Processing of the paper completed (except section 2)

[Sensornets15.git] / Example.tex

diff --git a/Example.tex b/Example.tex
index abb42a49883b295630f4fb006bc3ebb526bf9d43..08ab194a3105ffded45a1cadf830bca3d73fece1 100644 (file)
--- a/Example.tex
+++ b/Example.tex
@@ -64,50 +64,49 @@ Optimization, Scheduling.}
 \section{\uppercase{Introduction}}
 \label{sec:introduction}
 \noindent 
 \section{\uppercase{Introduction}}
 \label{sec:introduction}
 \noindent 

-Energy efficiency  is a crucial issue  in wireless sensor  networks since sensor
-nodes drain their  energy from batteries. In fact,  strong constraints on energy
+Energy efficiency is  a crucial issue in wireless  sensor networks since sensory

 consumption,  in order  to maximize  the network  lifetime, represent  the major
 difficulty when designing WSNs. As a consequence, one of the scientific research
 challenges in  WSNs, which has  been addressed by  a large amount  of literature
 during the  last few  years, is  the design of  energy efficient  approaches for
 consumption,  in order  to maximize  the network  lifetime, represent  the major
 difficulty when designing WSNs. As a consequence, one of the scientific research
 challenges in  WSNs, which has  been addressed by  a large amount  of literature
 during the  last few  years, is  the design of  energy efficient  approaches for

-coverage and  connectivity~\cite{conti2014mobile}. Coverage reflects  how well a
-sensor field  is monitored. The  most discussed coverage problems  in literature
+coverage and connectivity~\cite{conti2014mobile}.   Coverage reflects how well a
+sensor field is  monitored.  The most discussed coverage  problems in literature

 can  be classified into  three types  \cite{li2013survey}: area  coverage (where
 every point inside an area is  to be monitored), target coverage (where the main
 objective is to  cover only a finite number of  discrete points called targets),
 and  barrier coverage (to  prevent intruders  from entering  into the  region of
 interest). On the one  hand we want to monitor the area  of interest in the most
 efficient way~\cite{Nayak04}. On the other hand we want to use as less energy as
 can  be classified into  three types  \cite{li2013survey}: area  coverage (where
 every point inside an area is  to be monitored), target coverage (where the main
 objective is to  cover only a finite number of  discrete points called targets),
 and  barrier coverage (to  prevent intruders  from entering  into the  region of
 interest). On the one  hand we want to monitor the area  of interest in the most
 efficient way~\cite{Nayak04}. On the other hand we want to use as less energy as

-possible. % TO BE CONTINUED
-Sensor nodes runs on batteries with limited capacities~\cite{Sudip03}
-and  it  is  impossible,  difficult  or expensive  to  recharge  and/or  replace
-batteries  in remote,  hostile, or  unpractical environments.  Therefore,  it is
-desired that  the WSNs  are deployed with  high densities  so as to  exploit the
-overlapping sensing regions  of some sensor nodes to save  energy by turning off
-some of them during the sensing phase to prolong the network lifetime.
-
-In this paper we concentrate on  the area coverage problem with the objective of
-maximizing the network lifetime by using DiLCO protocol to maintain the coverage
-and  to improve  the lifetime  in WSNs.  The area  of interest  is  divided into
-subregions using divide-and-conquer method and an activity scheduling for sensor
-nodes is planned by the elected leader  in each subregion. In fact, the nodes in
-a subregion can be  seen as a cluster where each node  sends sensing data to the
-cluster   head  or   the  sink   node.    Furthermore,  the   activities  in   a
-subregion/cluster can  continue even  if another cluster  stops due to  too many
-node failures.  Our DiLCO protocol considers periods, where a period starts with
-a discovery phase  to exchange information between sensors  of the subregion, in
-order to choose in a suitable manner a sensor node (the leader) to carry out the
-coverage  strategy.  Our  DiLCO protocol  involves solving  an  integer program,
-which  provides the  activation of  the  sensors for  the sensing  phase of  the
-current period.
+possible.  Sensor nodes  are  battery-powered  with no  means  of recharging  or
+replacing, usually due to environmental (hostile or unpractical environments) or
+cost reasons.  Therefore, it  is desired  that the WSNs  are deployed  with high
+densities so as to exploit the  overlapping sensing regions of some sensor nodes
+to save energy by  turning off some of them during the  sensing phase to prolong
+the network lifetime.
+
+In this  paper we design  a protocol that  focuses on the area  coverage problem
+with  the objective  of maximizing  the network  lifetime. Our  proposition, the
+DiLCO protocol,  maintains the coverage and  improves the lifetime  in WSNs. The
+area of  interest is  first divided into  subregions using  a divide-and-conquer
+algorithm and  an activity scheduling  for sensor nodes  is then planned  by the
+elected leader in each subregion. In fact,  the nodes in a subregion can be seen
+as a cluster where each node sends  sensing data to the cluster head or the sink
+node.  Furthermore, the  activities in a subregion/cluster can  continue even if
+another cluster  stops due to too  many node failures.  Our Distributed Lifetime
+Coverage Optimization (DILCO) protocol  considers periods, where a period starts
+with  a  discovery phase  to  exchange information  between  sensors  of a  same
+subregion, in order to choose in a suitable manner a sensor node (the leader) to
+carry out the coverage strategy. In each subregion the activation of the sensors
+for the  sensing phase of the current  period is obtained by  solving an integer
+program.

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

-results. The paper ends with conclusions and some suggestions for futher work in
-Section~\ref{sec:Conclusion and Future Works}.
+results. The paper  ends with conclusions and some  suggestions for further work
+in Section~\ref{sec:Conclusion and Future Works}.

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


@@ -201,34 +200,55 @@ The main contributions of our DiLCO Protocol can be summarized as follows:
 
 \fi
 
 
 \fi
 

-\section{ The DiLCO Protocol Description}
+\section{\uppercase{Description of the DiLCO protocol}}

 \label{sec:The DiLCO Protocol Description}
 
 \label{sec:The DiLCO Protocol Description}
 

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

 \fi
 \fi

-\subsection{ Assumptions and Models}
-\noindent 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. 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.
-
-\indent Instead of working with the coverage area, we consider for each
-sensor a set of points called  primary points~\cite{idrees2014coverage}. We also assume that the
-sensing disk defined  by a sensor is covered if all the primary points of
-this sensor are covered.
+
+\subsection{Assumptions and models}
+
+\noindent  We consider  a sensor  network composed  of static  nodes distributed
+independently and uniformly at random.  A high density deployment ensures a high
+coverage ratio of the interested area at the starting. The nodes are supposed to
+have homogeneous characteristics from a  communication and a processing point of
+view, whereas they  have heterogeneous energy provisions.  Each  node has access
+to its location thanks,  either to a hardware component (like a  GPS unit), or a
+location discovery algorithm. 
+
+\indent We consider a boolean disk  coverage model which is the most widely used
+sensor coverage  model in the  literature. Thus, since  a sensor has  a constant
+sensing range $R_s$, every space points  within a disk centered at a sensor with
+the radius of  the sensing range is said  to be covered by this  sensor. We also
+assume  that  the communication  range  $R_c \geq  2R_s$.   In  fact, Zhang  and
+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.
+
+\indent  For  each  sensor  we  also  define a  set  of  points  called  primary
+points~\cite{idrees2014coverage} to  approximate the area  coverage it provides,
+rather  than  working  with  a   continuous  coverage.   Thus,  a  sensing  disk
+corresponding to  a sensor node is covered  by its neighboring nodes  if all its
+primary points are covered. Obviously,  the approximation of coverage is more or
+less accurate according to the number of primary points.

 
 \iffalse
 By  knowing the  position (point  center: ($p_x,p_y$))  of  a wireless
 
 \iffalse
 By  knowing the  position (point  center: ($p_x,p_y$))  of  a wireless


@@ -274,11 +294,13 @@ $X_{13}=( p_x + R_s * (0), p_y + R_s * (\frac{-\sqrt{2}}{2})) $.
 
 \fi
 
 
 \fi
 

-\subsection{The Main Idea}
-\noindent 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 periods fashion as shown in figure~\ref{fig2}.
+\subsection{The main idea}
+\label{main_idea}
+
+\noindent We start  by applying a divide-and-conquer algorithm  to partition the
+area of interest  into smaller areas called subregions and  then our protocol is
+executed   simultaneously  in   each   subregion.
+

 \begin{figure}[ht!]
 \centering
 \includegraphics[width=75mm]{FirstModel.pdf} % 70mm
 \begin{figure}[ht!]
 \centering
 \includegraphics[width=75mm]{FirstModel.pdf} % 70mm


@@ -286,54 +308,64 @@ simultaneously. Our DiLCO protocol works in periods fashion as shown in figure~\
 \label{fig2}
 \end{figure} 
 
 \label{fig2}
 \end{figure} 
 

-%Modifier la figure pour faire apparaitre des periodes et dans le schema en bleu, indiquer sensing round au lieu de sensing tout seul.
-
-Each period  is divided  into 4 phases  : Information  (INFO) Exchange,
-Leader  Election, Decision,  and  Sensing.  For  each  period 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 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, since a new set cover
-will take charge of the sensing task in the next period.  The energy
-consumption and some other constraints can easily be taken into
-account since the sensors can update and then exchange  their
-information (including their residual energy) at the beginning of each
-period.  However,   the  pre-sensing  phases   (INFO  Exchange,  Leader
-Election,  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.
+As  shown  in Figure~\ref{fig2},  the  proposed  DiLCO  protocol is  a  periodic
+protocol where  each period is  decomposed into 4~phases:  Information Exchange,
+Leader Election ,  Decision, and Sensing. For each period  there will be exactly
+one  cover  set  in charge  of  the  sensing  task.   A periodic  scheduling  is
+interesting  because it  enhances the  robustness  of the  network against  node
+failures. First,  a node  that has not  enough energy  to complete a  period, or
+which fails before  the decision is taken, will be  excluded from the scheduling
+process. Second,  if a node  fails later, whereas  it was supposed to  sense the
+region  of interest,  it will  only  affect the  quality of  coverage until  the
+definition of a new cover set  in the next period.  Constraints, like the energy
+consumption, can be easily taken into consideration since the sensors can update
+and exchange their  information during the first phase.  Let  us notice that the
+phases  before  the sensing  one  (Information  Exchange,  Leader Election,  and
+Decision) are  energy consuming for all the  nodes, even nodes that  will not be
+retained by the leader to keep watch over the corresponding area.
+
+During the execution of the DiLCO protocol, two kinds of packets will be used:

 %\begin{enumerate}[(a)]
 \begin{itemize} 
 %\begin{enumerate}[(a)]
 \begin{itemize} 

-\item INFO packet: sent by each sensor node to all the nodes inside a same subregion for information exchange.
-\item ActiveSleep packet: sent by the leader to all the nodes in its subregion to inform them to be Active or Sleep during the sensing phase.
+\item INFO  packet: sent  by each  sensor node to  all the  nodes inside  a same
+  subregion for information exchange.
+\item ActiveSleep packet:  sent by the leader to all the  nodes in its subregion
+  to inform them to be stay Active or to go Sleep during the sensing phase.

 \end{itemize}
 %\end{enumerate}
 \end{itemize}
 %\end{enumerate}

-
-There are five status for each sensor node in the network :
+and each sensor node will have five possible status in the network:

 %\begin{enumerate}[(a)] 
 \begin{itemize} 
 %\begin{enumerate}[(a)] 
 \begin{itemize} 

-\item LISTENING: Sensor is waiting for a decision (to be active or not)
-\item COMPUTATION: Sensor applies the optimization process as leader
-\item ACTIVE: Sensor is active
-\item SLEEP: Sensor is turned off
-\item COMMUNICATION: Sensor is transmitting or receiving packet
+\item LISTENING: sensor is waiting for a decision (to be active or not);
+\item COMPUTATION: sensor applies the optimization process as leader;
+\item ACTIVE: sensor is active;
+\item SLEEP: sensor is turned off;
+\item COMMUNICATION: sensor is transmitting or receiving packet.

 \end{itemize}
 %\end{enumerate}
 \end{itemize}
 %\end{enumerate}

-%Below, we describe each phase in more details.
-Algorithm 1 gives a brief description of the protocol applied by each sensor node (denoted by $s_j$ for a sensor node indexed by $j$).
-Initially, the sensor node checks its remaining energy in order to participate in the current period. After that, all the sensors collect position coordinates, remaining energy $RE_j$, sensor node id, and the number of its one-hop live neighbors during the information exchange. 
-Then all the sensor nodes in the same subregion will select the leader based on the received informations. The selection criteria for the leader in order  of priority  are: larger number  of neighbours,  larger remaining  energy, and  then in  case of equality, larger index. After that, if the sensor node is leader, it will execute the integer program algorithm (see section~\ref{cp}) which provides a set of sensors planned to be active in the sensing round. As leader, it will send an Active-Sleep packet to each sensor in the same subregion to indicate it if it has to be active or not. On the contrary, if the sensor is not the leader, it will wait for the Active-Sleep packet to know its state for the sensing round.
-

 
 

+An outline of the  protocol implementation is given by Algorithm~\ref{alg:DiLCO}
+which describes  the execution of  a period  by a node  (denoted by $s_j$  for a
+sensor  node indexed by  $j$). At  the beginning  a node  checks whether  it has
+enough energy to stay active during the next sensing phase. If yes, it exchanges
+information  with  all the  other  nodes belonging  to  the  same subregion:  it
+collects from each node its position coordinates, remaining energy ($RE_j$), ID,
+and  the number  of  one-hop neighbors  still  alive. Once  the  first phase  is
+completed, the nodes  of a subregion choose a leader to  take the decision based
+on  the  following  criteria   with  decreasing  importance:  larger  number  of
+neighbors, larger remaining energy, and  then in case of equality, larger index.
+After that,  if the sensor node is  leader, it will execute  the integer program
+algorithm (see Section~\ref{cp})  which provides a set of  sensors planned to be
+active in the next sensing phase. As leader, it will send an Active-Sleep packet
+to each sensor  in the same subregion to  indicate it if it has to  be active or
+not.  Alternately, if  the  sensor  is not  the  leader, it  will  wait for  the
+Active-Sleep packet to know its state for the coming sensing phase.

 
 \iffalse
 \subsubsection{Information Exchange Phase}
 
 Each sensor node $j$ sends its position, remaining energy $RE_j$, and
 
 \iffalse
 \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
+the number of neighbors  $NBR_j$ 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
 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


@@ -348,7 +380,7 @@ 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
 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
+number  of neighbors,  larger remaining  energy, and  then in  case of

 equality, larger index. 
 
 \subsubsection{Decision phase}
 equality, larger index. 
 
 \subsubsection{Decision phase}


@@ -359,23 +391,23 @@ sensor in the subregion based on the algorithm's results.
 
 
 \subsubsection{Sensing phase}
 
 
 \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. Algorithm 1, which
-will be executed by each node at the beginning of a round, explains how the
-Active-Sleep packet is obtained.
+
+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.  Algorithm 1, which will be executed by each
+node  at the  beginning of  a  round, explains  how the  Active-Sleep packet  is
+obtained.

 
 \fi
 
 
 \iffalse
 \subsection{DiLCO protocol Algorithm}
 
 \fi
 
 
 \iffalse
 \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. 
-\fi
+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.  \fi

 
 \begin{algorithm}[h!]                
  % \KwIn{all the parameters related to information exchange}
 
 \begin{algorithm}[h!]                
  % \KwIn{all the parameters related to information exchange}


@@ -425,22 +457,20 @@ The sensor node enter in listening mode waiting to receive ActiveSleep packet fr
 \fi
 
 
 \fi
 
 

-\section{Coverage problem formulation}
+\section{\uppercase{Coverage problem formulation}}

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

 \begin{equation}
 \alpha_{jp} = \left \{ 
 \begin{array}{l l}
 \begin{equation}
 \alpha_{jp} = \left \{ 
 \begin{array}{l l}


@@ -450,8 +480,8 @@ indicator function of whether the point $p$ is covered, that is:
 \end{array} \right.
 %\label{eq12} 
 \end{equation}
 \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:
+The  number of  active sensors  that cover  the primary  point $p$  can  then be
+computed by $\sum_{j \in J} \alpha_{jp} * X_{j}$ where:

 \begin{equation}
 X_{j} = \left \{ 
 \begin{array}{l l}
 \begin{equation}
 X_{j} = \left \{ 
 \begin{array}{l l}


@@ -483,7 +513,7 @@ U_{p} = \left \{
 \label{eq14} 
 \end{equation}
 
 \label{eq14} 
 \end{equation}
 

-\noindent Our coverage optimization problem can then be formulated as follows
+\noindent Our coverage optimization problem can then be formulated as follows:

 \begin{equation} \label{eq:ip2r}
 \left \{
 \begin{array}{ll}
 \begin{equation} \label{eq:ip2r}
 \left \{
 \begin{array}{ll}


@@ -500,38 +530,32 @@ X_{j} \in \{0,1\}, &\forall j \in J
 \right.
 \end{equation}
 
 \right.
 \end{equation}
 

-
-

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

   $p$ is being covered (1 if not covered and 0 if covered).
 \end{itemize}
 
   $p$ 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{\uppercase{Simulation Results and Analysis}}  
+The first group  of constraints indicates that some primary  point $p$ should be
+covered by at least  one sensor and, if it is not  always the case, overcoverage
+and undercoverage  variables help balancing the restriction  equations by taking
+positive values. Two objectives can be noticed in our model. First, we limit the
+overcoverage of primary  points to activate as few  sensors as possible. Second,
+to  avoid   a  lack  of  area   monitoring  in  a  subregion   we  minimize  the
+undercoverage. Both  weights $w_\theta$  and $w_U$ must  be carefully  chosen in
+order to  guarantee that the  maximum number of  points are covered  during each
+period.
+
+\section{\uppercase{Protocol evaluation}}  

 \label{sec:Simulation Results and Analysis}
 \label{sec:Simulation Results and Analysis}

-\noindent \subsection{Simulation Framework}
-In this subsection, we conducted a series of simulations to evaluate the
-efficiency and the relevance of our DiLCO protocol, using the discrete event
-simulator OMNeT++  \cite{varga}. The simulation parameters are summarized in
-Table~\ref{table3}.
+\noindent \subsection{Simulation framework}
+
+To assess the performance of our DiLCO protocol, we have used the discrete
+event simulator OMNeT++ \cite{varga} to run different series of simulations.
+Table~\ref{table3} gives the chosen parameters setting.

 
 \begin{table}[ht]
 \caption{Relevant parameters for network initializing.}
 
 \begin{table}[ht]
 \caption{Relevant parameters for network initializing.}


@@ -570,29 +594,36 @@ $w_{U}$ & $|P|^2$
 % is used to refer this table in the text
 \end{table}
 
 % is used to refer this table in the text
 \end{table}
 

-We  performed  simulations for five different densities varying from 50 to 250~nodes. Experimental results are the average obtained from 25 randomly generated networks (25 for each network density) 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 a high coverage ratio.\\
-
-We first concentrate on the required number of subregions making effective our protocol. Thus 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}. 
+Simulations with five  different node densities going from  50 to 250~nodes were
+performed  considering  each  time  25~randomly generated  networks,  to  obtain
+experimental results  which are relevant. The  nodes are deployed on  a field of
+interest of $(50 \times 25)~m^2 $ in such a way that they cover the field with a
+high coverage ratio.
+
+We chose as energy consumption model the one proposed proposed by~\cite{ChinhVu}
+and based on ~\cite{raghunathan2002energy} with slight modifications. The energy
+consumed by  the communications  is added  and the part  relative to  a variable
+sensing range is removed. We also assume that the nodes have the characteristics
+of the  Medusa II sensor  node platform \cite{raghunathan2002energy}.   A sensor
+node typically  consists of  four units: a  MicroController Unit, an  Atmels AVR
+ATmega103L in  case of Medusa II,  to perform the  computations; a communication
+(radio) unit able to send and  receive messages; a sensing unit to collect data;
+a power supply  which provides the energy consumed by  node. Except the battery,
+all the other unit  can be be switched off to save  energy according to the node
+status.   Table~\ref{table4} summarizes  the energy  consumed (in  milliWatt per
+second) by a node for each of its possible status.

 
 \begin{table}[ht]
 
 \begin{table}[ht]

-\caption{The Energy Consumption Model}
+\caption{Energy consumption model}

 % title of Table
 \centering
 % used for centering table
 % title of Table
 \centering
 % used for centering table

+{\scriptsize

 \begin{tabular}{|c|c|c|c|c|}
 % centered columns (4 columns)
       \hline
 %inserts double horizontal lines
 \begin{tabular}{|c|c|c|c|c|}
 % centered columns (4 columns)
       \hline
 %inserts double horizontal lines

-Sensor mode & MCU   & Radio & Sensing & Power (mW) \\ [0.5ex]
+Sensor status & MCU   & Radio & Sensing & Power (mW) \\ [0.5ex]

 \hline
 % inserts single horizontal line
 Listening & ON & ON & ON & 20.05 \\
 \hline
 % inserts single horizontal line
 Listening & ON & ON & ON & 20.05 \\


@@ -607,33 +638,50 @@ Computation & ON & ON & ON & 26.83 \\
 %\multicolumn{4}{|c|}{Energy needed to send/receive a 1-bit} & 0.2575\\
  \hline
 \end{tabular}
 %\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}
 
 
 \label{table4}
 % is used to refer this table in the text
 \end{table}
 

-For the sake of simplicity we ignore the energy needed to turn on the
-radio, to start up the sensor node, the transition 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 its 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 energy needed to send or receive a 1-bit is equal to $0.2575 mW$.
-
-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 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.\\ 
-
-
-In the simulations, we introduce the following performance metrics to evaluate the efficiency of our approach: 
+Less  influent  energy consumption  sources  like  when  turning on  the  radio,
+starting the sensor node, changing the status of a node, etc., will be neglected
+for the  sake of simplicity. Each node  saves energy by switching  off its radio
+once it has  received its decision status from the  corresponding leader (it can
+be itself).  As explained previously in subsection~\ref{main_idea}, two kinds of
+packets  for communication  are  considered  in our  protocol:  INFO packet  and
+ActiveSleep  packet. To  compute the  energy  needed by  a node  to transmit  or
+receive such  packets, we  use the equation  giving the  energy spent to  send a
+1-bit-content   message  defined   in~\cite{raghunathan2002energy}   (we  assume
+symmetric  communication costs), and  we set  their respective  size to  112 and
+24~bits. The energy required to send or receive a 1-bit is equal to $0.2575 mW$.
+
+Each node has an initial energy level, in Joules, which is randomly drawn in the
+interval  $[500-700]$.   If  it's  energy   provision  reaches  a   value  below
+$E_{th}=36$~Joules, the minimum  energy needed for a node  to stay active during
+one period,  it will no  more participate in  the coverage task. 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 active during at most 20 rounds.
+
+In the simulations,  we introduce the following performance  metrics to evaluate
+the efficiency of our approach:

 
 %\begin{enumerate}[i)]
 \begin{itemize}
   
 
 %\begin{enumerate}[i)]
 \begin{itemize}
   

-\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:
+\item {{\bf Coverage Ratio (CR)}:} it measures how well the WSN is able to 
+  observe the area of interest. In our case, we discretized the sensor field
+  as a regular grid, which yields the following equation to compute the
+  coverage ratio: 

 \begin{equation*}
 \scriptsize
 \mbox{CR}(\%) = \frac{\mbox{$n$}}{\mbox{$N$}} \times 100.
 \end{equation*}
 \begin{equation*}
 \scriptsize
 \mbox{CR}(\%) = \frac{\mbox{$n$}}{\mbox{$N$}} \times 100.
 \end{equation*}

-where  $n$ is the number of covered grid points by the active sensors of all subregions during the current sensing phase and $N$ is total number of grid points in the sensing field of the network. In our simulation $N = 51 \times 26 = 1326$ grid points.
+where  $n$ is  the number  of covered  grid points  by active  sensors  of every
+subregions during  the current  sensing phase  and $N$ is  total number  of grid
+points in  the sensing field. In  our simulations, we have  a layout of  $N = 51
+\times 26 = 1326$ grid points.

 %The accuracy of this method depends on the distance between grids. In our
 %simulations, the sensing field has been divided into 50 by 25 grid points, which means
 %there are $51 \times 26~ = ~ 1326$ points in total.
 %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.


@@ -652,35 +700,36 @@ Where: $A_r^t$ is the number of active sensors in the subregion $r$ during round
 
 \fi
 
 
 \fi
 

-\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 periods. The EC can be computed as follow: \\
- \begin{equation*}
-\scriptsize
-\mbox{EC} = \frac{\sum\limits_{m=1}^{M} \left( E^{\mbox{com}}_m+E^{\mbox{list}}_m+E^{\mbox{comp}}_m  + E^{a}+E^{s} \right)}{M_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 $M$ corresponds to the number of  periods.  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, $E^a_{m}$ and $E^s_{m}$  indicate the energy consumed by the whole network in the sensing round.
+\item {{\bf  Energy Consumption}:}  energy consumption (EC)  can be seen  as the
+  total   energy   consumed   by   the   sensors   during   $Lifetime_{95}$   or
+  $Lifetime_{50}$, divided  by the number of periods.  Formally, the computation
+  of EC can be expressed as follows:
+  \begin{equation*}
+    \scriptsize
+    \mbox{EC} = \frac{\sum\limits_{m=1}^{M} \left( E^{\mbox{com}}_m+E^{\mbox{list}}_m+E^{\mbox{comp}}_m  
+      + E^{a}_m+E^{s}_m \right)}{M},
+  \end{equation*}
+
+where $M$  corresponds to the number  of periods.  The total  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, $E^a_{m}$ and $E^s_{m}$ indicate  the energy consumed
+by the whole network in the sensing phase (active and sleeping nodes).
+
+\item {{\bf Network Lifetime}:} we define the network lifetime as the time until
+  the  coverage  ratio  drops  below  a  predefined  threshold.   We  denote  by
+  $Lifetime_{95}$ (respectively $Lifetime_{50}$) the amount of time during which
+  the  network can  satisfy an  area coverage  greater than  $95\%$ (respectively
+  $50\%$). We assume that the sensor  network can fulfill its task until all its
+  nodes have  been drained of their  energy or it  becomes disconnected. Network
+  connectivity  is crucial because  an active  sensor node  without connectivity
+  towards a base  station cannot transmit any information  regarding an observed
+  event in the area that it monitors.

 
 \iffalse 
 \item {{\bf Execution Time}:} a  sensor  node has  limited  energy  resources  and computing  power,
 
 \iffalse 
 \item {{\bf Execution Time}:} a  sensor  node has  limited  energy  resources  and computing  power,


@@ -698,84 +747,145 @@ disconnected (some nodes are dead and are not able to send information to the ba
 %\end{enumerate}
 
 
 %\end{enumerate}
 
 

-%\subsection{Performance Analysis for differnet subregions}
-\subsection{Performance Analysis}
+%\subsection{Performance Analysis for different subregions}
+\subsection{Performance analysis}

 \label{sub1}
 \label{sub1}

-In this subsection, we study the performance of our DiLCO protocol for  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 do not take into account the DiLC0-1 protocol in our simulation results because it requires  high execution time to solve the integer program and thus it is too costly in term of energy.
-
-Our method is compared with other two approaches. The first approach, called DESK and proposed by ~\cite{ChinhVu}  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. 

 
 

+In this subsection, we first focus  on the performance of our DiLCO protocol for
+different numbers  of subregions.  We consider partitions  of the WSN  area into
+$2$, $4$, $8$, $16$, and $32$ subregions. Thus the DiLCO protocol is declined in
+five versions:  DiLCO-2, DiLCO-4,  DiLCO-8, DiLCO-16, and  DiLCO-32. Simulations
+without  partitioning  the  area  of  interest,  case  which  corresponds  to  a
+centralized  approach, are  not presented  because they  require  high execution
+times to solve the integer program and therefore consume too much energy.
+
+We compare our protocol to two  other approaches. The first one, called DESK and
+proposed  by ~\cite{ChinhVu}  is a  fully distributed  coverage  algorithm.  The
+second one, called GAF  ~\cite{xu2001geography}, consists in dividing the region
+into fixed  squares.  During the decision  phase, in each square,  one sensor is
+chosen to remain active during the sensing phase.
+
+\subsubsection{Coverage ratio} 
+
+Figure~\ref{fig3} shows  the average coverage  ratio for 150 deployed  nodes. It
+can  be seen  that both  DESK and  GAF provide  a little  better  coverage ratio
+compared to DiLCO  in the first thirty periods. This can  be easily explained by
+the number of  active nodes: the optimization process  of our protocol activates
+less nodes  than DESK  or GAF, resulting  in a  slight decrease of  the coverage
+ratio. In case of DiLCO-2  (respectively DiLCO-4), the coverage ratio exhibits a
+fast decrease with  the number of periods and reaches zero  value in period {\bf
+  X} (respectively {\bf Y}), whereas the  other versions of DiLCO, DESK, and GAF
+ensure a coverage  ratio above 50\% for subsequent periods.  We believe that the
+results obtained with  these two methods can be explained  by a high consumption
+of energy and we will check this assumption in the next subsection.
+
+Concerning  DiLCO-8, DiLCO-16,  and  DiLCO-32,  these methods  seem  to be  more
+efficient than DESK  and GAF, since they can provide the  same level of coverage
+(except in the first periods where  DESK and GAF slightly outperform them) for a
+greater number  of periods. In fact, when  our protocol is applied  with a large
+number of subregions (from 8 to 32~regions), it activates a restricted number of
+nodes, and thus allow to extend the network lifetime.

 
 

-\subsubsection{Coverage Ratio} 
-Figure~\ref{fig3} shows the average coverage ratio for 150 deployed nodes.  

 \parskip 0pt    
 \parskip 0pt    

-\begin{figure}[h!]
+\begin{figure}[t!]

 \centering
  \includegraphics[scale=0.45] {R/CR.pdf} 
 \centering
  \includegraphics[scale=0.45] {R/CR.pdf} 

-\caption{The Coverage Ratio}
+\caption{Coverage ratio}

 \label{fig3}
 \end{figure} 
 
 \label{fig3}
 \end{figure} 
 

-Figure~\ref{fig3} shows that DESK and GAF provide a
-a little better coverage ratio compared to DiLCO in the first thirty periods. This is due to the fact that our DiLCO protocol versions  put in sleep mode some sensors through optimization process (which slightly decreases the coverage ratio) while there are more active nodes  with DESK or GAF. With DiLCO-2 (respectively DiLCO-4), the coverage ratio decreases rapidly to reach zero value in period ... (respectively in period ....) whereas other methods guarantee a coverage ratio greater than $50\%$ after this period. We believe that the results obtained with these two methods can be explained by a high consumption of energy
-and we will check this assumption in the next paragraph. Concerning DiLCO-8, DiLCO-16 and DiLCO-32, these methods seem to be more efficient than DESK and GAF because they can provide the same level of coverage (except in the first periods, slightly lower) for a greater number of periods. Unlike other methods, their strategy enables to activate a restricted number of nodes, and thus extends the lifetime of the network.

 %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 periods. Coverage ratio decreases when the number of periods 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.%
 
 %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 periods. Coverage ratio decreases when the number of periods 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{Energy consumption}

 
 

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

 
 

-\subsubsection{The Energy Consumption}
-Based on previous results in figure~\ref{fig3}, we keep DiLCO-16 and  DiLCO-32  and we compare their performances in terms of energy consumption with the two other approaches. We measure the energy consumed by the sensors during the communication, listening, computation, active, and sleep modes for different network densities.  Figure~\ref{fig95} illustrates the energy consumption for different network sizes.
-% for $Lifetime95$ and $Lifetime50$. 
-We denote by $DiLCO-/50$ (respectively  $DiLCO-/95$) as the amount of energy consumed during which the network can satisfy an area coverage greater than $50\%$ (repectively $95\%$) and we refer to the same definition for the two other approaches.

 \begin{figure}[h!]
 \centering
 \includegraphics[scale=0.45]{R/EC.pdf} 
 \begin{figure}[h!]
 \centering
 \includegraphics[scale=0.45]{R/EC.pdf} 

-\caption{The Energy Consumption}
+\caption{Energy consumption}

 \label{fig95}
 \end{figure} 
 
 \label{fig95}
 \end{figure} 
 

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

 
 
 %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. 
 %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.  
 
 
 
 %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. 
 %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.  
 

+\subsubsection{Execution time}

 
 

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

 
 \begin{figure}[h!]
 \centering
 \includegraphics[scale=0.45]{R/T.pdf}  
 
 \begin{figure}[h!]
 \centering
 \includegraphics[scale=0.45]{R/T.pdf}  

-\caption{Execution Time (in seconds)}
+\caption{Execution time in seconds}

 \label{fig8}
 \end{figure} 
 
 \label{fig8}
 \end{figure} 
 

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

 
 %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.
 
 
 %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{Network lifetime}

 
 

-\subsubsection{The Network Lifetime}
-In figure~\ref{figLT95}, network lifetime is illustrated for different network sizes. The term $/50$ (respectively  $/95$) next to the name of the method refers to the amount of time during which the network can satisfy an area coverage greater than $50\%$ ($Lifetime50$)(repectively $95\%$ ($Lifetime95$)) 
+In the next figure, the network lifetime is illustrated. Obviously, the lifetime
+increases with  the network  size, whatever the  considered protocol,  since the
+correlated node  density also  increases.  A high  network density means  a high
+node redundancy  which allows  to turn-off  many nodes and  thus to  prolong the
+network lifetime.

 
 \begin{figure}[h!]
 \centering
 \includegraphics[scale=0.45]{R/LT.pdf}  
 
 \begin{figure}[h!]
 \centering
 \includegraphics[scale=0.45]{R/LT.pdf}  

-\caption{The Network Lifetime}
+\caption{Network lifetime}

 \label{figLT95}
 \end{figure} 
 
 \label{figLT95}
 \end{figure} 
 

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

 
 % with  our DiLCO-16/50, DiLCO-32/50, DiLCO-16/95 and DiLCO-32/95 protocols
 % that leads to the larger lifetime improvement in comparison with other approaches. By choosing the best 
 
 % with  our DiLCO-16/50, DiLCO-32/50, DiLCO-16/95 and DiLCO-32/95 protocols
 % that leads to the larger lifetime improvement in comparison with other approaches. By choosing the best 


@@ -783,29 +893,33 @@ increases when the size of the network increases. For the same level of coverage
 % letting the other ones sleep in order to be used later in next rounds. Comparison shows that our DiLCO-16/50, DiLCO-32/50, DiLCO-16/95 and DiLCO-32/95 protocols, which are used distributed optimization over the subregions, are 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 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.
 
 % letting the other ones sleep in order to be used later in next rounds. Comparison shows that our DiLCO-16/50, DiLCO-32/50, DiLCO-16/95 and DiLCO-32/95 protocols, which are used distributed optimization over the subregions, are 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 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{\uppercase{Conclusion and future work}}
+\label{sec:Conclusion and Future Works} 
+
+A crucial problem in WSN is  to schedule the sensing activities of the different
+nodes  in order to  ensure both  coverage of  the area  of interest  and longest
+network lifetime. The inherent limitations of sensor nodes, in energy provision,
+communication and computing capacities,  require protocols that optimize the use
+of  the  available resources  to  fulfill the  sensing  task.   To address  this
+problem, this paper proposes a  two-step approach. Firstly, the field of sensing
+is  divided into  smaller  subregions using  the  concept of  divide-and-conquer
+method. Secondly,  a distributed  protocol called Distributed  Lifetime Coverage
+Optimization is applied in each  subregion to optimize the coverage and lifetime
+performances.   In a subregion,  our protocol  consists to  elect a  leader node
+which will then perform a sensor activity scheduling. The challenges include how
+to  select   the  most  efficient  leader   in  each  subregion   and  the  best
+representative set of active nodes to ensure a high level of coverage. To assess
+the performance of our approach, we  compared it with two other approaches using
+many performance metrics  like coverage ratio or network  lifetime. We have also
+study the  impact of the  number of subregions  chosen to subdivide the  area of
+interest,  considering  different  network  sizes.  The  experiments  show  that
+increasing the  number of subregions allows  to improves the  lifetime. The more
+there  are   subregions,  the  more   the  network  is  robust   against  random
+disconnection resulting from dead nodes.  However, for a given sensing field and
+network size  there is an optimal  number of subregions.  Therefore,  in case of
+our simulation  context a  subdivision in $16$~subregions  seems to be  the most
+relevant. The optimal number of subregions will be investigated in the future.

 
 

-
-
-\section{\uppercase{Conclusion and Future Works}}
-\label{sec:Conclusion and Future Works}
-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 set of active nodes to ensure a high level of coverage.
-We have compared this method with two other approaches using many metrics as coverage ratio, execution time, lifetime.
-Some experiments have been performed to study the choice of the number of
-subregions  which subdivide  the  sensing field,  considering different  network
-sizes. They show that as the number of subregions increases, so does the network
-lifetime. Moreover,  it makes  the DiLCO protocol  more robust  against random
-network  disconnection due  to node  failures.  However,  too  much subdivisions
-reduces the advantage  of the optimization. In fact, there  is a balance between
-the  benefit  from the  optimization  and the  execution  time  needed to  solve
-it. Therefore, the subdivision in $16$ subregions seems to be the most appropriate. 

 \iffalse
 \noindent 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
 \iffalse
 \noindent 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


@@ -836,18 +950,17 @@ optimization  methods. \iffalse The round  will still consist of 4 phases, but t
 The computation of all cover sets in one time is far more
 difficult, but will reduce the communication overhead. \fi
 \fi
 The computation of all cover sets in one time is far more
 difficult, but will reduce the communication overhead. \fi
 \fi

-\section*{\uppercase{Acknowledgements}}
-\noindent As a Ph.D. student, Ali Kadhum IDREES would like to gratefully acknowledge the University of Babylon - IRAQ for the financial support and Campus France for the received support.
-
-

 
 

+\section*{\uppercase{Acknowledgements}}

 
 

+\noindent  As  a Ph.D.  student,  Ali Kadhum  IDREES  would  like to  gratefully
+acknowledge  the University  of Babylon  - IRAQ  for the  financial  support and
+Campus France for the received support.

 
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