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\journal{Ad Hoc Networks}

\begin{document}

\begin{frontmatter}

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

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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.}

\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 Multiperiod 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 into periods
during which sets of sensor nodes are scheduled to remain active for a number of
rounds  during the  sensing phase,  to  ensure coverage  so as  to maximize  the
lifetime of  WSN.  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. 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 multirounds 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}.

\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 sensors during the sensing period.
\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  coverage  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}
%{\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} 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}  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.

\cite{cardei2005improving} 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  node
participations 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.

\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.

Some        distributed       algorithms        have        been       developed
in~\cite{Gallais06,Tian02,Ye03,Zhang05,HeinzelmanCB02,    yardibi2010distributed}
to perform  the scheduling so  as to preserve coverage.   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.    \cite{prasad2007distributed}  defines  a  model  for
capturing the  dependencies between different cover sets  and 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}  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  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} focuses  on coverage-aware,
distributed energy-efficient,  and distributed clustering  methods respectively,
which aims  to extend the network  lifetime, while the coverage  is ensured.  S.
Misra et al.  \cite{Misra} 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} 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. 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 Multiperiod 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.

%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 Multiperiod 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
working nodes in the active mode.

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.

%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}

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.

As can  be seen  in Figure~\ref{fig2}, our  protocol works  in periods
fashion, where  each is  divided into 4  phases: Information~Exchange,
Leader~Election,  Decision, and  Sensing.  Each  sensing phase  may be
itself divided  into $T$ rounds  and for each  round a set  of sensors
(said a cover set) is responsible for the sensing task.
\begin{figure}[ht!]
\centering
\includegraphics[width=95mm]{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 is reliable  against an unexpected node failure, because
it works  in periods. 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.

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:  a such 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 participate to 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 informations 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 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.  The integer program  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. The  integer program is based on the model
proposed  by  \cite{pedraza2006}  with  some modification,  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 activation  of
sensor $j$  during the  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  the 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{6 mm} \forall j \in J, t = 1,\dots,T
  \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}


\begin{itemize}
\item $X_{t,j}$: indicates  whether or not the sensor  $j$ is actively
  sensing during the 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 the round $t$;
\item $U_{t,p}$  - {\it undercoverage}:  indicates whether or  not the
  primary point  $p$ is being covered  during the 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. In  our simulations
priority is given to the  coverage by choosing $W_{\theta}$ very large
compared to $W_U$.
%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 node  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 $ActiveSleep()$ 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}

\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. Experimental results are obtained from randomly generated networks in
which  nodes  are 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.

\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
$w_{U}$ & $|P^2|$
%inserts single line
\end{tabular}
\label{table3}
% is used to refer this table in the text
\end{table}
  
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).  In  the following,  the general  case will  be
denoted by MuDiLCO-T.   We compare MuDiLCO-T 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.

\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, 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}.  The energy  needed to send or receive a 1-bit
packet is equal to $0.2575~mW$.

\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 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 already decides  it's 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 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 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 for
  calculating 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 total number
of grid points in the sensing field of the network.
%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 the 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  $Lifetime_{50}$) as  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:
 \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*}

%\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*}

where $M_L$ and  $T_L$ are respectively the number of  periods and rounds 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 $E^{\scriptsize \mbox{com}}_m$, represent  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 consummed 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}

%%%%%%%%%%%%%%%%%%%%%%%%VU JUSQU ICI**************************************************

\section{Results and analysis}

\subsection{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-T. This is due to the fact
that in  comparison with MuDiLCO that  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 nodes
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-T
save more  energy, with less  dead nodes, at  most for several rounds,  and thus
should extend the network lifetime.

\begin{figure}[h!]
\centering
 \includegraphics[scale=0.5] {R1/CR.pdf} 
\caption{Average coverage ratio for 150 deployed nodes}
\label{fig3}
\end{figure} 

\subsection{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-T clearly outperforms  them with only 24.8\% of  active nodes. After the
thirty  fifth round,  MuDiLCO-T exhibits  larger number  of active  nodes, 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-T activates the available
nodes in a more efficient manner.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{R1/ASR.pdf}  
\caption{Active sensors ratio for 150 deployed nodes}
\label{fig4}
\end{figure} 

\subsection{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 of
the methods.  DESK stops first,  after around 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.  MuDiLCO-T overcomes
DESK and GAF because 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}[h!]
\centering
\includegraphics[scale=0.5]{R1/SR.pdf} 
\caption{Cumulative percentage of stopped simulation runs for 150 deployed nodes }
\label{fig6}
\end{figure} 

\subsection{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]{R1/EC95.pdf}} & (a) \\
    \verb+ + \\
    \parbox{9.5cm}{\includegraphics[scale=0.5]{R1/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-T  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,  the larger  the time
computation to  solve the optimization  problem. 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.

%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. 


\subsection{Execution time}

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 (times  needed to  solve optimization problem)  for different  values of
$T$.   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}[h!]
\centering
\includegraphics[scale=0.5]{R1/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 for  scheduling of the sensing phase. The times
obtained for $T=1,3$ or $5$  seems 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  subregion 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.

\subsection{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
result in  more and more redundant  nodes that can  be deactivated and
thus  save energy.  Compared  to the  other approaches,  our MuDiLCO-T
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 bee
observed for MuDiLCO-7 in  case of $Lifetime_{95}$ with large wireless
sensor networks result 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.

\begin{figure}[h!]
  \centering
  \begin{tabular}{cl}
    \parbox{9.5cm}{\includegraphics[scale=0.5]{R1/LT95.pdf}} & (a) \\
    \verb+ + \\
    \parbox{9.5cm}{\includegraphics[scale=0.5]{R1/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-T protocol efficiently prolonges the network lifetime. 

%In Figure~\ref{fig8}, Comparison shows that our MuDiLCO-T 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}

In this paper,  we have addressed the problem of  the coverage and 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  (Multiperiod 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 time
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}

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