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\journal{Ad Hoc Networks}

\begin{document}

\begin{frontmatter}

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\title{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. In this paper, a Distributed Lifetime Coverage Optimization Protocol (DiLCO)
to maintain the coverage and to improve the lifetime in wireless sensor networks is proposed. The area  of interest is first divided into subregions using  a divide-and-conquer method and then the  DiLCO protocol is distributed on the sensor nodes in each subregion. The DiLCO combines two efficient techniques: Leader election for each subregion after that activity scheduling based optimization is planned for each subregion. The proposed
DiLCO  works into rounds during which a small number  of nodes,
remaining active for sensing, is  selected to ensure coverage so as to maximize the lifetime of wireless sensor network. Each round consists of four phases: (i)~Information Exchange, (ii)~Leader
Election, (iii)~Decision,  and (iv)~Sensing.  The  decision process is
carried  out  by a  leader  node,  which  solves an  integer  program. Compared with some existing
protocols, simulation  results show that the proposed protocol can  prolong the
network lifetime and improve the coverage performance effectively.

\end{abstract}

\begin{keyword}
Wireless   Sensor   Networks,   Area   Coverage,   Network   lifetime,
Optimization, Scheduling.

\end{keyword}

\end{frontmatter}

\section{Introduction}
\indent In the last years, there has been increasing development in wireless networking,
Micro-Electro-Mechanical Systems (MEMS), and embedded computing technologies, which are led to construct low-cost, small-sized and low-power sensor nodes that can perform detection, computation and data communication of surrounding environment. 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, cooperate together to monitor the area of interest, 
and the measured data can be reported to a monitoring center called sink  
for analysis it~\cite{Sudip03}. There are several applications used the
WSN including health, home, environmental, military, and industrial
applications~\cite{Akyildiz02}. 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 approches 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 WSNs
design~\cite{Sudip03}, where it is impossible or inconvenient to replace and/or recharge their batteries because the the area of interest nature (such
as remote, hostile or unpractical 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 adaptive  scheduling. The area of interest is
divided into subregions and an activity scheduling for sensor nodes is
planned for each subregion. In fact, the nodes in a subregion can be
seen as a cluster where each node sends sensing data to the cluster head or the sink node. Furthermore, the activities in a
subregion/cluster can continue even if another cluster stops  due to
too many node failures.  Our scheduling scheme considers rounds, where
a round starts with a  discovery phase to exchange information between
sensors of  the subregion, in order  to choose in a  suitable manner a
sensor node to  carry out a coverage strategy.  This coverage strategy
involves  the  solving  of  an  integer program,  which  provides  the
activation of the sensors for the sensing phase of the current round.

The remainder of the paper is organized as follows. The next section
% Section~\ref{rw}
reviews the related work in the field. In section~\ref{prel}, the problem definition and some background are described. Section~\ref{pd} is devoted to
the DiLCO Protocol Description. Section~\ref{cp} gives  the coverage model formulation,  which is used
to schedule the activation of sensors.  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}
\label{rw}

In this section, we summarize the related works regarding coverage lifetime maximization and scheduling, and distinguish our DiLCO protocol from the works presented in the literature. Many centralized algorithms ~\cite{Slijepcevic01powerefficient, abrams2004set, cardei2005improving, zorbas2010solving, pujari2011high, cardei2005energy, berman04} and distributed algorithms ~\cite{Gallais06,Tian02,Ye03, Zhang05,HeinzelmanCB02, yardibi2010distributed, ChinhVu} for activity scheduling have been proposed in the literature, and based on different assumptions and objectives.
In centralized algorithms, a central controller makes all decisions and distributes the results to sensor nodes. In the distributed algorithms, the decision process is localized in each individual sensor node, and only information from neighboring nodes are used for the activity decision. 

Zorbas  et  al.  \cite{zorbas2010solving} presented a centralised greedy
algorithm for the efficient production of both node disjoint
and non-disjoint cover sets. The algorithm  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 provide cover sets offering longer network lifetime and they require a smaller number of node participations in order to achieve these results.

Cardei et al.~\cite{cardei2005energy} presented 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.

In \cite{he2012leveraging}, the authors proposed efficient centralized and distributed truncated greedy to improve the coverage and lifetime in WSNs by exploiting temporal-spatial correlations among sensory data. The basic idea lies in that a sensor node can be turned off safely when its sensory information can be inferred through some prediction methods, like Bayesian inference.
   
Zhou et al. \cite{zhou2009variable} have presented a centralized and distributed algorithms to conserve energy by exploiting redundancy in the network. In particular, they are addressed the problem of constructing a connected sensor cover in a sensor network model wherein each sensor can adjust its sensing and transmission range.
 Wang et al. \cite{wang2009parallel} are  focused on the energy-efficient coverage optimization problem of WSNs. Based on the models of coverage and energy, stationary nodes are partitioned into clusters by entropy clustering  and then  a parallel particle swarm optimization  is implemented by the cluster heads to maximize the coverage area and minimize the communication energy in each cluster. They are combined the maximum entropy clustering and parallel optimization, in which the stationary and mobile nodes can be organized to achieve energy efficiency of WSNs.
In \cite{yan2008design}, the authors  have proposed a monitoring service for sensor networks based on a distributed energy-efficient sensing coverage protocol. Each node  is able to dynamically decide it's schedule to guarantee a certain degree of coverage with average energy consumption inversely proportional to the node density.

The works presented in \cite{Bang, Zhixin, Zhang}  focuses on a 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{ChinhVu},  the authors 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  1-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.   
 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.
 
 The work in~\cite{esnaashari2010learning} proposed SALA, a scheduling algorithm based on learning automata, to deal with the problem of dynamic point coverage. In SALA each node in the network is equipped with a set of learning automata. The learning automata residing in each node try to learn the maximum sleep duration for the node in such a way that the detection rate of target points by the node does not degrade dramatically. 
 
 In~\cite{misra2011connectivity}, They are addressed the problem of network coverage and connectivity and proposed an efficient solution to maintain coverage, while preserving the connectivity of the network. The proposed solution aims to cover the area of interest, while minimizing the number of the active nodes. The overlap region between two nodes varies according to the distance between them. If the distance between two nodes
is maximized, the total coverage area of these nodes will also be maximized. Also, to preserve the connectivity of the network, each node should be in the communication range of at least one other node. 

Rizvi et al.~\cite{rizvi2012a1} have investigated the problem of constructing a Connected Dominating Set (CDS) 
, which provides better sensing coverage in an energy efficient manner. The have presented a CDS based topology control algorithm, A1, which forms an energy
efficient virtual backbone. They are proven that a single phase topology construction with fewer number of messages lead towards an efficient algorithm.

In ~\cite{tran2009novel}, the authors are defined a maximum sensing coverage region (MSCR) problem and presented a novel gossip-based sensing-coverage-aware algorithm to solve the problem. In this approach,  nodes gossip with their neighbors about their sensing coverage region where nodes decide locally to be an active or a sleeping  node.  In this method, the redundant node can reduce its activities whenever its sensing region is covered by enough neighbors.  

The main contributions of our DiLCO 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, (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  to take the mission of the coverage in each subregion, (7) The energy consumption model (8) The very low energy consumption, (9) The higher network lifetime.



\section{Preliminaries:}
\label{prel}

There are some design issues, which should be taken into consideration for coverage problem such as:  coverage type, deployment method, coverage degree, coverage ratio, activity scheduling, network connectivity and network lifetime ~\cite{wang2011coverage}.

\subsection{Coverage Problem}
Coverage reflects how well a sensor field is monitored, is one of
the most important performance metrics to measure WSNs. The most discussed coverage problems in literature can be classified
into three types \cite{ghosh2008coverage}\cite{wang2011coverage}: area coverage \cite{mulligan2010coverage}(also called full or blanket
coverage), target coverage \cite{yang2014novel}, and barrier coverage \cite{HeShibo}.  An area coverage problem is to find a minimum number of sensors to work, such that each physical point in the area is within the sensing range  of at least one working sensor node.
Target coverage problem is to cover only a finite number of discrete
points  called targets. This type  of coverage  has  mainly military
applications. The problem of preventing an intruder from entering a region of interest is referred to as the barrier coverage .
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).

\subsection{Deployment Method}
Deployment reflects how a sensor network is constructed over the sensing field. There are two ways to deploy the sensor nodes over the sensing field, fixed and random.  The fixed sensor placement could be used in small sensing field while for a large sensor network, remote and hostile environment might the random sensor placement is 	
recommended. The deployment of wireless sensor network could be dense or sparse. A dense deployment has a larger number of sensor nodes over the area of interest while sparse deployment has lower number of sensor nodes over the sensing field. The dense deployment method is used in situations where it is very important for every event to be detected or when it is important to have multiple sensors cover an area. Sparse deployment might be used when the cost of the sensors make a dense deployment is very expensive or to achieve maximum coverage using the minimum number of sensor nodes. 
	
\subsection{Coverage Degree}
Coverage degree refers to the number of sensor nodes, which cover point in the sensing disk model. As the number of sensor nodes, which cover a point increase, the  robustness of coverage increases. Coverage degree is represented one of the QoS requirements in WSNs.

\subsection{Coverage Ratio}
Coverage ratio refers to how much area of the total area of interest or how many points of the total points in the sensing field, which satisfy the QoS requirement of coverage degree. Coverage ratio can be seen as one of the QoS requirement in WSNs.

\subsection{Activity Scheduling }
Activity scheduling is to schedule the activation and deac-
tivation of sensor nodes. The basic objective is to decide which
sensors are in what states (active or sleeping mode) and for
how long, so that the application coverage requirement can be
guaranteed and the network lifetime can be prolonged. Various
approaches, including centralized, distributed, and localized
algorithms, have been proposed for activity scheduling. In
distributed algorithms, each node in the network autonomously
makes decisions on whether to turn on or turn off itself only
using local neighbor information. In centralized algorithms, a
central controller (a node or base station) informs every sensors
of the time intervals to be activated. There are many sensor node scheduling methods are proposed in \cite{wang2010clique}, where they are grouped into two main categories:round-based sensor node scheduling in which, sensor nodes will execute the scheduling
algorithm during the initialization of each round and group-based sensor node scheduling in which, each node will performs the scheduling algorithm only once after its deployment and after
the execution of scheduling algorithm, all nodes will be allocated into different groups.

\subsection{Network Connectivity}
Network connectivity refers to ensure that the WSN connected with the sink. The connected WSN should be guarantee that every sensor node in WSN can send the sensed data to other sensor nodes and to the sink using multihop communication. So, by using the sensing disk coverage model, each sensor node can communicate with each other using the communication range of the sensor node.


\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{ The DiLCO Protocol Description}
\label{pd}

In this section, we introduce a Distributed Lifetime Coverage Optimization protocol, which is called DiLCO. 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 DiLCO 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 rounds, 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 set 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 and Models}
We consider  a randomly and  uniformly deployed network  consisting of
static  wireless sensors. The  wireless 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 in term of energy provision.
The  location  information is  available  to  the  sensor node  either
through hardware such as embedded GPS or through location discovery
algorithms.   
\indent We consider a boolean  disk coverage model which is the most
widely used sensor coverage model in the literature. Each sensor has a
constant sensing range $R_s$. All  space points within a 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 $R_c \geq 2R_s$.
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.



%\begin{figure}[h!]
%\centering
%\begin{tabular}{cc}
%%\includegraphics[scale=0.25]{fig1.pdf}\\ %& \includegraphics[scale=0.10]{1.pdf} \\
%%(A) Figure 1 & (B) Figure 2
%\end{tabular}
%\caption{Unit Circle in radians. }
%\label{fig:cluster1}
%\end{figure}

%By using the Unit Circle in figure~\ref{fig:cluster1}, 
%We choose to representEach wireless sensor node will be represented into a selected number of primary points by which we can know if the sensor node is covered or not.
% Figure ~\ref{fig:cluster2} shows the selected primary points that represents the area of the sensor node and according to the sensing range of the wireless sensor node.

\indent Instead of working with the coverage area, we consider for each
sensor a set of points called  primary points. We also assume that the
sensing disk defined  by a sensor is covered if all the primary points of
this sensor are covered.
%\begin{figure}[h!]
%\centering
%\begin{tabular}{cc}
%%\includegraphics[scale=0.25]{fig2.pdf}\\ %& \includegraphics[scale=0.10]{1.pdf} \\
%%(A) Figure 1 & (B) Figure 2
%\end{tabular}
%\caption{Wireless Sensor Node Area Coverage Model.}
%\label{fig:cluster2}
%\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.

\indent  We can  calculate  the positions of the selected primary
points in the circle disk of the sensing range of a wireless sensor
node (see figure~\ref{fig1}) 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 * (0)) $\\
$X_7=( p_x + R_s *  (\frac{\sqrt{2}}{2}), p_y + R_s * (0))$\\
$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 * (\frac{\sqrt{2}}{2})) $\\
$X_{11}=( p_x + R_s * (\frac{\sqrt{2}}{2}), p_y + R_s * (\frac{\sqrt{2}}{2})) $\\
$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})) $.

 \begin{figure}[h!]
\centering
 \begin{multicols}{3}
\centering
\includegraphics[scale=0.20]{fig21.pdf}\\~ ~ ~ ~ ~(a)
\includegraphics[scale=0.20]{fig22.pdf}\\~ ~ ~ ~ ~(b)
\includegraphics[scale=0.20]{principles13.pdf}\\~ ~ ~ ~ ~(c)
%\includegraphics[scale=0.10]{fig25.pdf}\\~ ~ ~(d)
%\includegraphics[scale=0.10]{fig26.pdf}\\~ ~ ~(e)
%\includegraphics[scale=0.10]{fig27.pdf}\\~ ~ ~(f)
\end{multicols} 
\caption{Wireless Sensor Node represented by (a)5, (b)9 and (c)13 primary points respectively}
\label{fig1}
\end{figure}

\subsection{The Main Idea}
The   area  of  interest   can  be  divided using the
divide-and-conquer strategy into smaller areas  called subregions and
then  our coverage  protocol  will be  implemented  in each  subregion
simultaneously. Our DiLCO protocol works in rounds fashion as shown in figure~\ref{fig2}.
\begin{figure}[ht!]
\centering
\includegraphics[width=95mm]{FirstModel.pdf} % 70mm
\caption{DiLCO protocol}
\label{fig2}
\end{figure} 

Each round  is divided  into 4 phases  : Information  (INFO) Exchange,
Leader  Election, Decision,  and  Sensing.  For  each  round there  is
exactly one set cover responsible  for the sensing task.  This protocol is
more reliable  against an unexpected node failure  because it works
in rounds. 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 round starts, since a new set cover
will take  charge of the  sensing task in  the next round.  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
round.  However,   the  pre-sensing  phases   (INFO  Exchange,  Leader
Election,  Decision) are energy  consuming for  some nodes,  even when
they do not  join the network to monitor the  area. 
We define two types of packets to be used by our DiLCO protocol.
\begin{enumerate}[(a)] 
\item INFO packet: sent by each sensor node to all the nodes of it's subregion for information exchange.
\item ActiveSleep packet: sent by the leader to all the nodes in the same of it's subregion to inform them to be Active or Sleep during the sensing phase.
\end{enumerate}

There are four status for each sensor node in the network
\begin{enumerate}[(a)] 
\item LISTENING: Sensor is waiting for a decision (to be active or not)
\item COMPUTATION: Sensor applies the optimization process as leader
\item ACTIVE: Sensor is active
\item SLEEP: Sensor is turned off
\item COMMUNICATION: Sensor is transmitting or receiving packet
\end{enumerate}
Below, we describe each phase in more details.

\subsubsection{Information Exchange Phase}

Each sensor node $j$ sends its position, remaining energy $RE_j$, and
the number of neighbours  $NBR_j$ to all wireless sensor nodes in
its subregion by using an INFO packet and then listens to the packets
sent from  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 :

\subsubsection{Leader Election Phase}
This  step includes 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  round.  All the  sensor  nodes cooperate  to
select 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  in order  of priority  are: larger
number  of neighbours,  larger remaining  energy, and  then in  case of
equality, larger index. The pseudo-code for leader election phase is provided in Algorithm~1.
\begin{algorithm}                
  \KwIn{all the parameters related to information exchange}
  \KwOut{$node-id$ (: the id of the winner sensor node, which is the leader of current round)}
  \BlankLine
  \emph{Select the node(s) with higher $NBR_j$ and $ RE_j \geqslant E_{th}$} \; 
  
  \If{ there are more than two nodes with the same maximum $NBR_j$ }{
         \If{ there are more than two nodes with the same maximum $NBR_j$ and the same $RE_j$}{
         	\emph{ Select the node with higher id} \;
         }	 
         \Else{
             \emph{Select the node with maximum $RE_j$} \;
         }
   }
   \Else{
            \emph{ Select the node with higher $NBR_j$ } \;
    }
  
  \emph{return node-id} \;
\caption{LEADER ELECTION}
\label{alg:LEADER}

\end{algorithm}

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.  


\subsubsection{Decision phase}
The  WSNL will  solve an  integer  program (see  section~\ref{cp})  to
select which sensors will be  activated in the following sensing phase
to cover  the subregion.  WSNL will send  Active-Sleep packet  to each
sensor in the subregion based on the algorithm's results.


\subsubsection{Sensing phase}
Active  sensors  in the  round  will  execute  their sensing  task  to
preserve maximal  coverage in the  region of interest. We  will assume
that the cost  of keeping a node awake (or asleep)  for sensing task is
the same  for all wireless sensor  nodes in the  network.  Each sensor
will receive  an Active-Sleep  packet from WSNL  informing it  to stay
awake or to go to sleep  for a time  equal to  the period of  sensing until
starting a new round.

\subsection{DiLCO protocol Algorithm}
we first show the pseudo-code of DiLCO protocol, which is executed by each sensor in the subregion and then describe it in more detail. 

\begin{algorithm}                
 % \KwIn{all the parameters related to information exchange}
%  \KwOut{$winer-node$ (: the id of the winner sensor node, which is the leader of current round)}
  \BlankLine
  \emph{Initialize the sensor node and determine it's position and it's subregion} \; 
  
  \If{ $RE_j \geq E_{th}$ }{
     \emph{ Send and Receive INFO Packet to and from other nodes in the subregion}\;
      \emph{ Collect information and construct the list L for all nodes in the subregion}\;
      \emph{ $s_j.status$ = LISTENING}\;
       \If{ the received INFO Packet = No. of nodes in it's subregion -1  }{
         \emph{ LeaderID $\leftarrow$ \bf Algorithm~\ref{alg:LEADER}}\;
         \If{ $ s_j.ID = LeaderID $}{ 
          	\emph{Execute Integer Program Algorithm (Gbest) }\;
          	\For{$k\leftarrow 1$ \KwTo No. of nodes in subregion}{
          	    \If{$ s_j.ID \neq L_k$ }{
              		 \If{$ Gbest_k = 1$ }{
                		   \emph{ Send ActiveSleep() Packet with status = ACTIVE  }\;
             		  }
            		  \Else{\emph{Send ActiveSleep() Packet with status = SLEEP}\;}
            	}
            	\Else{
            	    	\If{$ Gbest_k = 1$ }{
            		    \emph{ $s_j.status$ = ACTIVE}\; 
            		    \emph{UPDATE Remaining Energy $RE_j $}\;
              		}
              		\Else{ 
              		      \emph{ $s_j.status$ = SLEEP}\;
              		      \emph{UPDATE Remaining Energy $RE_j $}\;
                		 }
            	  }
            		  
            }	  
           
         }
         \Else{ 
                \emph{Wait ActiveSleep() Packet from the Leader}\;
                 \If{received ActiveSleep().status = ACTIVE  }{ 
                  \emph{ $s_j.status$ = ACTIVE}\; 
            	  \emph{UPDATE Remaining Energy $RE_j $}\;
                 }
                 \Else{
                  \emph{ $s_j.status$ = SLEEP}\;
            	  \emph{UPDATE Remaining Energy $RE_j $}\;
                 }
         }  
       }

   }
   \Else { Exclude me from entering in the current round}
   
 %   \emph{return X} \;
\caption{DiLCO($s_j$)}
\label{alg:DMRCLOP}

\end{algorithm}

The DiLCO protocol work in rounds 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 DiLCO 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 to take the decision. 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 select the set of sensor nodes to take the mission of coverage during the sensing phase. The leader will send ActiveSleep packet to each sensor node in the subregion to inform him to it's status during the period of sensing, either Active or sleep until the starting of next round. Based on the decision, the leader as other nodes in subregion, either go to be active or go to be sleep 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, all the sensor nodes in the same subregion will start new round by executing the DiLCO protocol and the lifetime in the subregion will continue until all the sensor nodes are died or the network becomes disconnected in the subregion.

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

\indent   Our   model   is   based   on  the   model   proposed   by
\cite{pedraza2006} where the objective is  to find a maximum number of
disjoint  cover sets.   To accomplish  this goal,  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 $X_{j}$,  which  determine  the
activation of  sensor $j$ in the  sensing phase of the  round. We also
consider  primary points  as targets.   The set  of primary  points is
denoted by $P$ and the set of sensors by $J$.

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

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



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

The first group  of constraints indicates that some  primary point $p$
should be covered by at least one  sensor and, if it is not always the
case,  overcoverage  and  undercoverage  variables  help  balancing  the
restriction  equations by taking  positive values.  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.
 

\section{Simulation Results and Analysis}
\label{exp}
\subsection{Simulation framework, energy consumption model and performance metrics}
In this subsection, we conducted  a series of simulations to evaluate the
efficiency and the relevance of our protocol DiLCO,  using the  discrete event
simulator  OMNeT++  \cite{varga}. The simulation parameters are summarized in
Table~\ref{table3}

\begin{table}[ht]
\caption{Relevant parameters for network initializing.}
% title of Table
\centering
% used for centering table
\begin{tabular}{c|c}
% centered columns (4 columns)
      \hline
%inserts double horizontal lines
Parameter & Value  \\ [0.5ex]
   
%Case & Strategy (with Two Leaders) & Strategy (with One Leader) & Simple Heuristic \\ [0.5ex]
% inserts table
%heading
\hline
% inserts single horizontal line
Sensing  Field  & $(50 \times 25)~m^2 $   \\
% inserting body of the table
%\hline
Nodes Number &  50, 100, 150, 200 and 250~nodes   \\
%\hline
Initial Energy  & 500-700~joules  \\  
%\hline
Sensing Period & 60 Minutes \\
$E_{thr}$ & 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}

25 simulation runs are performed with different network topologies. The results presented hereafter are the average of these 25 runs. 
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.\\
  
Our DiLCO protocol is declined into five versions: DiLCO-2, DiLCO-4, DiLCO-8, DiLCO-16, and DiLCO-32, corresponding  to $2$, $4$, $8$, $16$ or $32$ subregions (leaders).  

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 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 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 mode & MCU   & Radio & Sensing & Power (mWs) \\ [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, the transition from mode 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. DiLCO protocol uses two types of packets for communication. The size of the INFO-Packet and Status-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 $[500-700]$.  Each  sensor  node will  not participate in the next round if its remaining energy is less than $E_{th}=36 Joules$, 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 mWs) 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.\\ 

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

\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, we treated the sensing fields as a grid, and used 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{$S$}} \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, $S$ 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 denoted by $Lifetime95$ (respectively  $Lifetime50$) as the amount of  time during which  the network  can  satisfy an area  coverage greater than $95\%$ (repectively $50\%$). 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}:}

 Energy Consumption (EC) can be seen as the total energy consumed by the sensors during the $Lifetime95$ or $Lifetime50$ divided by the number of rounds. The EC can be computed as follow: \\
 \begin{equation*}
\scriptsize
\mbox{EC} =  \frac{\mbox{$\sum\limits_{d=1}^D \left( E^c_d + E^l_d + E^a_d + E^s_d + E^p_d \right)$ }}{\mbox{$D$}}   .
\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: D is the number of rounds during $Lifetime95$ or $Lifetime50$. 
The total energy consumed by the sensors (EC) comes through taking into consideration four main energy factors, which are $E^c_d$, $E^l_d$, $E^a_d$, $E^s_d$ and $E^p_d$.
The energy consumption $E^c_d$ for wireless  communications  is  calculated by taking into account the  energy spent by  all the nodes while  transmitting and
receiving  packets during round $d$. The $E^l_d$ represents the energy consumed by all the sensors during the listening mode before taking the decision to go Active or Sleep in round $d$. $E^a_d$ and $E^s_d$  refer to energy consumed in the active mode or in the sleeping mode. The $E^p_d$ refers to energy consumed by the computation (processing) to solve the integer program.

%\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 Comparison for differnet subregions}
\label{sub1}
In this subsection, we are studied the performance of our DiLCO protocol for a different number of subregions (Leaders).
The DiLCO-1 protocol is a centralized approach on all the area of the interest, while  DiLCO-2, DiLCO-4, DiLCO-8, DiLCO-16 and DiLCO-32 are distributed on two, four, eight, sixteen, and thirty-two subregions respectively. We did not take the DiLCO-1 protocol in our simulation results because it need high execution time to give the decision leading to consume all it's energy before producing the solution for optimization problem.

\subsubsection{Coverage Ratio} 
In this experiment, Figure~\ref{fig3} shows the average coverage ratio for 150 deployed nodes.  
\parskip 0pt    
\begin{figure}[h!]
\centering
 \includegraphics[scale=0.5] {R1/CR.pdf} 
\caption{The impact of the number of rounds on the coverage ratio for 150 deployed nodes}
\label{fig3}
\end{figure} 

It can be seen that the DiLCO protocol (with 4, 8, 16 and 32 subregions) gives nearly similar coverage ratios during  the first thirty rounds.  
DiLCO-2 protocol gives near similar coverage ratio with other ones for first 10 rounds and then decreased until the died of the network in the round $18^{th}$ because it consume more energy with the effect of the network disconnection. 
As shown in the figure ~\ref{fig3}, as the number of subregions increases ,  the coverage preservation for area of interest increases for a larger number of rounds. Coverage ratio decreases when the number of rounds increases due to dead nodes. Although  some nodes are dead,
thanks to  DiLCO-8,  DiLCO-16 and  DiLCO-32 protocols,  other nodes are  preserved to  ensure the coverage. Moreover, when  we have a dense sensor network, it leads to maintain the  coverage for a larger number of rounds. DiLCO-8,  DiLCO-16 and  DiLCO-32 protocols are
slightly more efficient than other protocols, because they subdivides
the area of interest into 8, 16 and 32~subregions if one of the subregions becomes disconnected, the coverage may be still ensured in the remaining subregions.

\subsubsection{Active Sensors Ratio} 
 Figure~\ref{fig4} shows the average active nodes ratio for 150 deployed nodes.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{R1/ASR.pdf}  
\caption{The impact of the number of rounds on the active sensors ratio for 150 deployed nodes }
\label{fig4}
\end{figure} 
The results presented in figure~\ref{fig4} show  the increase in the number of subregions led to increase in the number of active nodes. The DiLCO-16 and DiLCO-32 protocols have a larger number of active nodes but it preserve the coverage for a larger number of rounds. The  advantage of the DiLCO-16, and DiLCO-32 protocols are that even if a network is disconnected in one subregion, the other ones usually continues the optimization process, and this extends the lifetime of the network.

\subsubsection{The percentage of stopped simulation runs}
Figure~\ref{fig6} illustrates the percentage of stopped simulation runs per round for 150 deployed nodes. 
\begin{figure}[h!]
\centering
\includegraphics[scale=0.43]{R1/SR.pdf} 
\caption{The percentage of stopped simulation runs compared to the number of rounds for 150 deployed nodes }
\label{fig6}
\end{figure} 

It can be observed that the DiLCO-2  is the approach which stops first because it applied the optimization on only two subregions for the area of interest that is why it is first exhibits network disconnections.
Thus, as explained previously, in case of the DiLCO-16 and DiLCO-32 with several subregions the optimization effectively continues as long as a network in a subregion is still connected. This longer partial coverage optimization participates in extending the network lifetime. 

\subsubsection{The Energy Consumption}
We measure the energy consumed by the sensors during the communication, listening, computation, active, and sleep modes for different network densities and compare it for different subregions.  Figures~\ref{fig95} and ~\ref{fig50} illustrate the energy consumption for different network sizes for $Lifetime95$ and $Lifetime50$. 

\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{R1/EC95.pdf} 
\caption{The Energy Consumption for Lifetime95}
\label{fig95}
\end{figure} 


The results show that  DiLCO-16 and DiLCO-32 are the most competitive from the energy consumption point of view but as the network size increase the energy consumption increase compared with DiLCO-2,  DiLCO-4 and DiLCO-8. The other approaches have a  high energy  consumption  due  to  the energy consumed during the different modes of the sensor node.\\
 
As shown in Figures~\ref{fig95} and ~\ref{fig50} , DiLCO-2 consumes more energy than the other versions of DiLCO, especially for large sizes of network. This is easy to understand since the bigger the number of sensors involved in the integer program, the larger the time computation to solve the optimization problem as well as the higher energy consumed during the communication.  
\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{R1/EC50.pdf} 
\caption{The Energy Consumption for Lifetime50}
\label{fig50}
\end{figure} 
In fact,  a distributed  method on the subregions greatly reduces the number of communications, the time of listening and computation so thanks to the partitioning of the initial network in several independent subnetworks. 

\subsubsection{Execution Time}
In this experiment, we study the the impact of the size of the network on the excution time of the our distributed optimization approach. Figure~\ref{fig8} gives the average execution times in seconds for the decision phase (solving of the optimization problem) during one round. They are given for the different approaches and various numbers of sensors. 
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 $\left( \frac{35330}{2} \times 6\right)$ and reported on Figure~\ref{fig8} for different network sizes.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{R1/T.pdf}  
\caption{Execution Time (in seconds)}
\label{fig8}
\end{figure} 


We can see from figure~\ref{fig8}, that the DiLCO-32 has very low execution times in comparison with other DiLCO versions, because it distributed on larger number of small subregions.  Conversely, the DiLCO-2 which requires to solve an optimization problem considering half  the nodes  in each subregion presents  high execution times.

The DiLCO-32 has more suitable times in the same time it turn on redundent nodes more.  We think that in distributed fashion the solving of the  optimization problem 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{The Network Lifetime}
In figure~\ref{figLT95} and \ref{figLT50}, network lifetime, $Lifetime95$ and $Lifetime50$ respectively, are illustrated for different  network sizes. 

\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{R1/LT95.pdf}  
\caption{The Network Lifetime for $Lifetime95$}
\label{figLT95}
\end{figure} 
We see that the DiLCO-2 results in execution times that quickly become unsuitable for a sensor network as well as the energy consumed during the communication seems to be huge because it is distributed over only two subregions.

As highlighted by figures~\ref{figLT95} and \ref{figLT50}, the network lifetime obviously
increases when the size of the network increases, with  our DiLCO-16 protocol
that leads to the larger lifetime improvement. By choosing the best 
suited nodes, for each round, to cover the area of interest and by
letting the other ones sleep in order to be used later in next rounds,
our DiLCO-16 protocol efficiently extends  the network lifetime because the benefit from the optimization with 16 subregions is better than the DiLCO-32  with 32 subregion.  DilCO-32 protocol puts in active mode a larger number of sensor nodes especially near the bordes of the subdivisions.

Comparison shows that the DiLCO-16 protocol, which uses 16 leaders, is the best one because it is used less number of active nodes during the network lifetime compared with DiLCO-32. It also  means that distributing the protocol in each 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.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{R1/LT50.pdf}  
\caption{The Network Lifetime for $Lifetime50$}
\label{figLT50}
\end{figure} 

\subsection{Performance Study for Primary Point Models}
\label{sub2}
In this subsection, we are studied the performance of the DiLCO~16 approach for a different primary point models. The objective of this comparison is to select the suitable primary point model to be used by our DiLCO protocol. 

In this comparisons, our DiLCO-16 protocol are used with five models which are called Model~1( With 5 Primary Points), Model~2 ( With 9 Primary Points), Model~3 ( With 13 Primary Points), Model~4 ( With 17 Primary Points), and Model~5 ( With 21 Primary Points). 
\subsubsection{Coverage Ratio} 
In this experiment, we Figure~\ref{fig33} shows the average coverage ratio for 150 deployed nodes.  
\parskip 0pt    
\begin{figure}[h!]
\centering
 \includegraphics[scale=0.5] {R2/CR.pdf} 
\caption{The impact of the number of rounds on the coverage ratio for 150 deployed nodes}
\label{fig33}
\end{figure} 
It is shown that all models provides a very near coverage ratios during the network lifetime, with very small superiority for the models with higher number of primary points.
Moreover, when the number of rounds increases, coverage
ratio produced by Model~3, Model~4 and Model~5 decreases in comparison with Model~1 and Model~2 due to the high energy consumption during the listening to take the decision after finishing optimization process for larger number of primary points. As shown in figure ~\ref{fig33}, Coverage ratio decreases when the number of rounds increases  due to dead nodes. Although  some nodes are dead,
thanks to  Model~2, which is slightly more efficient than other Models, because it is balanced between the number of rounds and the better coverage ratio in comparison with other Models.

\subsubsection{Active Sensors Ratio} 
 Figure~\ref{fig44} shows the average active nodes ratio for 150 deployed nodes.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{R2/ASR.pdf}  
\caption{The impact of the number of rounds on the active sensors ratio for 150 deployed nodes }
\label{fig44}
\end{figure} 

The  results presented  in figure~\ref{fig44}  show the  superiority of
the proposed  Model 1, in comparison with the other Models. The
model with less number of primary points uses less active nodes than the other models, which uses a more number of primary points to represent the area of the sensor. According to the results that presented in figure~\ref{fig33}, we observe that although the Model~1 continue to a larger number of rounds, but it has less coverage ratio compared with other models.The advantage of the Model~2 approach is to use less number of active nodes for each round compared with Model~3,  Model~4 and Model~5, and this led to continue for a larger number of rounds with extending the network lifetime. Model~2 has a better coverage ratio compared to Model~1 and acceptable number of rounds.


\subsubsection{The percentage of stopped simulation runs}
In this study, we want to show the effect of increasing the primary points on the number of stopped simulation runs for each round. Figure~\ref{fig66} illustrates the percentage of stopped simulation
runs per round for 150 deployed nodes. 

\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{R2/SR.pdf} 
\caption{The percentage of stopped simulation runs compared to the number of rounds for 150 deployed nodes }
\label{fig66}
\end{figure} 

As shown in Figure~\ref{fig66}, when the number of primary points are increased, the percentage of the stopped simulation runs per rounds is increased. The reason behind the increase is the increase in the sensors dead when the primary points increases. We are observed that the Model~1 is a better than other models because it conserve more energy by turn on less number of sensors during the sensing phase, but in the same time it preserve the coverage with a less coverage ratio in comparison with other models. Model~2 seems to be more suitable to be used in wireless sensor networks.


\subsubsection{The Energy Consumption}
In this experiment, we study the effect of increasing the primary points to represent the area of the sensor on the energy consumed by the wireless sensor network for different network densities.  Figures~\ref{fig2EC95} and ~\ref{fig2EC50} illustrate the energy consumption for different network sizes for $Lifetime95$ and $Lifetime50$.
\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{R2/EC95.pdf} 
\caption{The Energy Consumption with $95\%-Lifetime$}
\label{fig2EC95}
\end{figure} 
 
\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{R2/EC50.pdf} 
\caption{The Energy Consumption with $Lifetime50$}
\label{fig2EC50}
\end{figure} 

We see from the results presented in Figures~\ref{fig2EC95} and \ref{fig2EC50}, The energy consumed by the network for each round increases when the primary points increases, because the decision for optimization process will takes more time leads to consume more energy during the listening mode. The  results show  that the Model~1 is  the  most  competitive  from  the energy consumption point of view but the worst one from coverage ratio point of view. The other Models  have a high energy  consumption  due  to the increase in the primary points, which are led to increase the energy consumption during the listening mode before producing the solution by solving the optimization process. In fact, we see that the Model~2 is a good candidate to be used by wireless sensor network because It preserve a good coverage ratio and a suitable energy consumption in comparison with other models. 


\subsubsection{Execution Time}
In this experiment, we study the impact of the increase in primary points on the excution time of our DiLCO protocol. Figure~\ref{figt} gives the average execution times in seconds for the decision phase (solving of the optimization problem) during one round.  

\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{R2/T.pdf}  
\caption{The Execution Time(s) vs The Number of Sensors }
\label{figt}
\end{figure} 

They  are given for  the different primary point models and
various numbers of sensors. We can see from Figure~\ref{figt}, that the Model~1 has  lower execution time in comparison with other Models, because it used smaller number of primary points to represent the area of the sensor.  Conversely, the other primary point models  presents  higher  execution  times.
Moreover, the Model~2 has more suitable times, coverage ratio, and saving energy ratio leads to continue for a larger number of rounds extending the network lifetime. We  think that a good primary point model, this one that balances between the coverage ratio and the number of rounds during the lifetime of the network.

\subsubsection{The Network Lifetime}
Finally, we will study the effect of increasing the primary points on the lifetime of the network. In Figure~\ref{fig2LT95} and in Figure~\ref{fig2LT50}, network lifetime, $Lifetime95$ and $Lifetime50$ respectively, are illustrated for different network sizes. 

\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{R2/LT95.pdf}  
\caption{The Network Lifetime for $Lifetime95$}
\label{fig2LT95}
\end{figure} 


\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{R2/LT50.pdf}  
\caption{The Network Lifetime for $Lifetime50$}
\label{fig2LT50}
\end{figure} 


As  highlighted by figures~\ref{fig2LT95} and \ref{fig2LT50}, the network lifetime obviously
increases when  the size  of the network  increases, with  our Model~1 that leads to the larger lifetime improvement.
Comparison shows that the Model~1, which uses less number of primary points , is the best one because it is less energy consumption during the network lifetime. It is also the worst one from the point of view of coverage ratio. Our proposed Model~2 efficiently prolongs the network lifetime with a good coverage ratio in comparison with other models.
 
\subsection{Performance Comparison for Different Approaches}
Based on the results, which are conducted from previous two subsections, ~\ref{sub1} and \ref{sub2}, we are  found that Our DiLCO-16, and DiLCO-32 protocols with Model~2 are the best candidates to be compared with other two approches. The first approach, called DESK that proposed by ~\cite{ChinhVu}, which is a full distributed coverage algorithm. The second approach, called GAF ~\cite{xu2001geography}, consists in dividing the region into fixed squares.   During the  decision phase,  in  each square,  one sensor  is chosen to remain on during the sensing phase time. 

\subsubsection{Coverage Ratio} 
In this experiment, Figure~\ref{fig333} shows  the average coverage ratio for 150 deployed nodes. 
 
\parskip 0pt    
\begin{figure}[h!]
\centering
 \includegraphics[scale=0.5] {R3/CR.pdf} 
\caption{The coverage ratio for 150 deployed nodes}
\label{fig333}
\end{figure} 

It is shown that DESK and GAF provides a
a little better coverage ratio with 99.99\% and 99.91\% against 99.1\% and 99.2\% produced by DiLCO-16 and DiLCO-32 for the lowest number of rounds. This is due to the fact that our DiLCO protocol versions  put in sleep mode redundant sensors using optimization (which lightly decreases the coverage ratio) while there are more nodes are active in the case of DESK and GAF.

Moreover, when the number of rounds increases, coverage ratio produced by DESK and GAF protocols decreases. This is due to dead nodes. However, Our DiLCO-16 and DiLCO-32 protocols maintains almost a good coverage. This is because it optimize the coverage and the lifetime in wireless sensor network by selecting the best representative sensor nodes to take the reponsibilty of coverage during the sensing phase and this will leads to continue for a larger number of rounds and prolonging the network lifetime; although some nodes are dead, sensor activity scheduling of our protocol chooses other nodes to ensure the coverage of the area of interest. 

\subsubsection{Active Sensors Ratio} 
It is important to have as few active nodes as possible in each round,
in  order to  minimize the energy consumption and maximize the network lifetime. Figure~\ref{fig444} shows the average active nodes ratio for 150 deployed nodes. 

\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{R3/ASR.pdf}  
\caption{The active sensors ratio for 150 deployed nodes }
\label{fig444}
\end{figure} 

The results presented in figure~\ref{fig444} show the superiority of the proposed DiLCO-16 and DiLCO-32 protocols, in comparison with the other approaches.  We can observe that DESK and GAF have 37.5 \% and 44.5 \% active nodes and our DiLCO-16 and DiLCO-32 protocols competes perfectly with only 17.4 \%, 24.8 \% and 26.8 \%  active nodes for the first 14 rounds. Then as the number of rounds increases our DiLCO-16 and DiLCO-32 protocols have larger number of active nodes in comparison with DESK and GAF, especially from round $35^{th}$ because they give a better coverage ratio than other approaches. We see that the DESK and GAF have less number of active nodes beginning at the rounds $35^{th}$ and $32^{th}$ because there are many nodes are died due to the high energy consumption by the redundant nodes during the sensing phase. 

\subsubsection{The percentage of stopped simulation runs}
The results presented in this experiment, is to show the comparison of our DiLCO-16 and DiLCO-32 protocols with other two approaches from the point of view the stopped simulation runs per round.
Figure~\ref{fig666} illustrates the percentage of stopped simulation
runs per round for 150 deployed nodes. 
\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{R3/SR.pdf} 
\caption{The percentage of stopped simulation runs compared to the number of rounds for 150 deployed nodes }
\label{fig666}
\end{figure} 
It can be observed that the DESK is the approach, which stops first because it consumes more energy for communication as well as it turn on a large number of redundant nodes during the sensing phase. Our DiLCO-16 and DiLCO-32 protocols have less stopped simulation runs in comparison with DESK and GAF because it distributed the optimization on several subregions in order to optimize the coverage and the lifetime of the network by activating a less number of nodes during the sensing phase leading to extend the network lifetime and coverage preservation.The optimization effectively continues as long as a network in a subregion is still connected.

\subsubsection{The Energy Consumption}
In this experiment, we study the effect of the energy consumed by the wireless sensor network during the communication, computation, listening, active, and sleep modes for different network densities and compare it with other approaches. Figures~\ref{fig3EC95} and ~\ref{fig3EC50} illustrate the energy consumption for different network sizes for $Lifetime95$ and $Lifetime50$. 

\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{R3/EC95.pdf} 
\caption{The Energy Consumption with $95\%-Lifetime$}
\label{fig3EC95}
\end{figure} 

\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{R3/EC50.pdf} 
\caption{The Energy Consumption with $Lifetime50$}
\label{fig3EC50}
\end{figure} 

The results show that our DiLCO-16 and DiLCO-32 protocols 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 as well as the energy consumed during the different modes of sensor nodes. In fact,  a distributed  method on the subregions greatly reduces the number of communications and the time of listening so thanks to the partitioning of the initial network into several independent subnetworks. 

\subsubsection{The Network Lifetime}
In this experiment, we are observed the superiority of our DiLCO-16 and DiLCO-32 protocols against other two approaches in prolonging the network lifetime. In figures~\ref{fig3LT95} and \ref{fig3LT50}, network lifetime, $Lifetime95$ and $Lifetime50$ respectively, are illustrated for different network sizes.  

\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{R3/LT95.pdf}  
\caption{The Network Lifetime for $Lifetime95$}
\label{fig3LT95}
\end{figure}


\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{R3/LT50.pdf}  
\caption{The Network Lifetime for $Lifetime50$}
\label{fig3LT50}
\end{figure} 

As highlighted by figures~\ref{fig3LT95} and \ref{fig3LT50}, the network lifetime obviously
increases when the size of the network increases, with our DiLCO-16 and DiLCO-32 protocols
that leads to maximize the lifetime of the network compared with other approaches. 
By choosing the best suited nodes, for each round, by optimizing the coverage and lifetime of the network to cover the area of interest and by letting the other ones sleep in order to be used later in next rounds, our DiLCO-16 and DiLCO-32 protocols efficiently prolonges the network lifetime. 
Comparison shows that our DiLCO-16 and DiLCO-32 protocols, which are used distributed optimization over the subregions, 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 algorithm in each 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.

\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 a DiLCO protocol for optimizing the coverage and lifetime performances in each subregion.
The proposed protocol 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 representative active nodes that will optimize the network lifetime
while  taking the responsibility of covering the corresponding
subregion. The network lifetime in each subregion is divided into
rounds, each round consists  of four phases: (i) Information Exchange,
(ii) Leader Election, (iii) an optimization-based Decision in order to
select the  nodes remaining  active for  the  last phase,  and  (iv)
Sensing.  The  simulations show the relevance  of the proposed DiLCO
protocol in terms of lifetime, coverage ratio, active sensors ratio, energy consumption, execution time, and the number of stopped simulation runs due to network disconnection. Indeed, when
dealing with large and dense wireless sensor networks, a distributed
approach like the one we are proposed 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.

In future work, we plan to study  and propose a coverage optimization protocol, which
computes  all active sensor schedules in one time, using
optimization  methods. The round  will still consist of 4 phases, but the
  decision phase will compute the schedules for several sensing phases
  which, aggregated together, define a kind of meta-sensing phase.
The computation of all cover sets in one time is far more
difficult, but will reduce the communication overhead.
% use section* for acknowledgement
%\section*{Acknowledgment}




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