained by the limited power of each sensor node ~\cite{Akyildiz02}. 

%In this paper,  we concentrate on the area coverage  problem, with the objective
%of maximizing the network lifetime by using an optimized multirounds scheduling.
%The area of interest is divided into subregions.

% Each period includes four phases starts with a discovery phase to exchange information among the sensors of the subregion, in order  to choose in a  suitable manner a sensor node as leader to carry out a coverage strategy.  This coverage strategy involves the solving of an integer program by the leader,  to optimize the coverage and the lifetime in the subregion by producing a sets of sensor nodes in order to take the mission of coverage preservation during several rounds in the sensing phase. 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.  

The remainder of the paper is organized as follows. The next section
% Section~\ref{rw}
reviews the  related works  in the  field.  Section~\ref{pd}  is devoted  to the
description of MuDiLCO protocol.  Section~\ref{exp} shows the simulation results
obtained using  the discrete event  simulator OMNeT++ \cite{varga}.   They fully
demonstrate  the  usefulness  of  the   proposed  approach.   Finally,  we  give
concluding    remarks   and    some    suggestions   for    future   works    in
Section~\ref{sec:conclusion}.


%%RC : Related works good for a phd thesis but too long for a paper. Ali you  need to learn to .... summarize :-)
\section{Related works} % Trop proche de l'etat de l'art de l'article de Zorbas ?
\label{rw}

\indent  This section is  dedicated to  the various  approaches proposed  in the
literature for  the coverage lifetime maximization problem,  where the objective
is to optimally schedule sensors' activities in order to extend network lifetime
in WSNs. Cardei  and Wu \cite{cardei2006energy} provide a  taxonomy for coverage
algorithms in WSNs according to several design choices:
\begin{itemize}
\item  Sensors   scheduling  algorithm  implementation,   i.e.   centralized  or
  distributed/localized algorithms.
\item The objective of sensor coverage, i.e. to maximize the network lifetime or
  to minimize the number of active sensors during a sensing round.
\item The homogeneous or heterogeneous nature  of the nodes, in terms of sensing
  or communication capabilities.
\item The node deployment method, which may be random or deterministic.
\item  Additional  requirements  for  energy-efficient and  connected coverage.
\end{itemize}

The choice of non-disjoint or disjoint cover sets (sensors participate or not in
many cover sets) can be added to the above list.
% The independency in the cover set (i.e. whether the cover sets are disjoint or non-disjoint) \cite{zorbas2010solving} is another design choice that can be added to the above list.

\subsection{Centralized approaches}

The major approach  is to divide/organize the sensors into  a suitable number of
cover sets where  each set completely covers an interest  region and to activate
these cover sets successively.  The centralized algorithms always provide nearly
or close to  optimal solution since the  algorithm has global view  of the whole
network. Note that  centralized algorithms have the advantage  of requiring very
low  processing  power  from  the  sensor  nodes,  which  usually  have  limited
processing  capabilities. The  main drawback  of this  kind of  approach is  its
higher cost in communications, since the  node that will make the decision needs
information from  all the sensor  nodes.  \textcolor{blue} {Exact  or heuristics
  approaches are designed to provide cover sets.
%(Moreover, centralized approaches usually
%suffer from the scalability problem, making them less competitive as the network
%size increases.) 
Contrary to exact methods, heuristic ones  can handle very large and centralized
problems.  They are  proposed to  reduce computational  overhead such  as energy
consumption, delay, and generally allow to increase the network lifetime.}

The first algorithms proposed in the literature consider that the cover sets are
disjoint:  a  sensor  node  appears  in  exactly  one  of  the  generated  cover
sets~\cite{abrams2004set,cardei2005improving,Slijepcevic01powerefficient}.    In
the  case   of  non-disjoint   algorithms  \cite{pujari2011high},   sensors  may
participate in  more than one  cover set.  In some  cases, this may  prolong the
lifetime of the network in comparison  to the disjoint cover set algorithms, but
designing  algorithms for  non-disjoint cover  sets generally  induces a  higher
order  of complexity.   Moreover, in  case of  a sensor's  failure, non-disjoint
scheduling policies  are less  resilient and  reliable because  a sensor  may be
involved in more than one cover sets.
%For instance, the proposed work in ~\cite{cardei2005energy, berman04}    

In~\cite{yang2014maximum},  the authors  have  considered  a linear  programming
approach  to select  the minimum  number of  working sensor  nodes, in  order to
preserve a  maximum coverage and  to extend lifetime  of the network.   Cheng et
al.~\cite{cheng2014energy} have defined a  heuristic algorithm called Cover Sets
Balance  (CSB), which  chooses  a set  of  active nodes  using  the tuple  (data
coverage range, residual  energy).  Then, they have introduced  a new Correlated
Node Set Computing (CNSC) algorithm to find  the correlated node set for a given
node.   After that,  they  proposed a  High Residual  Energy  First (HREF)  node
selection algorithm to minimize the number of  active nodes so as to prolong the
network  lifetime.   Various  centralized  methods based  on  column  generation
approaches                   have                    also                   been
proposed~\cite{gentili2013,castano2013column,rossi2012exact,deschinkel2012column}.
\textcolor{blue}{In~\cite{gentili2013}, authors highlight  the trade-off between
  the  network lifetime  and the  coverage  percentage. They  show that  network
  lifetime can be hugely improved by decreasing the coverage ratio.}

\subsection{Distributed approaches}
%{\bf Distributed approaches}
In distributed  and localized coverage  algorithms, the required  computation to
schedule the  activity of  sensor nodes  will be done  by the  cooperation among
neighboring nodes. These  algorithms may require more computation  power for the
processing by the cooperating sensor nodes, but they are more scalable for large
WSNs.  Localized and distributed algorithms generally result in non-disjoint set
covers.

Many distributed algorithms have been  developed to perform the scheduling so as
to          preserve         coverage,          see          for         example
\cite{Gallais06,Tian02,Ye03,Zhang05,HeinzelmanCB02,       yardibi2010distributed,
  prasad2007distributed,Misra}.   Distributed  algorithms  typically operate  in
rounds for  a predetermined duration. At  the beginning of each  round, a sensor
exchanges information with  its neighbors and makes a  decision to either remain
turned on or  to go to sleep for  the round. This decision is  basically made on
simple     greedy     criteria    like     the     largest    uncovered     area
\cite{Berman05efficientenergy}      or       maximum      uncovered      targets
\cite{lu2003coverage}.   The  Distributed  Adaptive Sleep  Scheduling  Algorithm
(DASSA) \cite{yardibi2010distributed}  does not require  location information of
sensors while  maintaining connectivity and  satisfying a user  defined coverage
target.  In  DASSA, nodes use the  residual energy levels and  feedback from the
sink for  scheduling the activity  of their neighbors.  This  feedback mechanism
reduces  the randomness  in scheduling  that would  otherwise occur  due  to the
absence of location information.  In  \cite{ChinhVu}, the author have designed a
novel distributed heuristic,  called Distributed Energy-efficient Scheduling for
k-coverage (DESK), which  ensures that the energy consumption  among the sensors
is  balanced  and the  lifetime  maximized  while  the coverage  requirement  is
maintained.   This heuristic  works in  rounds, requires  only  one-hop neighbor
information, and each  sensor decides its status (active or  sleep) based on the
perimeter coverage model from~\cite{Huang:2003:CPW:941350.941367}.

%Our Work, which is presented in~\cite{idrees2014coverage} proposed a coverage optimization protocol to improve the lifetime in
%heterogeneous energy wireless sensor networks. 
%In this work, the coverage protocol distributed in each sensor node in the subregion but the optimization take place over the the whole subregion. We consider only distributing the coverage protocol over two subregions. 

The  works presented  in  \cite{Bang, Zhixin,  Zhang}  focus on  coverage-aware,
distributed energy-efficient,  and distributed clustering  methods respectively,
which  aim at extending  the network  lifetime, while  the coverage  is ensured.
More recently, Shibo et al.  \cite{Shibo} have expressed the coverage problem as
a  minimum  weight submodular  set  cover  problem  and proposed  a  Distributed
Truncated Greedy  Algorithm (DTGA) to solve  it.  They take  advantage from both
temporal and spatial correlations between  data sensed by different sensors, and
leverage prediction, to improve  the lifetime.  In \cite{xu2001geography}, Xu et
al.  have  described an algorithm, called Geographical  Adaptive Fidelity (GAF),
which uses geographic  location information to divide the  area of interest into
fixed square grids.   Within each grid, it keeps only one  node staying awake to
take the responsibility of sensing and communication.

Some  other  approaches (outside  the  scope  of our  work)  do  not consider  a
synchronized and  predetermined time-slot where  the sensors are active  or not.
Indeed, each sensor  maintains its own timer and its  wake-up time is randomized
\cite{Ye03} or regulated \cite{cardei2005maximum} over time.

The MuDiLCO protocol (for  Multiround Distributed Lifetime Coverage Optimization
protocol) presented  in this paper  is an  extension of the  approach introduced
in~\cite{idrees2014coverage}.   In~\cite{idrees2014coverage},  the  protocol  is
deployed over  only two subregions.  Simulation results  have shown that  it was
more  interesting  to  divide  the  area  into  several  subregions,  given  the
computation complexity. Compared to our previous paper, in this one we study the
possibility of dividing  the sensing phase into multiple rounds  and we also add
an  improved  model of  energy  consumption  to  assess  the efficiency  of  our
approach. In fact, in this paper we make a multiround optimization, while it was
a single round  optimization in our previous work.  \textcolor{blue}{The idea is
  to take advantage  of the pre-sensing phase to plan  the sensor's activity for
  several  rounds instead  of one,  thus saving  energy. In  addition, when  the
  optimization problem becomes  more complex, its resolution is  stopped after a
  given time threshold}.

\iffalse
   
\subsection{Centralized Approaches}
%{\bf Centralized approaches}
The major approach  is to divide/organize the sensors into  a suitable number of
set covers where  each set completely covers an interest  region and to activate
these set covers successively.  The centralized algorithms always provide nearly
or close  to optimal solution since the  algorithm has global view  of the whole
network. Note that  centralized algorithms have the advantage  of requiring very
low  processing  power  from  the  sensor  nodes,  which  usually  have  limited
processing  capabilities. The  main drawback  of this  kind of  approach  is its
higher cost in communications, since the  node that will take the decision needs
information from all the  sensor nodes. Moreover, centralized approaches usually
suffer from the scalability problem, making them less competitive as the network
size increases.

The first algorithms proposed in the literature consider that the cover sets are
disjoint: a sensor node appears in exactly one of the generated cover sets.  For
instance,  Slijepcevic  and  Potkonjak  \cite{Slijepcevic01powerefficient}  have
proposed an algorithm, which allocates sensor nodes in mutually independent sets
to monitor an area divided into  several fields.  Their algorithm builds a cover
set by including in priority the  sensor nodes which cover critical fields, that
is to say fields  that are covered by the smallest number  of sensors.  The time
complexity of  their heuristic is $O(n^2)$  where $n$ is the  number of sensors.
Abrams et al.~\cite{abrams2004set}  have designed three approximation algorithms
for a variation of the set  k-cover problem, where the objective is to partition
the sensors  into covers such  that the number  of covers that include  an area,
summed  over all  areas, is  maximized.  Their  work builds  upon  previous work
in~\cite{Slijepcevic01powerefficient}  and  the  generated  cover  sets  do  not
provide complete coverage of the monitoring zone.

In \cite{cardei2005improving}, the authors have proposed a method to efficiently
compute the maximum number of disjoint set covers such that each set can monitor
all targets. They first transform the problem into a maximum flow problem, which
is formulated  as a mixed integer  programming (MIP). Then  their heuristic uses
the output  of the MIP to compute  disjoint set covers.  Results  show that this
heuristic  provides  a  number  of   set  covers  slightly  larger  compared  to
\cite{Slijepcevic01powerefficient}, but with a  larger execution time due to the
complexity of the mixed integer programming resolution.

Zorbas et al.  \cite{zorbas2010solving} presented a centralized greedy algorithm
for the efficient production of  both node disjoint and non-disjoint cover sets.
Compared    to    algorithm's    results    of   Slijepcevic    and    Potkonjak
\cite{Slijepcevic01powerefficient}, their heuristic produces more disjoint cover
sets with a  slight growth rate in execution  time.  When producing non-disjoint
cover sets,  both Static-CCF  and Dynamic-CCF algorithms,  where CCF  means that
they  use a cost  function called  Critical Control  Factor, provide  cover sets
offering longer network lifetime than those produced by \cite{cardei2005energy}.
Also, they require  a smaller number of participating nodes  in order to achieve
these results.

In  the  case  of  non-disjoint algorithms  \cite{pujari2011high},  sensors  may
participate in  more than one  cover set.  In  some cases, this may  prolong the
lifetime of the network in comparison  to the disjoint cover set algorithms, but
designing  algorithms for  non-disjoint cover  sets generally  induces  a higher
order  of complexity.   Moreover, in  case of  a sensor's  failure, non-disjoint
scheduling policies are less resilient and less reliable because a sensor may be
involved   in   more  than   one   cover   sets.    For  instance,   Cardei   et
al.~\cite{cardei2005energy}  present a  linear programming  (LP) solution  and a
greedy approach to extend the  sensor network lifetime by organizing the sensors
into a maximal number of  non-disjoint cover sets.  Simulation results show that
by  allowing sensors  to  participate  in multiple  sets,  the network  lifetime
increases     compared     with     related     work~\cite{cardei2005improving}.
In~\cite{berman04},  the  authors  have  formulated  the  lifetime  problem  and
suggested another (LP) technique to  solve this problem.  A centralized solution
based  on  the  Garg-K\"{o}nemann  algorithm~\cite{garg98},  provably  near  the
optimal solution, is also proposed.

In~\cite{yang2014maximum},  the  authors  have  proposed  a  linear  programming
approach for selecting  the minimum number of working sensor  nodes, in order to
as to preserve  a maximum coverage and extend lifetime of  the network. Cheng et
al.~\cite{cheng2014energy} have defined a  heuristic algorithm called Cover Sets
Balance (CSB), which choose a set of active nodes using the tuple (data coverage
range, residual energy).   Then, they have introduced a  new Correlated Node Set
Computing (CNSC)  algorithm to find  the correlated node  set for a  given node.
After that,  they proposed  a High Residual  Energy First (HREF)  node selection
algorithm to  minimize the number of active  nodes so as to  prolong the network
lifetime. Various centralized methods based on column generation approaches have
also been proposed~\cite{castano2013column,rossi2012exact,deschinkel2012column}.

\subsection{Distributed approaches}
%{\bf Distributed approaches}
In distributed  and localized coverage  algorithms, the required  computation to
schedule the  activity of  sensor nodes  will be done  by the  cooperation among
neighboring nodes. These  algorithms may require more computation  power for the
processing by the cooperating sensor nodes, but they are more scalable for large
WSNs.  Localized and distributed algorithms generally result in non-disjoint set
covers.

Many distributed algorithms have been  developed to perform the scheduling so as
to          preserve         coverage,          see          for         example
\cite{Gallais06,Tian02,Ye03,Zhang05,HeinzelmanCB02,yardibi2010distributed}.
Distributed  algorithms   typically  operate  in  rounds   for  a  predetermined
duration. At  the beginning of each  round, a sensor  exchanges information with
its neighbors and makes a decision to  either remain turned on or to go to sleep
for the  round. This decision is  basically made on simple  greedy criteria like
the largest  uncovered area \cite{Berman05efficientenergy}  or maximum uncovered
targets  \cite{lu2003coverage}.   In  \cite{Tian02},  the scheduling  scheme  is
divided into rounds, where each round  has a self-scheduling phase followed by a
sensing phase.  Each sensor broadcasts  a message containing the node~ID and the
node  location to  its  neighbors at  the  beginning of  each  round.  A  sensor
determines its status by a rule named off-duty eligible rule, which tells him to
turn off if its  sensing area is covered by its neighbors.  A back-off scheme is
introduced to let each sensor delay the decision process with a random period of
time, in  order to  avoid simultaneous conflicting  decisions between  nodes and
lack  of coverage  on any  area.   In \cite{prasad2007distributed}  a model  for
capturing  the dependencies  between  different  cover sets  is  defined and  it
proposes localized heuristic based on this dependency. The algorithm consists of
two  phases,  an initial  setup  phase during  which  each  sensor computes  and
prioritizes  the covers  and  a sensing  phase  during which  each sensor  first
decides  its on/off  status, and  then remains  on or  off for  the rest  of the
duration. 

The  authors  in  \cite{yardibi2010distributed}  have  developed  a  Distributed
Adaptive  Sleep Scheduling  Algorithm (DASSA)  for WSNs  with  partial coverage.
DASSA  does  not  require  location  information of  sensors  while  maintaining
connectivity and satisfying a user defined coverage target.  In DASSA, nodes use
the  residual  energy levels  and  feedback from  the  sink  for scheduling  the
activity of their neighbors.  This  feedback mechanism reduces the randomness in
scheduling  that  would   otherwise  occur  due  to  the   absence  of  location
information.  In  \cite{ChinhVu}, the author  have proposed a  novel distributed
heuristic, called Distributed Energy-efficient Scheduling for k-coverage (DESK),
which ensures that the energy consumption  among the sensors is balanced and the
lifetime maximized while the coverage requirement is maintained.  This heuristic
works in  rounds, requires  only one-hop neighbor  information, and  each sensor
decides  its status  (active or  sleep) based  on the  perimeter  coverage model
proposed in \cite{Huang:2003:CPW:941350.941367}.

%Our Work, which is presented in~\cite{idrees2014coverage} proposed a coverage optimization protocol to improve the lifetime in
%heterogeneous energy wireless sensor networks. 
%In this work, the coverage protocol distributed in each sensor node in the subregion but the optimization take place over the the whole subregion. We consider only distributing the coverage protocol over two subregions. 

The  works presented in  \cite{Bang, Zhixin,  Zhang} focus  on coverage-aware,
distributed energy-efficient,  and distributed clustering  methods respectively,
which aim  to extend the network  lifetime, while the coverage  is ensured.  S.
Misra et al.   \cite{Misra} have proposed a localized  algorithm for coverage in
sensor networks.  The  algorithm conserve the energy while  ensuring the network
coverage by activating the subset of  sensors with the minimum overlap area. The
proposed method preserves  the network connectivity by formation  of the network
backbone.  More recently, Shibo et  al. \cite{Shibo} have expressed the coverage
problem  as  a  minimum weight  submodular  set  cover  problem and  proposed  a
Distributed Truncated Greedy Algorithm (DTGA)  to solve it.  They take advantage
from both  temporal and  spatial correlations between  data sensed  by different
sensors,   and    leverage   prediction,   to   improve    the   lifetime.    In
\cite{xu2001geography},   Xu  et   al.  have   proposed  an   algorithm,  called
Geographical Adaptive Fidelity (GAF), which uses geographic location information
to divide  the area of  interest into fixed  square grids. Within each  grid, it
keeps only  one node  staying awake  to take the  responsibility of  sensing and
communication.

Some  other  approaches (outside  the  scope  of our  work)  do  not consider  a
synchronized and  predetermined period of time  where the sensors  are active or
not.   Indeed, each  sensor maintains  its  own timer  and its  wake-up time  is
randomized \cite{Ye03} or regulated \cite{cardei2005maximum} over time.

The MuDiLCO protocol (for Multiround Distributed Lifetime Coverage Optimization
protocol) presented  in this  paper is an  extension of the  approach introduced
in~\cite{idrees2014coverage}.   In~\cite{idrees2014coverage},  the  protocol  is
deployed over  only two  subregions. Simulation results  have shown that  it was
more  interesting  to  divide  the  area  into  several  subregions,  given  the
computation complexity. Compared to our previous paper, in this one we study the
possibility of dividing  the sensing phase into multiple rounds  and we also add
an  improved  model  of energy  consumption  to  assess  the efficiency  of  our
approach.




\fi
%The main contributions of our MuDiLCO Protocol can be summarized as follows:
%(1) The high coverage ratio, (2) The reduced number of active nodes, (3) The distributed optimization over the subregions in the area of interest, (4) The distributed dynamic leader election at each round based on some priority factors that led to energy consumption balancing among the nodes in the same subregion, (5) The primary point coverage model to represent each sensor node in the network, (6) The activity scheduling based optimization on the subregion, which are based on the primary point coverage model to activate as less number as possible of sensor nodes for a multirounds to take the mission of the coverage in each subregion, (7) The very low energy consumption, (8) The higher network lifetime.
%\section{Preliminaries}
%\label{Pr}

%Network Lifetime

%\subsection{Network Lifetime}
%Various   definitions   exist   for   the   lifetime   of   a   sensor
%network~\cite{die09}.  The main definitions proposed in the literature are
%related to the  remaining energy of the nodes or  to the coverage percentage. 
%The lifetime of the  network is mainly defined as the amount
%of  time during which  the network  can  satisfy its  coverage objective  (the
%amount of  time that the network  can cover a given  percentage of its
%area or targets of interest). In this work, we assume that the network
%is alive  until all  nodes have  been drained of  their energy  or the
%sensor network becomes disconnected, and we measure the coverage ratio
%during the WSN lifetime.  Network connectivity is important because an
%active sensor node without  connectivity towards a base station cannot
%transmit information on an event in the area that it monitors.

\section{MuDiLCO protocol description}
\label{pd}

%Our work will concentrate on the area coverage by design
%and implementation of a  strategy, which efficiently selects the active
%nodes   that  must   maintain  both   sensing  coverage   and  network
%connectivity and at the same time improve the lifetime of the wireless
%sensor  network. But,  requiring  that  all physical  points  of  the
%considered region are covered may  be too strict, especially where the
%sensor network is not dense.   Our approach represents an area covered
%by a sensor as a set of primary points and tries to maximize the total
%number  of  primary points  that  are  covered  in each  round,  while
%minimizing  overcoverage (points  covered by  multiple  active sensors
%simultaneously).

%In this section, we introduce a Multiround Distributed Lifetime Coverage Optimization protocol, which is called MuDiLCO. It 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 continuously and efficiently to maximize the lifetime in the network.  
%The main features of our MuDiLCO protocol:
%i)It divides the area of interest into subregions by using divide-and-conquer concept, ii)It requires only the information of the nodes within the subregion, iii) it divides the network lifetime into periods, which consists in round(s), iv)It based on the autonomous distributed decision by the nodes in the subregion to elect the Leader, v)It apply the activity scheduling based optimization on the subregion, vi)  it achieves an energy consumption balancing among the nodes in the subregion by selecting different nodes as a leader during the network lifetime, vii) It uses the optimization to select the best representative non-disjoint sets of sensors in the subregion by optimize the coverage and the lifetime over the area of interest, viii)It uses our proposed primary point coverage model, which represent the sensing range of the sensor as a set of points, which are used by the our optimization algorithm, ix) It uses a simple energy model that takes communication, sensing and computation energy consumptions into account to evaluate the performance of our Protocol.

\subsection{Assumptions}

We  consider a  randomly and  uniformly  deployed network  consisting of  static
wireless sensors.  The sensors are  deployed in high density to ensure initially
a high  coverage ratio  of the interested  area.  We  assume that all  nodes are
homogeneous  in   terms  of  communication  and   processing  capabilities,  and
heterogeneous  from the  point  of view  of  energy provision.   Each sensor  is
supposed  to get information  on its  location either  through hardware  such as
embedded GPS or through location discovery algorithms.
   
To model  a sensor node's coverage  area, we consider the  boolean disk coverage
model   which  is  the   most  widely   used  sensor   coverage  model   in  the
literature. Thus, each  sensor has a constant sensing range  $R_s$ and all space
points within  the disk centered  at the sensor  with the radius of  the sensing
range  is  said  to  be  covered  by  this sensor.   We  also  assume  that  the
communication   range  satisfies   $R_c  \geq   2R_s$.   In   fact,   Zhang  and
Zhou~\cite{Zhang05} proved that if  the transmission range fulfills the previous
hypothesis, a complete coverage of  a convex area implies connectivity among the
active nodes.

%Instead  of working  with a  continuous coverage  area, we  make it  discrete by considering for each sensor a set of points called primary points. Consequently, we assume  that the sensing disk  defined by a sensor  is covered if  all of its primary points are covered. The choice of number and locations of primary points is the subject of another study not presented here.

\indent Instead of working with the coverage area, we consider for each sensor a
set of  points called  primary points~\cite{idrees2014coverage}. We  assume that
the sensing  disk defined by a  sensor is covered  if all the primary  points of
this  sensor are  covered.   By knowing  the position  of  wireless sensor  node
(centered at  the the  position $\left(p_x,p_y\right)$)  and it's  sensing range
$R_s$,  we define  up to  25 primary  points $X_1$  to $X_{25}$  as decribed  on
Figure~\ref{fig1}. The optimal number of primary points is investigated in
section~\ref{ch4:sec:04:06}.

The coordinates of the primary points are defined as follows:\\
%$(p_x,p_y)$ = point center of wireless sensor node\\  
$X_1=(p_x,p_y)$ \\ 
$X_2=( p_x + R_s * (1), p_y + R_s * (0) )$\\           
$X_3=( p_x + R_s * (-1), p_y + R_s * (0)) $\\
$X_4=( p_x + R_s * (0), p_y + R_s * (1) )$\\
$X_5=( p_x + R_s * (0), p_y + R_s * (-1 )) $\\
$X_6=( p_x + R_s * (\frac{-\sqrt{2}}{2}), p_y + R_s * (\frac{\sqrt{2}}{2})) $\\
$X_7=( p_x + R_s * (\frac{\sqrt{2}}{2}), p_y + R_s * (\frac{\sqrt{2}}{2})) $\\
$X_8=( p_x + R_s * (\frac{-\sqrt{2}}{2}), p_y + R_s * (\frac{-\sqrt{2}}{2})) $\\
$X_9=( p_x + R_s * (\frac{\sqrt{2}}{2}), p_y + R_s * (\frac{-\sqrt{2}}{2})) $\\
$X_{10}= ( p_x + R_s * (\frac{-\sqrt{2}}{2}), p_y + R_s * (0)) $\\
$X_{11}=( p_x + R_s *  (\frac{\sqrt{2}}{2}), p_y + R_s * (0))$\\
$X_{12}=( p_x + R_s * (0), p_y + R_s * (\frac{\sqrt{2}}{2})) $\\
$X_{13}=( p_x + R_s * (0), p_y + R_s * (\frac{-\sqrt{2}}{2})) $\\
$X_{14}=( p_x + R_s * (\frac{\sqrt{3}}{2}), p_y + R_s * (\frac{1}{2})) $\\
$X_{15}=( p_x + R_s * (\frac{-\sqrt{3}}{2}), p_y + R_s * (\frac{1}{2})) $\\
$X_{16}=( p_x + R_s * (\frac{\sqrt{3}}{2}), p_y + R_s * (\frac{- 1}{2})) $\\
$X_{17}=( p_x + R_s * (\frac{-\sqrt{3}}{2}), p_y + R_s * (\frac{- 1}{2})) $\\
$X_{18}=( p_x + R_s * (\frac{\sqrt{3}}{2}), p_y + R_s * (0)) $\\
$X_{19}=( p_x + R_s * (\frac{-\sqrt{3}}{2}), p_y + R_s * (0)) $\\
$X_{20}=( p_x + R_s * (0), p_y + R_s * (\frac{1}{2})) $\\
$X_{21}=( p_x + R_s * (0), p_y + R_s * (-\frac{1}{2})) $\\
$X_{22}=( p_x + R_s * (\frac{1}{2}), p_y + R_s * (\frac{\sqrt{3}}{2})) $\\
$X_{23}=( p_x + R_s * (\frac{- 1}{2}), p_y + R_s * (\frac{\sqrt{3}}{2})) $\\
$X_{24}=( p_x + R_s * (\frac{- 1}{2}), p_y + R_s * (\frac{-\sqrt{3}}{2})) $\\
$X_{25}=( p_x + R_s * (\frac{1}{2}), p_y + R_s * (\frac{-\sqrt{3}}{2})) $.


%\begin{figure} %[h!]
%\centering
% \begin{multicols}{2}
%\centering
%\includegraphics[scale=0.28]{fig21.pdf}\\~ (a)
%\includegraphics[scale=0.28]{principles13.pdf}\\~(c) 
%\hfill \hfill
%\includegraphics[scale=0.28]{fig25.pdf}\\~(e)
%\includegraphics[scale=0.28]{fig22.pdf}\\~(b)
%\hfill \hfill
%\includegraphics[scale=0.28]{fig24.pdf}\\~(d)
%\includegraphics[scale=0.28]{fig26.pdf}\\~(f)
%\end{multicols} 
%\caption{Wireless Sensor Node represented by (a) 5, (b) 9, (c) 13, (d) 17, (e) 21 and (f) 25 primary points respectively}
%\label{fig1}
%\end{figure}
    
\begin{figure}[h]
  \centering
  \includegraphics[scale=0.375]{fig26.pdf}
  \label{fig1}
  \caption{Wireless sensor node represented by up to 25~primary points}
\end{figure}

%By  knowing the  position (point  center: ($p_x,p_y$))  of  a wireless
%sensor node  and its $R_s$,  we calculate the primary  points directly
%based on the proposed model. We  use these primary points (that can be
%increased or decreased if necessary)  as references to ensure that the
%monitored  region  of interest  is  covered  by  the selected  set  of
%sensors, instead of using all the points in the area.

%The MuDiLCO protocol works in periods and executed at each sensor node in the network, each sensor node can still sense data while being in
%LISTENING mode. Thus, by entering the LISTENING mode at the beginning of each round,
%sensor nodes still executing sensing task while participating in the leader election and decision phases. More specifically, The MuDiLCO protocol algorithm works as follow: 
%Initially, the sensor node check it's remaining energy in order to participate in the current round. Each sensor node determines it's position and it's subregion based Embedded GPS  or Location Discovery Algorithm. After that, All the sensors collect position coordinates, current remaining energy, sensor node id, and the number of its one-hop live neighbors during the information exchange. It stores this information into a list $L$.
%The sensor node enter in listening mode waiting to receive ActiveSleep packet from the leader after the decision to apply multi-round activity scheduling during the sensing phase. Each sensor node will execute the Algorithm~1 to know who is the leader. After that, if the sensor node is leader, It will execute the integer program algorithm ( see section~\ref{cp}) to optimize the coverage and the lifetime in it's subregion. After the decision, the optimization approach will produce the cover sets of sensor nodes to take the mission of coverage during the sensing phase for $T$ rounds. The leader will send ActiveSleep packet to each sensor node in the subregion to inform him to it's schedule for $T$ rounds during the period of sensing, either Active or sleep until the starting of next period. Based on the decision, the leader as other nodes in subregion, either go to be active or go to be sleep based on it's schedule for $T$ rounds during current sensing phase. the other nodes in the same subregion will stay in listening mode waiting the ActiveSleep packet from the leader. After finishing the time period for sensing, which are includes $T$ rounds, all the sensor nodes in the same subregion will start new period by executing the MuDiLCO protocol and the lifetime in the subregion will continue until all the sensor nodes are died or the network becomes disconnected in the subregion.

\subsection{Background idea}
%%RC : we need to clarify the difference between round and period. Currently it seems to be the same (for me at least).
%The area of  interest can be divided using  the divide-and-conquer strategy into
%smaller  areas,  called  subregions,  and  then our MuDiLCO  protocol will be
%implemented in each subregion in a distributed way.

\textcolor{blue}{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. Sensor nodes are assumed to be deployed almost uniformly and with high
  density over the region. The regular  subdivision is made such that the number
  of hops between any pairs of sensors  inside a subregion is less than or equal
  to 3.}

As can  be seen  in Figure~\ref{fig2},  our protocol  works in  periods fashion,
where   each   period   is    divided   into   4~phases:   Information~Exchange,
Leader~Election,  Decision,  and Sensing.   Each  sensing  phase may  be  itself
divided into $T$ rounds \textcolor{blue} {of  equal duration} and for each round
a set of sensors (a cover set) is  responsible for the sensing task. In this way
a  multiround  optimization  process  is  performed  during  each  period  after
Information~Exchange and Leader~Election  phases, in order to  produce $T$ cover
sets that will take the mission of sensing for $T$ rounds.
\begin{figure}[t!]
\centering \includegraphics[width=125mm]{Modelgeneral.pdf} % 70mm
\caption{The MuDiLCO protocol scheme executed on each node}
\label{fig2}
\end{figure} 

%Each period is divided into 4 phases: Information  Exchange,
%Leader  Election, Decision,  and  Sensing.  Each sensing phase may be itself divided into $T$ rounds.
% set cover responsible for the sensing task.  
%For each round a set of sensors (said a cover set) is responsible for the sensing task.

This  protocol minimizes  the  impact of  unexpected node  failure  (not due  to
batteries running out of energy), because it works in periods.
%This protocol is reliable against an unexpected node failure, because it works in periods. 
%%RC : why? I am not convinced
 On the one hand, if a node  failure is detected before making the decision, the
 node will not  participate to this phase,  and, on the other hand,  if the node
 failure occurs  after the  decision, the  sensing task of  the network  will be
 temporarily affected:  only during  the period  of sensing  until a  new period
 starts.   \textcolor{blue}{The   duration   of  the   rounds   are   predefined
   parameters. Round duration  should be long enough to hide  the system control
   overhead and  short enough to minimize  the negative effects in  case of node
   failure.}

%%RC so if there are at least one failure per period, the coverage is bad...
%%MS if we want to be reliable against many node failures we need to have an
%% overcoverage...  

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 (Information  Exchange, Leader  Election, and  Decision) are
energy  consuming for some  nodes, even  when they  do not  join the  network to
monitor the area.

%%%%%%%%%%%%%%%%%parler optimisation%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We define two types of packets that will be used by the proposed protocol:
\begin{enumerate}[(a)] 
\item INFO  packet: such a  packet  will be sent by  each sensor node  to all the
  nodes inside a subregion for information exchange.
\item  Active-Sleep  packet: sent  by  the  leader to  all  the  nodes inside  a
  subregion to  inform them to remain Active  or to go Sleep  during the sensing
  phase.
\end{enumerate}

There are five status for each sensor node in the network:
\begin{enumerate}[(a)] 
\item LISTENING: sensor node is waiting for a decision (to be active or not);
\item  COMPUTATION: sensor  node  has been  elected  as leader  and applies  the
  optimization process;
\item ACTIVE: sensor node is taking part in the monitoring of the area;
\item SLEEP: sensor node is turned off to save energy;
\item COMMUNICATION: sensor node is transmitting or receiving packet.
\end{enumerate}

Below, we describe each phase in more details.

\subsection{Information Exchange Phase}

Each sensor node $j$ sends its position, remaining energy $RE_j$, and the number
of neighbors $NBR_j$  to all wireless sensor nodes in its  subregion by using an
INFO packet  (containing information on position  coordinates, current remaining
energy, sensor node ID, number of its one-hop live neighbors) and then waits for
packets sent by other nodes.  After  that, each node will have information about
all  the sensor  nodes in  the subregion.   In our  model, the  remaining energy
corresponds to the time that a sensor can live in the active mode.

%\subsection{\textbf Working Phase:}

%The working phase works in rounding fashion. Each round include 3 steps described as follow :

\subsection{Leader Election phase}

This step  consists in choosing  the Wireless  Sensor Node Leader  (WSNL), which
will be responsible for executing the coverage algorithm.  Each subregion in the
area of  interest will select its  own WSNL independently for  each period.  All
the sensor  nodes cooperate to  elect a WSNL.  The  nodes in the  same subregion
will select the leader based on the received information from all other nodes in
the same subregion.  The selection criteria  are, in order of importance: larger
number of  neighbors, larger  remaining energy,  and then  in case  of equality,
larger index. Observations on previous simulations  suggest to use the number of
one-hop neighbors as  the primary criterion to reduce energy  consumption due to
the communications.

%the more priority selection factor is the number of $1-hop$ neighbors, $NBR j$, which can  minimize the energy consumption during the communication Significantly.  
%The pseudo-code for leader election phase is provided in Algorithm~1.

%Where $E_{th}$ is the minimum energy needed to stay active during the sensing phase. As shown in Algorithm~1, the more priority selection factor is the number of $1-hop$ neighbours, $NBR j$, which can  minimize the energy consumption during the communication Significantly.  

\subsection{Decision phase}

Each WSNL will  \textcolor{blue}{solve an integer program to  select which cover
  sets will be  activated in the following sensing phase  to cover the subregion
  to which it belongs.  $T$ cover sets will be produced, one for each round. The
  WSNL will send an Active-Sleep packet to each sensor in the subregion based on
  the algorithm's results,  indicating if the sensor should be  active or not in
  each round of the sensing phase.}
%Each  WSNL will \textcolor{red}{ execute an optimization algorithm (see section \ref{oa})} to  select which  cover sets  will be
%activated in  the following  sensing phase  to cover the  subregion to  which it
%belongs.  The \textcolor{red}{optimization algorithm} will produce $T$ cover sets,  one for each round. The WSNL will send an Active-Sleep  packet to each sensor in the subregion based on the algorithm's results, indicating if  the sensor should be active or not in
%each round  of the  sensing phase.  


%solve  an integer  program







%\section{\textcolor{red}{ Optimization Algorithm for Multiround Lifetime Coverage Optimization}}
%\label{oa}
As shown in Algorithm~\ref{alg:MuDiLCO}, the leader will execute an optimization
algorithm based on an integer program. The integer program is based on the model
proposed by \cite{pedraza2006}  with some modifications, where  the objective is
to find  a maximum  number of disjoint  cover sets.  To  fulfill 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 binary variables  $X_{t,j}$ to determine the  possibility of activating
sensor $j$ during round $t$ of a  given sensing phase.  We also consider primary
points as targets.  The  set of primary points is denoted by $P$  and the set of
sensors by  $J$. Only sensors  able to  be alive during  at least one  round are
involved in the integer program.

%parler de la limite en energie Et pour un round

For a  primary point  $p$, let $\alpha_{j,p}$  denote the indicator  function of
whether the point $p$ is covered, that is:
\begin{equation}
\alpha_{j,p} = \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 $p$ during
round $t$ is equal to $\sum_{j \in J} \alpha_{j,p} * X_{t,j}$ where:
\begin{equation}
X_{t,j} = \left \{ 
\begin{array}{l l}
  1& \mbox{if sensor $j$  is active during round $t$,} \\
  0 &  \mbox{otherwise.}\\
\end{array} \right.
%\label{eq11} 
\end{equation}
We define the Overcoverage variable $\Theta_{t,p}$ as:
\begin{equation}
 \Theta_{t,p} = \left \{ 
\begin{array}{l l}
  0 & \mbox{if the primary point $p$}\\
    & \mbox{is not covered during round $t$,}\\
  \left( \sum_{j \in J} \alpha_{jp} * X_{tj} \right)- 1 & \mbox{otherwise.}\\
\end{array} \right.
\label{eq13} 
\end{equation}
More  precisely, $\Theta_{t,p}$  represents the  number of  active  sensor nodes
minus  one  that  cover  the  primary  point $p$  during  round  $t$.   The
Undercoverage variable  $U_{t,p}$ of the primary  point $p$ during  round $t$ is
defined by:
\begin{equation}
U_{t,p} = \left \{ 
\begin{array}{l l}
  1 &\mbox{if the primary point $p$ is not covered during round $t$,} \\
  0 & \mbox{otherwise.}\\
\end{array} \right.
\label{eq14} 
\end{equation}

Our coverage optimization problem can then be formulated as follows:
\begin{equation}
 \min \sum_{t=1}^{T} \sum_{p=1}^{|P|} \left(W_{\theta}* \Theta_{t,p} + W_{U} * U_{t,p}  \right)  \label{eq15} 
\end{equation}

Subject to
\begin{equation}
  \sum_{j=1}^{|J|} \alpha_{j,p} * X_{t,j}   = \Theta_{t,p} - U_{t,p} + 1 \label{eq16} \hspace{6 mm} \forall p \in P, t = 1,\dots,T
\end{equation}

\begin{equation}
  \sum_{t=1}^{T}  X_{t,j}   \leq  \floor*{RE_{j}/E_{R}} \hspace{10 mm}\forall j \in J\hspace{6 mm} 
  \label{eq144} 
\end{equation}

\begin{equation}
X_{t,j} \in \lbrace0,1\rbrace,   \hspace{10 mm} \forall j \in J, t = 1,\dots,T \label{eq17} 
\end{equation}

\begin{equation}
U_{t,p} \in \lbrace0,1\rbrace, \hspace{10 mm}\forall p \in P, t = 1,\dots,T  \label{eq18} 
\end{equation}

\begin{equation}
 \Theta_{t,p} \geq 0 \hspace{10 mm}\forall p \in P, t = 1,\dots,T \label{eq178}
\end{equation}

%\begin{equation}
%(W_{\theta}+W_{\psi} = P)    \label{eq19} 
%\end{equation}

%%RC why W_{\theta} is not defined (only one sentence)? How to define in practice Wtheta and Wu?

\begin{itemize}
\item $X_{t,j}$:  indicates whether  or not the  sensor $j$ is  actively sensing
  during round $t$ (1 if yes and 0 if not);
\item $\Theta_{t,p}$ - {\it overcoverage}:  the number of sensors minus one that
  are covering the primary point $p$ during round $t$;
\item  $U_{t,p}$ -  {\it undercoverage}:  indicates whether  or not  the primary
  point $p$  is being covered during round $t$ (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. The constraint  given by equation~(\ref{eq144}) guarantees that
the sensor has enough energy ($RE_j$  corresponds to its remaining energy) to be
alive during  the selected rounds knowing  that $E_{R}$ is the  amount of energy
required to be alive during one round.

There are  two main  objectives.  First,  we limit  the overcoverage  of primary
points in order to activate a minimum  number of sensors.  Second we prevent the
absence  of  monitoring  on  some  parts of  the  subregion  by  minimizing  the
undercoverage.  The weights  $W_\theta$ and $W_U$ must be properly  chosen so as
to guarantee that the maximum number of points are covered during each round.
%% MS W_theta is smaller than W_u => problem with the following sentence
In our simulations,  priority is given to the coverage  by choosing $W_{U}$ very
large compared to $W_{\theta}$.

\textcolor{blue}{The size of the problem depends  on the number of variables and
  constraints. The number of variables is  linked to the number of alive sensors
  $A \subseteq J$,  the number of rounds  $T$, and the number  of primary points
  $P$.  Thus  the integer  program contains $A*T$  variables of  type $X_{t,j}$,
  $P*T$ overcoverage variables and $P*T$  undercoverage variables. The number of
  constraints  is equal  to $P*T$  (for constraints  (\ref{eq16})) $+$  $A$ (for
  constraints (\ref{eq144})).}
%The Active-Sleep packet includes the schedule vector with the number of rounds that should be applied by the receiving sensor node during the sensing phase

\subsection{Sensing phase}

The sensing phase consists of $T$ rounds. Each sensor node in the subregion will
receive an Active-Sleep packet from WSNL, informing it to stay awake or to go to
sleep for each  round of the sensing  phase.  Algorithm~\ref{alg:MuDiLCO}, which
will  be executed  by  each sensor  node~$s_j$  at the  beginning  of a  period,
explains how the Active-Sleep packet is obtained.

% In each round during the sensing phase, there is a cover set of sensor nodes,  in which  the active  sensors will  execute  their sensing  task  to preserve maximal  coverage and lifetime in the subregion and this will continue until finishing the round $T$ and starting new period. 

\begin{algorithm}[h!]                
 % \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} \; 
  
  \If{ $RE_j \geq E_{R}$ }{
      \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{$\left\{\left(X_{1,k},\dots,X_{T,k}\right)\right\}_{k \in J}$ =
          Execute Integer Program Algorithm($T,J$)}\;
        \emph{$s_j.status$ = COMMUNICATION}\;
        \emph{Send $ActiveSleep()$ packet to each node $k$ in subregion: a packet \\
          with vector of activity scheduling $(X_{1,k},\dots,X_{T,k})$}\;
        \emph{Update $RE_j $}\;
      }   
      \Else{
        \emph{$s_j.status$ = LISTENING}\;
        \emph{Wait $ActiveSleep()$ packet from the Leader}\;
        % \emph{After receiving Packet, Retrieve the schedule and the $T$ rounds}\;
        \emph{Update $RE_j $}\;
      }  
      %  }
  }
  \Else { Exclude $s_j$ from entering in the current sensing phase}
  
 %   \emph{return X} \;
\caption{MuDiLCO($s_j$)}
\label{alg:MuDiLCO}

\end{algorithm}

\iffalse
\textcolor{red}{This integer program can be solved using two approaches:}

\subsection{\textcolor{red}{Optimization solver for Multiround Lifetime Coverage Optimization}}
\label{glpk}
\textcolor{red}{The modeling language for Mathematical Programming (AMPL)~\cite{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) \cite{glpk} through a Branch-and-Bound method. We named the protocol which is based on GLPK solver in the decision phase as MuDiLCO.}
\fi

\iffalse

\subsection{\textcolor{red}{Genetic Algorithm for Multiround Lifetime Coverage Optimization}}
\label{GA}
\textcolor{red}{Metaheuristics  are a generic search strategies for exploring search spaces for solving the complex problems. These strategies have to dynamically balance between the exploitation of the accumulated search experience and the exploration of the search space. On one hand, this balance can find regions in the search space with high-quality solutions. On the other hand, it prevents waste too much time in regions of the search space which are either already explored or don’t provide high-quality solutions. Therefore,  metaheuristic provides an enough good solution to an optimization problem, especially with incomplete  information or limited computation capacity \cite{bianchi2009survey}. Genetic Algorithm (GA) is one of the population-based metaheuristic methods that simulates the process of natural selection \cite{hassanien2015applications}.  GA starts with a population of random candidate solutions (called individuals or phenotypes) . GA uses genetic operators inspired by natural evolution, such as selection, mutation, evaluation, crossover, and replacement so as to improve the initial population of candidate solutions. This process repeated until a stopping criterion is satisfied. In comparison with GLPK optimization solver, GA provides a near optimal solution with acceptable execution time, as well as it requires a less amount of memory especially for large size problems. GLPK provides optimal solution, but it requires higher execution time and amount of memory for large problem.}

\textcolor{red}{In this section, we present a metaheuristic based GA to solve our multiround lifetime coverage optimization problem. The proposed GA provides a near optimal sechedule for multiround sensing per period. The proposed GA is based on the mathematical model which is presented in Section \ref{oa}. Algorithm \ref{alg:GA} shows the proposed GA to solve the coverage lifetime optimization problem. We named the new protocol which is based on GA in the decision phase as GA-MuDiLCO. The proposed GA can be explained in more details as follow:}

\begin{algorithm}[h!]    
       
 \small
 \SetKwInput{Input}{\textcolor{red}{Input}}
 \SetKwInput{Output}{\textcolor{red}{Output}}
 \Input{ \textcolor{red}{$ P, J, T, S_{pop}, \alpha_{j,p}^{ind}, X_{t,j}^{ind}, \Theta_{t,p}^{ind}, U_{t,p}^{ind}, Child_{t,j}^{ind}, Ch.\Theta_{t,p}^{ind}, Ch.U_{t,p}^{ind_1}$}}
 \Output{\textcolor{red}{$\left\{\left(X_{1,1},\dots, X_{t,j}, \dots, X_{T,J}\right)\right\}_{t \in T, j \in J}$}}

  \BlankLine
  %\emph{Initialize the sensor node and determine it's position and subregion} \; 
  \ForEach {\textcolor{red}{Individual $ind$ $\in$ $S_{pop}$}} {
     \emph{\textcolor{red}{Generate Randomly Chromosome $\left\{\left(X_{1,1},\dots, X_{t,j}, \dots, X_{T,J}\right)\right\}_{t \in T, j \in J}$}}\;
     
     \emph{\textcolor{red}{Update O-U-Coverage $\left\{(P, J, \alpha_{j,p}^{ind}, X_{t,j}^{ind}, \Theta_{t,p}^{ind}, U_{t,p}^{ind})\right\}_{p \in P}$}}\;
     
  
     \emph{\textcolor{red}{Evaluate Individual $(P, J, X_{t,j}^{ind}, \Theta_{t,p}^{ind}, U_{t,p}^{ind})$}}\;  
  }
  
  \While{\textcolor{red}{ Stopping criteria is not satisfied} }{
  
  \emph{\textcolor{red}{Selection $(ind_1, ind_2)$}}\;
    \emph{\textcolor{red}{Crossover $(P_c, X_{t,j}^{ind_1}, X_{t,j}^{ind_2}, Child_{t,j}^{ind_1}, Child_{t,j}^{ind_2})$}}\;
    \emph{\textcolor{red}{Mutation $(P_m, Child_{t,j}^{ind_1}, Child_{t,j}^{ind_2})$}}\;
   
   
   \emph{\textcolor{red}{Update O-U-Coverage $(P, J, \alpha_{j,p}^{ind}, Child_{t,j}^{ind_1}, Ch.\Theta_{t,p}^{ind_1}, Ch.U_{t,p}^{ind_1})$}}\;
  \emph{\textcolor{red}{Update O-U-Coverage $(P, J, \alpha_{j,p}^{ind}, Child_{t,j}^{ind_2}, Ch.\Theta_{t,p}^{ind_2}, Ch.U_{t,p}^{ind_2})$}}\;  
 
\emph{\textcolor{red}{Evaluate New Individual$(P, J, Child_{t,j}^{ind_1}, Ch.\Theta_{t,p}^{ind_1}, Ch.U_{t,p}^{ind_1})$}}\;  
 \emph{\textcolor{red}{Replacement $(P, J, T, Child_{t,j}^{ind_1}, Ch.\Theta_{t,p}^{ind_1}, Ch.U_{t,p}^{ind_1}, X_{t,j}^{ind}, \Theta_{t,p}^{ind}, U_{t,p}^{ind}  )$ }}\;
 
 \emph{\textcolor{red}{Evaluate New Individual$(P, J, Child_{t,j}^{ind_2}, Ch.\Theta_{t,p}^{ind_2}, Ch.U_{t,p}^{ind_2})$}}\;  
  
 \emph{\textcolor{red}{Replacement $(P, J, T, Child_{t,j}^{ind_2}, Ch.\Theta_{t,p}^{ind_2}, Ch.U_{t,p}^{ind_2}, X_{t,j}^{ind}, \Theta_{t,p}^{ind}, U_{t,p}^{ind}  )$ }}\;
  
      
  }
  \emph{\textcolor{red}{$\left\{\left(X_{1,1},\dots,X_{t,j},\dots,X_{T,J}\right)\right\}$ =
            Select Best Solution ($S_{pop}$)}}\;
 \emph{\textcolor{red}{return X}} \;
\caption{\textcolor{red}{GA($T, J$)}}
\label{alg:GA}

\end{algorithm}


\begin{enumerate} [I)]

\item \textcolor{red}{\textbf{Representation:} Since the proposed GA's goal is to find the optimal schedule of the sensor nodes which take the responsibility of monitoring the subregion for $T$ rounds in the sensing phase, the chromosome is defined as a schedule for alive  sensors and each chromosome contains $T$ rounds. The proposed GA uses binary representation, where each round in the schedule includes J genes, the total alive sensors in the subregion. Therefore, the gene of such a chromosome is a schedule of a sensor. In other words, The genes corresponding to active nodes have the value of one, the others are zero. Figure \ref{chromo} shows solution representation in the proposed GA.}
%[scale=0.3]
\begin{figure}[h!]
\centering
 \includegraphics [scale=0.35] {rep.pdf} 
\caption{Candidate Solution representation by the proposed GA. }
\label{chromo}
\end{figure} 



\item \textcolor{red}{\textbf{Initialize Population:} The initial population is randomly generated and each chromosome  in the GA population represents a possible sensors schedule solution to cover the entire subregion for $T$ rounds during current period. Each sensor in the chromosome is given a random value (0 or  1) for all rounds. If the random value is 1, the remaining  energy of this sensor should be adequate to activate this sensor during the current round. Otherwise, the value is set to 0. The energy constraint is applied for each sensor during all rounds. }


\item \textcolor{red}{\textbf{Update O-U-Coverage:} 
After creating the initial population, The overcoverage $\Theta_{t,p}$ and undercoverage $U_{t,p}$ for each candidate solution are computed (see Algorithm \ref{OU}) so as to use them in the next step.}

\begin{algorithm}[h!]                
  
 \SetKwInput{Input}{\textcolor{red}{Input}}
 \SetKwInput{Output}{\textcolor{red}{Output}}
 \Input{ \textcolor{red}{parameters $P, J, ind, \alpha_{j,p}^{ind}, X_{t,j}^{ind}$}}
 \Output{\textcolor{red}{$U^{ind} = \left\lbrace U_{1,1}^{ind}, \dots, U_{t,p}^{ind}, \dots, U_{T,P}^{ind} \right\rbrace$ and $\Theta^{ind} = \left\lbrace \Theta_{1,1}^{ind}, \dots, \Theta_{t,p}^{ind}, \dots, \Theta_{T,P}^{ind} \right\rbrace$}}

  \BlankLine

  \For{\textcolor{red}{$t\leftarrow 1$ \KwTo $T$}}{
  \For{\textcolor{red}{$p\leftarrow 1$ \KwTo $P$}}{
     
 %    \For{$i\leftarrow 0$ \KwTo $I_j$}{
        \emph{\textcolor{red}{$SUM\leftarrow 0$}}\;
         \For{\textcolor{red}{$j\leftarrow 1$ \KwTo $J$}}{
              \emph{\textcolor{red}{$SUM \leftarrow SUM + (\alpha_{j,p}^{ind} \times X_{t,j}^{ind})$ }}\;
         }
         
         \If { \textcolor{red}{SUM = 0}} {
         \emph{\textcolor{red}{$U_{t,p}^{ind} \leftarrow 0$}}\;
         \emph{\textcolor{red}{$\Theta_{t,p}^{ind} \leftarrow 1$}}\;
         }
         \Else{
         \emph{\textcolor{red}{$U_{t,p}^{ind} \leftarrow SUM -1$}}\;
         \emph{\textcolor{red}{$\Theta_{t,p}^{ind} \leftarrow 0$}}\;
         }
     
     }
     
  }
\emph{\textcolor{red}{return $U^{ind}, \Theta^{ind}$ }} \;
\caption{O-U-Coverage}
\label{OU}

\end{algorithm}



\item \textcolor{red}{\textbf{Evaluate Population:}
After creating the initial population, each individual is evaluated and assigned a fitness value according to the fitness function is illustrated in Eq. \eqref{eqf}. In the proposed GA, the optimal (or near optimal) candidate solution, is the one with the minimum value for the fitness function. The lower the fitness values been assigned to an individual, the better opportunity it gets survived.  In our works, the function rewards  the decrease in the sensor nodes which cover the same primary point and penalizes the decrease to zero in the sensor nodes which cover the primary point. }

\begin{equation}
 F^{ind} \leftarrow  \sum_{t=1}^{T} \sum_{p=1}^{P} \left(W_{\theta}* \Theta_{t,p} + W_{U} * U_{t,p}  \right)    \label{eqf} 
\end{equation}


\item \textcolor{red}{\textbf{Selection:} In order to generate a new generation, a portion of the existing population is elected based on a fitness function that ranks the fitness of each candidate solution and preferentially select the best solutions. Two parents should be selected to the mating pool.  In the proposed GA-MuDiLCO algorithm, the first parent is selected by using binary tournament selection to select one of the parents \cite{goldberg1991comparative}. In this method,  two individuals are chosen at random from the population and the better of the two
individuals is selected. If they have similar fitness values, one of them will be selected randomly. The best individual in the population is selected as a second parent.}



\item \textcolor{red}{\textbf{Crossover:} Crossover is a genetic operator used to take more than one parent solutions and produce a child solution from them. If crossover probability $P_c$ is 100$\%$, then the crossover operation takes place between two individuals. If it is 0$\%$, the  two selected individuals in the mating pool will be the new chromosomes without crossover. In the proposed GA, a two-point crossover is used. Figure \ref{cross} gives an example for a two-point crossover for 8 sensors in the subregion and the schedule for 3 rounds.}


\begin{figure}[h!]
\centering
 \includegraphics [scale = 0.3] {crossover.pdf} 
\caption{Two-point crossover. }
\label{cross}
\end{figure} 


\item \textcolor{red}{\textbf{Mutation:}
Mutation is a divergence operation which introduces random modifications.  The purpose of the mutation is to maintain diversity within the population and prevent premature convergence. Mutation is used to add new genetic information (divergence) in order to achieve a global search over the solution search space and avoid to fall in local optima. The mutation operator in the proposed GA-MuDiLCO works as follow: If mutation probability $P_m$ is 100$\%$, then the mutation operation takes place on the new individual. The round number is selected randomly within (1..T) in the schedule solution. After that one sensor within this round is selected randomly within (1..J). If the sensor is scheduled as active "1", it should be rescheduled to sleep "0". If the sensor is scheduled as sleep, it rescheduled to active only if it has adequate remaining energy.}


\item \textcolor{red}{\textbf{Update O-U-Coverage for children:}
Before evaluating each new individual, Algorithm \ref{OU} is called for each new individual to compute the new undercoverage $Ch.U$ and overcoverage $Ch.\Theta$ parameters. }
 
\item \textcolor{red}{\textbf{Evaluate New Individuals:}
Each new individual is evaluated using Eq. \ref{eqf} but with using the new undercoverage $Ch.U$ and overcoverage $Ch.\Theta$ parameters of the new children.}

\item \textcolor{red}{\textbf{Replacement:}
After evaluation of new children, Triple Tournament Replacement (TTR) will be applied for each new individual. In TTR strategy, three individuals are selected
randomly from the population. Find the worst from them and then check its fitness with the new individual fitness. If the fitness of the new individual is better than the fitness of  the worst individual, replace the new individual with the worst individual. Otherwise, the replacement is not done. }

 
\item \textcolor{red}{\textbf{Stopping criteria:}
The proposed GA-MuDiLCO stops when the stopping criteria is met. It stops after running for an amount of time in seconds equal to \textbf{Time limit}. The \textbf{Time limit} is the execution time obtained by the optimization solver GLPK for solving the same size of problem. The best solution will be selected as a schedule of sensors for $T$ rounds during the sensing phase in the current period.}



\end{enumerate} 

\fi

%% EXPERIMENTAL STUDY

\section{Experimental study}
\label{exp}
\subsection{Simulation setup}

We  conducted  a series  of  simulations  to  evaluate  the efficiency  and  the
relevance  of  our   approach,  using  the  discrete   event  simulator  OMNeT++
\cite{varga}.  The  simulation parameters are summarized  in Table~\ref{table3}.
Each experiment for a network is run over 25~different random topologies and the
results presented hereafter are the average of these 25 runs.
%Based on the results of our proposed work in~\cite{idrees2014coverage}, we found as the region of interest are divided into larger subregions as the network lifetime increased. In this simulation, the network are divided into 16 subregions. 
We  performed  simulations for  five  different  densities  varying from  50  to
250~nodes deployed  over a $50 \times  25~m^2 $ sensing field.   More precisely,
the deployment  is controlled  at a  coarse scale  in order  to ensure  that the
deployed nodes can cover the sensing field with the given sensing range.

%%RC these parameters are realistic?
%% maybe we can increase the field and sensing range. 5mfor Rs it seems very small... what do the other good papers consider ?

\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 size & $(50 \times 25)~m^2 $   \\
% inserting body of the table
%\hline
Network size &  50, 100, 150, 200 and 250~nodes   \\
%\hline
Initial energy  & 500-700~joules  \\  
%\hline
Sensing time for one round & 60 Minutes \\
$E_{R}$ & 36 Joules\\
$R_s$ & 5~m   \\     
%\hline
$W_{\theta}$ & 1   \\
% [1ex] adds vertical space
%\hline
$W_{U}$ & $|P|^2$ \\
%$P_c$ & 0.95   \\ 
%$P_m$ & 0.6 \\
%$S_{pop}$ & 50
%inserts single line
\end{tabular}
\label{table3}
% is used to refer this table in the text
\end{table}

\textcolor{blue}{Our  protocol  is  declined   into  four  versions:  MuDiLCO-1,
  MuDiLCO-3, MuDiLCO-5, and MuDiLCO-7, corresponding respectively to $T=1,3,5,7$
  ($T$ the  number of rounds in  one sensing period). Since  the time resolution
  may  be prohibitive  when the  size  of the  problem increases,  a time  limit
  threshold has  been fixed when  solving large  instances. In these  cases, the
  solver returns  the best solution  found, which  is not necessary  the optimal
  one. In practice, we only set time  limit values for the three largest network
  sizes when $T=7$, using the following  respective values (in second): 0.03 for
  150~nodes, 0.06 for 200~nodes, and 0.08 for 250~nodes.
% Table \ref{tl} shows time limit values.
  These time limit threshold have been  set empirically. The basic idea consists
  in considering  the average execution  time to  solve the integer  programs to
  optimality, then by  dividing this average time by three  to set the threshold
  value.  After that,  this threshold value is increased if  necessary such that
  the solver is able  to deliver a feasible solution within  the time limit.  In
  fact, selecting the optimal values for the time limits will be investigated in
  future.}
%In Table \ref{tl},  "NO" indicates  that  the  problem has  been  solved to  optimality without time limit.}

%\begin{table}[ht]
%\caption{Time limit values for MuDiLCO protocol versions }
%\centering
%\begin{tabular}{|c|c|c|c|c|}
% \hline
% WSN size & MuDiLCO-1 & MuDiLCO-3 & MuDiLCO-5 & MuDiLCO-7 \\ [0.5ex]
%\hline
% 50 & NO & NO & NO & NO \\
% \hline
%100 & NO & NO & NO & NO \\
%\hline
%150 & NO & NO & NO & 0.03 \\
%\hline
%200 & NO & NO & NO & 0.06 \\
% \hline
% 250 & NO & NO & NO & 0.08 \\
% \hline
%\end{tabular}

%\label{tl}

%\end{table}

 In the  following, we will make  comparisons with two other  methods. The first
 method,  called DESK  and proposed  by  \cite{ChinhVu}, is  a full  distributed
 coverage  algorithm.   The  second method,  called  GAF~\cite{xu2001geography},
 consists in dividing the region into fixed squares.  During the decision phase,
 in each square, one  sensor is then chosen to remain  active during the sensing
 phase time.

Some preliminary experiments were performed to study the choice of the number of
subregions  which subdivides  the  sensing field,  considering different  network
sizes. They show that as the number of subregions increases, so does the network
lifetime. Moreover,  it makes  the MuDiLCO protocol  more robust  against random
network  disconnection due  to node  failures.  However,  too  many subdivisions
reduce the advantage  of the optimization. In fact, there  is a balance between
the  benefit  from the  optimization  and the  execution  time  needed to  solve
it. Therefore, we have set the number of subregions to 16 rather than 32.

\subsection{Energy model}

We  use an  energy consumption  model  proposed by~\cite{ChinhVu}  and based  on
\cite{raghunathan2002energy} with slight  modifications.  The energy consumption
for  sending/receiving the packets  is added,  whereas the  part related  to the
sensing range is removed because we consider a fixed sensing range.

% We are took into account the energy consumption needed for the high computation during executing the algorithm on the sensor node. 
%The new energy consumption model will take into account the energy consumption for communication (packet transmission/reception), the radio of the sensor node, data sensing, computational energy of Micro-Controller Unit (MCU) and high computation energy of MCU. 
%revoir la phrase

For our  energy consumption model, we  refer to the sensor  node Medusa~II which
uses an Atmels  AVR ATmega103L microcontroller~\cite{raghunathan2002energy}. The
typical  architecture  of a  sensor  is composed  of  four  subsystems: the  MCU
subsystem which is capable of computation, communication subsystem (radio) which
is responsible  for transmitting/receiving messages, the  sensing subsystem that
collects  data, and  the  power supply  which  powers the  complete sensor  node
\cite{raghunathan2002energy}. Each  of the first three subsystems  can be turned
on or  off depending on  the current status  of the sensor.   Energy consumption
(expressed in  milliWatt per second) for  the different status of  the sensor is
summarized in Table~\ref{table4}.

\begin{table}[ht]
\caption{The Energy Consumption Model}
% title of Table
\centering
% used for centering table
\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}

For the sake of simplicity we ignore the  energy needed to turn on the radio, to
start up the sensor node, to move from one status to another, etc.
%We also do not consider the need of collecting sensing data. PAS COMPRIS
Thus, when a sensor becomes active (i.e.,  it has already chosen its status), it
can turn its radio  off to save battery.  MuDiLCO uses two  types of packets for
communication. The size of the INFO  packet and Active-Sleep packet are 112~bits
and 24~bits  respectively.  The value  of energy  spent to send  a 1-bit-content
message is  obtained by using  the equation in  ~\cite{raghunathan2002energy} to
calculate the  energy cost  for transmitting  messages and  we propose  the same
value for receiving  the packets. The energy  needed to send or  receive a 1-bit
packet is equal to 0.2575~mW.

The initial energy of each node is  randomly set in the interval $[500;700]$.  A
sensor node will  not participate in the  next round if its  remaining energy is
less than  $E_{R}=36~\mbox{Joules}$, the minimum  energy needed for the  node to
stay alive  during one round.  This  value has been computed  by multiplying the
energy consumed in  active state (9.72 mW)  by the time in second  for one round
(3600 seconds).   According to the interval  of initial energy, a  sensor may be
alive during at most 20 rounds.

\subsection{Metrics}

To evaluate our approach we consider the following performance metrics:

\begin{enumerate}[i]
  
\item {{\bf Coverage Ratio (CR)}:} the coverage ratio measures how much of the area
  of a sensor field is covered. In our case, the sensing field is represented as
  a connected grid  of points and we use  each grid point as a  sample point to
  compute the coverage. The coverage ratio can be calculated by:
\begin{equation*}
\scriptsize
\mbox{CR}(\%) = \frac{\mbox{$n^t$}}{\mbox{$N$}} \times 100,
\end{equation*}
where $n^t$ is  the number of covered  grid points by the active  sensors of all
subregions during round $t$ in the current sensing phase and $N$ is the total number
of grid points  in the sensing field of  the network. In our simulations $N = 51
\times 26 = 1326$ grid points.
%The accuracy of this method depends on the distance between grids. In our
%simulations, the sensing field has been divided into 50 by 25 grid points, which means
%there are $51 \times 26~ = ~ 1326$ points in total.
% Therefore, for our simulations, the error in the coverage calculation is less than ~ 1 $\% $.

\item{{\bf Number  of Active Sensors Ratio  (ASR)}:} it is important  to have as
  few  active  nodes  as  possible  in  each  round, in  order  to  minimize  the
  communication overhead  and maximize the network lifetime.  The Active Sensors
  Ratio is defined as follows:
\begin{equation*}
\scriptsize  \mbox{ASR}(\%) = \frac{\sum\limits_{r=1}^R
  \mbox{$A_r^t$}}{\mbox{$|J|$}} \times 100,
\end{equation*}
where $A_r^t$ is the number of  active sensors in the subregion $r$ during round
$t$ in the  current sensing phase, $|J|$  is the total number of  sensors in the
network, and $R$ is the total number of subregions in the network.

\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 network is alive until all nodes have
  been   drained    of   their   energy   or   the    sensor   network   becomes
  disconnected. Network connectivity is  important because an active sensor node
  without connectivity towards a base  station cannot transmit information on an
  event in the area that it monitors.

\item {{\bf  Energy Consumption  (EC)}:} the average  energy consumption  can be
  seen as the total energy consumed by the sensors during the $Lifetime_{95}$ or
  $Lifetime_{50}$  divided  by the  number  of rounds.  EC  can  be computed  as
  follows:

  % New version with global loops on period
  \begin{equation*}
    \scriptsize
    \mbox{EC} = \frac{\sum\limits_{m=1}^{M} \left[ \left( E^{\mbox{com}}_m+E^{\mbox{list}}_m+E^{\mbox{comp}}_m \right) +\sum\limits_{t=1}^{T_m} \left( E^{a}_t+E^{s}_t \right) \right]}{\sum\limits_{m=1}^{M} T_m},
  \end{equation*}


% Old version with loop on round outside the loop on period
%  \begin{equation*}
%    \scriptsize
%    \mbox{EC} = \frac{\sum\limits_{m=1}^{M_L} \left( E^{\mbox{com}}_m+E^{\mbox{list}}_m+E^{\mbox{comp}}_m \right) +\sum\limits_{t=1}^{T_L} \left( E^{a}_t+E^{s}_t \right)}{T_L},
%  \end{equation*}

% Ali version 
%\begin{equation*}
%\scriptsize
%\mbox{EC} =  \frac{\mbox{$\sum\limits_{d=1}^D E^c_d$}}{\mbox{$D$}} + \frac{\mbox{$\sum\limits_{d=1}^D %E^l_d$}}{\mbox{$D$}} + \frac{\mbox{$\sum\limits_{d=1}^D E^a_d$}}{\mbox{$D$}} + %\frac{\mbox{$\sum\limits_{d=1}^D E^s_d$}}{\mbox{$D$}}.
%\end{equation*}

% Old version -> where $M_L$ and  $T_L$ are respectively the number of  periods and rounds during
%$Lifetime_{95}$ or  $Lifetime_{50}$. 
% New version
where  $M$ is  the  number  of periods  and  $T_m$ the  number  of  rounds in  a
period~$m$, both  during $Lifetime_{95}$  or $Lifetime_{50}$.  The  total 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, $E^a_t$ and $E^s_t$
indicate the energy consumed by the whole network in round $t$.

%\item {Network Lifetime:} we  have defined the network  lifetime as the  time until all
%nodes  have  been drained  of  their  energy  or each  sensor  network monitoring  an area has become  disconnected.

\item {{\bf  Execution Time}:}  a sensor node  has limited energy  resources and
  computing power, therefore it is important that the proposed algorithm has the
  shortest possible execution  time. The energy of a sensor  node must be mainly
  used for the sensing phase, not for the pre-sensing ones.
  
\item {{\bf Stopped simulation runs}:} a simulation ends when the sensor network
  becomes disconnected (some nodes are dead and are not able to send information
  to the base station). We report the number of simulations that are stopped due
  to network disconnections and for which round it occurs.

\end{enumerate}

\subsection{Performance analysis for different number of primary points}
\label{ch4:sec:04:06}

In this  section, we study the  performance of MuDiLCO-1 approach  for different
numbers of  primary points. The  objective of this  comparison is to  select the
suitable number  of primary points  to be used by  a MuDiLCO protocol.   In this
comparison,  MuDiLCO-1 protocol  is used  with five  primary point  models, each
model corresponding to a number of  primary points, which are called Model-5 (it
uses 5 primary points), Model-9, Model-13, Model-17, and Model-21.

%\begin{enumerate}[i)]

%\item {{\bf Coverage Ratio}}
\subsubsection{Coverage ratio} 

Figure~\ref{Figures/ch4/R2/CR} shows the average coverage ratio for 150 deployed
nodes.  As can be seen, at the beginning the models which use a larger number of
primary points provide slightly better coverage  ratios, but latter they are the
worst.
%Moreover, when the number of periods increases, coverage ratio produced by Model-9, Model-13, Model-17, and Model-21 decreases in comparison with Model-5 due to a larger time computation for the decision process for larger number of primary points.
Moreover, when the  number of periods increases, the coverage  ratio produced by
all models  decrease due  to dead nodes.  However, Model-5 is  the one  with the
slowest decrease due to lower numbers of active sensors in the earlier periods.
% smaller time computation of decision process for a smaller number of primary points.
Overall this  model is slightly more  efficient than the other  ones, because it
offers a good coverage ratio for a larger number of periods.
%\parskip 0pt
\begin{figure}[t!]
\centering
 \includegraphics[scale=0.5] {R2/CR.pdf} 
\caption{Coverage ratio for 150 deployed nodes}
\label{Figures/ch4/R2/CR}
\end{figure} 


%\item {{\bf Network Lifetime}}
\subsubsection{Network lifetime}

Finally, we study the effect of increasing the number of primary points on the lifetime of the network. 
%In Figure~\ref{Figures/ch4/R2/LT95} and in Figure~\ref{Figures/ch4/R2/LT50}, network lifetime, $Lifetime95$ and $Lifetime50$ respectively, are illustrated for different network sizes. 
As       highlighted       by       Figures~\ref{Figures/ch4/R2/LT}(a)       and
\ref{Figures/ch4/R2/LT}(b), the  network lifetime  obviously increases  when the
size of the network increases, with  Model-5 which leads to the largest lifetime
improvement.

\begin{figure}[h!]
\centering
\centering
\includegraphics[scale=0.5]{R2/LT95.pdf}\\~ ~ ~ ~ ~(a) \\

\includegraphics[scale=0.5]{R2/LT50.pdf}\\~ ~ ~ ~ ~(b)

\caption{Network lifetime for (a) $Lifetime_{95}$ and (b) $Lifetime_{50}$}
  \label{Figures/ch4/R2/LT}
\end{figure}

Comparison shows that Model-5, which uses  less number of primary points, is the
best one because it is less energy  consuming during the network lifetime. It is
also  the better  one  from the  point  of  view of  coverage  ratio, as  stated
before. Therefore, we have chosen the model with five primary points for all the
experiments presented thereafter.

%\end{enumerate}

% MICHEL => TO BE CONTINUED

\subsection{Experimental results and analysis}

\subsubsection{Coverage ratio} 

Figure~\ref{fig3} shows  the average coverage  ratio for 150 deployed  nodes. We
can notice that for the first thirty rounds both DESK and GAF provide a coverage
which is a little bit better than the one of MuDiLCO.  
%%RC : need to uniformize MuDiLCO or MuDiLCO-T? 
%%MS : MuDiLCO everywhere
%%RC maybe increase the size of the figure for the reviewers, no?
This is due  to the fact that, in comparison with  MuDiLCO which uses optimization
to put in  SLEEP status redundant sensors, more sensor  nodes remain active with
DESK and GAF.   As a consequence, when the number of  rounds increases, a larger
number of node failures  can be observed in DESK and GAF,  resulting in a faster
decrease of the coverage ratio.   Furthermore, our protocol allows to maintain a
coverage ratio  greater than  50\% for far  more rounds.  Overall,  the proposed
sensor  activity scheduling based  on optimization  in MuDiLCO  maintains higher
coverage ratios of the  area of interest for a larger number  of rounds. It also
means that MuDiLCO saves more energy,  with less dead nodes, at most for several
rounds, and thus should extend the network lifetime.

\begin{figure}[ht!]
\centering
 \includegraphics[scale=0.5] {F/CR.pdf} 
\caption{Average coverage ratio for 150 deployed nodes}
\label{fig3}
\end{figure} 

\iffalse
\textcolor{red}{ We
can see that for the first thirty nine rounds GA-MuDiLCO provides a little bit better coverage ratio  than MuDiLCO. Both DESK and GAF provide a coverage
which is a little bit better than the one of MuDiLCO and GA-MuDiLCO for the first thirty rounds because they activate a larger number of nodes during sensing phase. After that GA-MuDiLCO provides a coverage ratio near to the  MuDiLCO and better than DESK and GAF. GA-MuDiLCO gives approximate solution with activation a larger number of nodes than MuDiLCO during sensing phase while it activates a less number of nodes in comparison with both DESK and GAF. MuDiLCO and GA-MuDiLCO clearly outperform DESK and GAF for
a number of periods between 31 and 103. This is because they optimize the coverage and the lifetime in a wireless sensor network by selecting the best representative sensor nodes to take the responsibility of coverage during the sensing phase.}
\fi


\subsubsection{Active sensors ratio} 

It is crucial to have as few active nodes as possible in each round, in order to
minimize the communication overhead and maximize    the network lifetime. Figure~\ref{fig4}  presents the active  sensor ratio for  150 deployed
nodes all along the network lifetime. It appears that up to round thirteen, DESK
and GAF have  respectively 37.6\% and 44.8\% of nodes  in ACTIVE status, whereas
MuDiLCO clearly outperforms them  with only 24.8\%  of active nodes. 
%\textcolor{red}{GA-MuDiLCO activates a number of sensor nodes larger than MuDiLCO but lower than both DESK and GAF. GA-MuDiLCO-1, GA-MuDiLCO-3, and GA-MuDiLCO-5 continue in providing a larger number of active sensors until the forty-sixth round after that it provides less number of active nodes due to the died nodes. GA-MuDiLCO-7 provides a larger number of sensor nodes and maintains a better coverage ratio compared to MuDiLCO-7 until the fifty-seventh round.  After the thirty-fifth round, MuDiLCO exhibits larger numbers of active nodes compared with DESK  and GAF, which agrees with  the  dual  observation  of  higher  level  of  coverage  made  previously}.
Obviously, in that case DESK  and GAF have less active nodes, since  they have activated many nodes  at the beginning. Anyway, MuDiLCO  activates the available nodes in a more efficient manner. 
%\textcolor{red}{GA-MuDiLCO activates near optimal number of sensor nodes also in efficient manner compared with both DESK  and GAF}.

\begin{figure}[ht!]
\centering
\includegraphics[scale=0.5]{F/ASR.pdf}  
\caption{Active sensors ratio for 150 deployed nodes}
\label{fig4}
\end{figure} 

%\textcolor{red}{GA-MuDiLCO activates a sensor nodes larger than MuDiLCO but lower than both DESK and GAF }


\subsubsection{Stopped simulation runs}
%The results presented in this experiment, is to show the comparison of our MuDiLCO protocol with other two approaches from the point of view the stopped simulation runs per round. Figure~\ref{fig6} illustrates the percentage of stopped simulation
%runs per round for 150 deployed nodes. 

Figure~\ref{fig6} reports the cumulative  percentage of stopped simulations runs
per round for  150 deployed nodes. This figure gives the  breakpoint for each method.  DESK stops first,  after approximately 45~rounds, because it consumes the
more energy by  turning on a large number of redundant  nodes during the sensing
phase. GAF  stops secondly for the  same reason than  DESK. 
%\textcolor{red}{GA-MuDiLCO  stops thirdly for the  same reason than  DESK and GAF.} \textcolor{red}{MuDiLCO and GA-MuDiLCO overcome}
%DESK and GAF because \textcolor{red}{they activate less number of sensor nodes, as well as }the optimization process distributed on several subregions leads to coverage  preservation and  so extends  the network  lifetime.  
Let us emphasize that the  simulation continues as long as a network  in a subregion is still connected. 

%%% The optimization effectively continues as long as a network in a subregion is still connected. A VOIR %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{figure}[ht!]
\centering
\includegraphics[scale=0.5]{F/SR.pdf} 
\caption{Cumulative percentage of stopped simulation runs for 150 deployed nodes }
\label{fig6}
\end{figure} 

\subsubsection{Energy consumption} \label{subsec:EC}

We  measure  the  energy  consumed  by the  sensors  during  the  communication,
listening, computation, active, and sleep status for different network densities
and   compare   it   with   the  two   other   methods.    Figures~\ref{fig7}(a)
and~\ref{fig7}(b)  illustrate  the  energy  consumption,  considering  different
network sizes, for $Lifetime_{95}$ and $Lifetime_{50}$.

\begin{figure}[h!]
  \centering
  \begin{tabular}{cl}
    \parbox{9.5cm}{\includegraphics[scale=0.5]{F/EC95.pdf}} & (a) \\
    \verb+ + \\
    \parbox{9.5cm}{\includegraphics[scale=0.5]{F/EC50.pdf}} & (b)
  \end{tabular}
  \caption{Energy consumption for (a) $Lifetime_{95}$ and 
    (b) $Lifetime_{50}$}
  \label{fig7}
\end{figure} 

The  results  show  that  MuDiLCO  is  the  most  competitive  from  the  energy
consumption point of view.  The  other approaches have a high energy consumption
due  to activating a  larger number  of redundant  nodes as  well as  the energy consumed during  the different  status of the  sensor node.
% Among  the different versions of our protocol, the MuDiLCO-7  one consumes more energy than the other
%versions. This is  easy to understand since the bigger the  number of rounds and the number of  sensors involved in the integer program are,  the larger the time computation to solve the optimization problem is. To improve the performances of MuDiLCO-7, we  should increase the  number of subregions  in order to  have less sensors to consider in the integer program.
%\textcolor{red}{As shown in Figure~\ref{fig7}, GA-MuDiLCO consumes less energy than both DESK and GAF, but a little bit higher than MuDiLCO  because it provides a near optimal solution by activating a larger number of nodes during the sensing phase.  GA-MuDiLCO consumes less energy in comparison with MuDiLCO-7 version, especially for the dense networks. However, MuDiLCO protocol and GA-MuDiLCO protocol are the most competitive from the energy
%consumption point of view. The other approaches have a high energy consumption
%due to activating a larger number of redundant nodes.}
%In fact,  a distributed optimization decision, which produces T rounds, on the subregions is  greatly reduced the cost of communications and the time of listening as well as the energy needed for sensing phase and computation so thanks to the partitioning of the initial network into several independent subnetworks and producing T rounds for each subregion periodically. 


\subsubsection{Execution time}
\label{et}
We observe  the impact of the  network size and of  the number of  rounds on the
computation  time.   Figure~\ref{fig77} gives  the  average  execution times  in
seconds (needed to solve optimization problem) for different values of $T$. The modeling language for Mathematical Programming (AMPL)~\cite{AMPL} is  employed to generate the Mixed Integer Linear 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) \cite{glpk} through a Branch-and-Bound method. The
original execution time  is computed on a laptop  DELL with Intel Core~i3~2370~M
(2.4 GHz)  processor (2  cores) and the  MIPS (Million Instructions  Per Second)
rate equal to 35330. To be consistent  with the use of a sensor node with Atmels
AVR ATmega103L  microcontroller (6 MHz) and  a MIPS rate  equal to 6 to  run the
optimization   resolution,   this  time   is   multiplied   by  2944.2   $\left(
\frac{35330}{2} \times  \frac{1}{6} \right)$ and  reported on Figure~\ref{fig77}
for different network sizes.

\begin{figure}[ht!]
\centering
\includegraphics[scale=0.5]{F/T.pdf}  
\caption{Execution Time (in seconds)}
\label{fig77}
\end{figure} 

As expected,  the execution time increases  with the number of  rounds $T$ taken
into account to schedule the sensing phase. The times obtained for $T=1,3$
or $5$ seem bearable, but for $T=7$ they become quickly unsuitable for a sensor
node, especially when  the sensor network size increases.   Again, we can notice
that if we want  to schedule the nodes activities for a  large number of rounds,
we need to choose a relevant number of subregions in order to avoid a complicated
and cumbersome optimization.  On the one hand, a large value  for $T$ permits to
reduce the  energy-overhead due  to the three  pre-sensing phases, on  the other
hand  a leader  node may  waste a  considerable amount  of energy  to  solve the
optimization problem.

%While MuDiLCO-1, 3, and 5 solves the optimization process with suitable execution times to be used on wireless sensor network because it distributed on larger number of small subregions as well as it is used acceptable number of round(s) T.  We think that in distributed fashion the solving of the optimization problem to produce T rounds in a subregion can be tackled by sensor nodes. Overall, to be able to deal with very large networks, a distributed method is clearly required.

\subsubsection{Network lifetime}

The next  two figures,  Figures~\ref{fig8}(a) and \ref{fig8}(b),  illustrate the
network lifetime  for different network sizes,  respectively for $Lifetime_{95}$
and  $Lifetime_{50}$.  Both  figures show  that the  network  lifetime increases
together with the  number of sensor nodes, whatever the  protocol, thanks to the
node  density  which  results in  more  and  more  redundant  nodes that  can  be
deactivated and thus save energy.  Compared to the other approaches, our MuDiLCO
protocol  maximizes the  lifetime of  the network.   In particular  the  gain in
lifetime for a  coverage over 95\% is greater than 38\%  when switching from GAF
to MuDiLCO-3.  The  slight decrease that can be observed  for MuDiLCO-7 in case
of  $Lifetime_{95}$  with  large  wireless  sensor  networks  results  from  the
difficulty  of the optimization  problem to  be solved  by the  integer program.
This  point was  already noticed  in subsection  \ref{subsec:EC} devoted  to the
energy consumption,  since network lifetime and energy  consumption are directly
linked. 
%\textcolor{red}{As can be seen in these figures, the lifetime increases with the size of the network, and it is clearly largest for the MuDiLCO
%and the GA-MuDiLCO protocols. GA-MuDiLCO prolongs the network lifetime obviously in comparison with both DESK and GAF, as well as the MuDiLCO-7 version for $lifetime_{95}$.  However, comparison shows that MuDiLCO protocol and GA-MuDiLCO protocol, which use distributed optimization over the subregions are the best ones because they are robust to network disconnection during the network lifetime as well as they consume less energy in comparison with other approaches.}
\begin{figure}[t!]
  \centering
  \begin{tabular}{cl}
    \parbox{9.5cm}{\includegraphics[scale=0.5]{F/LT95.pdf}} & (a) \\
    \verb+ + \\
    \parbox{9.5cm}{\includegraphics[scale=0.5]{F/LT50.pdf}} & (b)
  \end{tabular}
  \caption{Network lifetime for (a) $Lifetime_{95}$ and 
    (b) $Lifetime_{50}$}
  \label{fig8}
\end{figure} 

% By choosing the best suited nodes, for each round, by optimizing the coverage and lifetime of the network to cover the area of interest with a maximum number rounds and by letting the other nodes sleep in order to be used later in next rounds, our MuDiLCO protocol efficiently prolonges the network lifetime. 

%In Figure~\ref{fig8}, Comparison shows that our MuDiLCO protocol, which are used distributed optimization on the subregions with the ability of producing T rounds, is the best one because it is robust to network disconnection during the network lifetime as well as it consume less energy in comparison with other approaches. It also means that distributing the protocol in each sensor node and subdividing the sensing field into many subregions, which are managed independently and simultaneously, is the most relevant way to maximize the lifetime of a network.


%We see that our MuDiLCO-7 protocol results in execution times that quickly become unsuitable for a sensor network as well as the energy consumption seems to be huge because it used a larger number of rounds T during performing the optimization decision in the subregions, which is led to decrease the network lifetime. On the other side, our MuDiLCO-1, 3, and 5 protocol seems to be more efficient in comparison with other approaches because they are prolonged the lifetime of the network more than DESK and GAF.


\section{Conclusion and future works}
\label{sec:conclusion}

We have addressed  the problem of the coverage and of the lifetime optimization in
wireless  sensor networks.  This is  a key  issue as  sensor nodes  have limited
resources in terms of memory, energy, and computational power. To cope with this
problem,  the field  of sensing  is divided  into smaller  subregions  using the
concept  of divide-and-conquer  method, and  then  we propose  a protocol  which
optimizes coverage  and lifetime performances in each  subregion.  Our protocol,
called MuDiLCO (Multiround  Distributed Lifetime Coverage Optimization) combines
two  efficient   techniques:  network   leader  election  and   sensor  activity
scheduling.
%,  where the challenges
%include how to select the  most efficient leader in each subregion and
%the best cover sets %of active nodes that will optimize the network lifetime
%while taking the responsibility of covering the corresponding
%subregion using more than one cover set during the sensing phase. 
The activity  scheduling in each subregion  works in periods,  where each period
consists of four  phases: (i) Information Exchange, (ii)  Leader Election, (iii)
Decision Phase to plan the activity  of the sensors over $T$ rounds, (iv) Sensing
Phase itself divided into $T$ rounds.

Simulations  results show the  relevance of  the proposed  protocol in  terms of
lifetime, coverage  ratio, active  sensors ratio, energy  consumption, execution
time. Indeed,  when dealing with  large wireless sensor networks,  a distributed
approach, like  the one we  propose, allows to  reduce the difficulty of  a single
global optimization problem by partitioning it in many smaller problems, one per
subregion, that can be solved  more easily. Nevertheless, results also show that
it is not possible to plan the activity of sensors over too many rounds, because
the resulting optimization problem leads to too high resolution times and thus to
an excessive energy consumption.

%In  future work, we plan  to study and propose adjustable sensing range coverage optimization protocol, which computes  all active sensor schedules in one time, by using
%optimization  methods. This protocol can prolong the network lifetime by minimizing the number of the active sensor nodes near the borders by optimizing the sensing range of sensor nodes.
% use section* for acknowledgement

\section*{Acknowledgment}
This work is  partially funded by the Labex ACTION program (contract ANR-11-LABX-01-01).
As a Ph.D.  student, Ali Kadhum IDREES would like to gratefully acknowledge the
University  of Babylon  - Iraq  for the  financial support,  Campus  France (The
French  national agency  for the  promotion of  higher  education, international
student   services,  and   international  mobility).%,   and  the   University  ofFranche-Comt\'e - France for all the support in France. 




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