X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/LiCO.git/blobdiff_plain/9803dc4f10026362650bd86d02cf61c7f3424b24..1c2762bf6f7508f4f418890d2903ad6f008233f0:/LiCO_Journal.tex?ds=sidebyside

diff --git a/LiCO_Journal.tex b/LiCO_Journal.tex
index ead2af0..e9d10ac 100644
--- a/LiCO_Journal.tex
+++ b/LiCO_Journal.tex
@@ -64,10 +64,10 @@ region of interest is first subdivided  into subregions and our protocol is then
 distributed among sensor nodes in each  subregion. A sensor node which runs LiCO
 protocol  repeats   periodically  four  stages:  information   exchange,  leader
 election, optimization decision, and sensing.  More precisely, the scheduling of
-nodes activities (sleep/wake up duty cycles)  is achieved in each subregion by a
+nodes' activities (sleep/wake up duty cycles) is achieved in each subregion by a
 leader selected after  cooperation between nodes within the  same subregion. The
 novelty of  approach lies essentially in  the formulation of a  new mathematical
-optimization  model  based  on  perimeter coverage  level  to  schedule  sensors
+optimization  model  based on  perimeter  coverage  level to  schedule  sensors'
 activities.  Extensive simulation experiments have been performed using OMNeT++,
 the  discrete event  simulator, to  demonstrate that  LiCO is  capable to  offer
 longer lifetime coverage for WSNs in comparison with some other protocols.
@@ -88,7 +88,7 @@ wireless communication hardware  has given rise to the opportunity  to use large
 networks    of     tiny    sensors,    called    Wireless     Sensor    Networks
 (WSN)~\cite{akyildiz2002wireless,puccinelli2005wireless}, to  fulfill monitoring
 tasks.   A  WSN  consists  of  small low-powered  sensors  working  together  by
-communicating with one another through  multihop radio communications. Each node
+communicating with one another through multi-hop radio communications. Each node
 can send the data  it collects in its environment, thanks to  its sensor, to the
 user by means of  sink nodes. The features of a WSN made  it suitable for a wide
 range of application  in areas such as business,  environment, health, industry,
@@ -125,7 +125,7 @@ This paper makes the following contributions.
   temporal subdivision.   On the one hand  the area of interest  if divided into
   several smaller subregions and on the other hand the time line is divided into
   periods of equal length. In each subregion the sensor nodes will cooperatively
-  choose a  leader which will  schedule nodes  activities, and this  grouping of
+  choose a  leader which will schedule  nodes' activities, and this  grouping of
   sensors is similar to typical cluster architecture.
 \item We  propose a new mathematical  optimization model.  Instead of  trying to
   cover a set of specified points/targets as  in most of the methods proposed in
@@ -214,7 +214,7 @@ algorithms~\cite{cardei2005improving,zorbas2010solving,pujari2011high}    always
 provide nearly  or close to  optimal solution since  the algorithm has  a global
 view of the whole network. The disadvantage of a centralized method is obviously
 its high cost  in communications needed to  transmit to a single  node, the base
-station which will  globally schedule nodes activities, data from  all the other
+station which will globally schedule nodes'  activities, data from all the other
 sensor nodes in  the area.  The price  in communications can be  very huge since
 long range communications will be needed. In fact the larger the WNS, the higher
 the  communication and  thus energy  cost.   {\it In  order to  be suitable  for
@@ -293,6 +293,7 @@ executed by each node.
 %Before proceeding in the presentation of the main ideas of the protocol, we will briefly describe the perimeter coverage model and give some necessary assumptions and definitions.
 
 \subsection{Assumptions and Models}
+\label{CI}
 
 \noindent A WSN consisting of $J$ stationary sensor nodes randomly and uniformly
 distributed in  a bounded sensor field  is considered. The wireless  sensors are
@@ -396,25 +397,25 @@ above is thus given by the sixth line of the table.
  \caption{Coverage intervals and contributing sensors for sensor node 0.}
 \begin{tabular}{|c|c|c|c|c|c|c|c|c|}
 \hline
-\begin{tabular}[c]{@{}c@{}}The angle \\ $\alpha$ \end{tabular} & \begin{tabular}[c]{@{}c@{}}Segment \\ Left (L) or\\  Right (R)\end{tabular} & \begin{tabular}[c]{@{}c@{}}Sensor \\ Node Id\end{tabular} & \begin{tabular}[c]{@{}c@{}}Interval \\ Coverage\\  Level\end{tabular} & \multicolumn{5}{c|}{\begin{tabular}[c]{@{}c@{}}The Set of Sensors\\ Involved in Interval \\ Coverage\end{tabular}} \\ \hline
-0.0291    & L                                                                         & 1                                                         & 4                                                                     & 0                     & 1                     & 3                    & 4                    &                      \\ \hline
-0.104     & L                                                                         & 2                                                         & 5                                                                     & 0                     & 1                     & 3                    & 4                    & 2                    \\ \hline
-0.3168    & R                                                                         & 3                                                         & 4                                                                     & 0                     & 1                     & 4                    & 2                    &                      \\ \hline
-0.6752    & R                                                                         & 4                                                         & 3                                                                     & 0                     & 1                     & 2                    &                      &                      \\ \hline
-1.8127    & R                                                                         & 1                                                         & 2                                                                     & 0                     & 2                     &                      &                      &                      \\ \hline
-1.9228    & L                                                                         & 5                                                         & 3                                                                     & 0                     & 2                     & 5                    &                      &                      \\ \hline
-2.3959    & L                                                                         & 6                                                         & 4                                                                     & 0                     & 2                     & 5                    & 6                    &                      \\ \hline
-2.4258    & R                                                                         & 2                                                         & 3                                                                     & 0                     & 5                     & 6                    &                      &                      \\ \hline
-2.7868    & L                                                                         & 7                                                         & 4                                                                     & 0                     & 5                     & 6                    & 7                    &                      \\ \hline
-2.8358    & L                                                                         & 8                                                         & 5                                                                     & 0                     & 5                     & 6                    & 7                    & 8                    \\ \hline
-2.9184    & R                                                                         & 5                                                         & 4                                                                     & 0                     & 6                     & 7                    & 8                    &                      \\ \hline
-3.3301    & R                                                                         & 7                                                         & 3                                                                     & 0                     & 6                     & 8                    &                      &                      \\ \hline
-3.9464    & L                                                                         & 9                                                         & 4                                                                     & 0                     & 6                     & 8                    & 9                    &                      \\ \hline
-4.767     & R                                                                         & 6                                                         & 3                                                                     & 0                     & 8                     & 9                    &                      &                      \\ \hline
-4.8425    & L                                                                         & 3                                                         & 4                                                                     & 0                     & 3                     & 8                    & 9                    &                      \\ \hline
-4.9072    & R                                                                         & 8                                                         & 3                                                                     & 0                     & 3                     & 9                    &                      &                      \\ \hline
-5.3804    & L                                                                         & 4                                                         & 4                                                                     & 0                     & 3                     & 4                    & 9                    &                      \\ \hline
-5.9157    & R                                                                         & 9                                                         & 3                                                                     & 0                     & 3                     & 4                    &                      &                      \\ \hline
+\begin{tabular}[c]{@{}c@{}}Left \\ point \\ angle~$\alpha$ \end{tabular} & \begin{tabular}[c]{@{}c@{}}Interval \\ left \\ point\end{tabular} & \begin{tabular}[c]{@{}c@{}}Interval \\ right \\ point\end{tabular} & \begin{tabular}[c]{@{}c@{}}Maximum \\ coverage\\  level\end{tabular} & \multicolumn{5}{c|}{\begin{tabular}[c]{@{}c@{}}Set of sensors\\ involved \\ in interval coverage\end{tabular}} \\ \hline
+0.0291    & 1L                                                                        & 2L                                                        & 4                                                                     & 0                     & 1                     & 3                    & 4                    &                      \\ \hline
+0.104     & 2L                                                                        & 3R                                                        & 5                                                                     & 0                     & 1                     & 3                    & 4                    & 2                    \\ \hline
+0.3168    & 3R                                                                        & 4R                                                        & 4                                                                     & 0                     & 1                     & 4                    & 2                    &                      \\ \hline
+0.6752    & 4R                                                                        & 1R                                                        & 3                                                                     & 0                     & 1                     & 2                    &                      &                      \\ \hline
+1.8127    & 1R                                                                        & 5L                                                        & 2                                                                     & 0                     & 2                     &                      &                      &                      \\ \hline
+1.9228    & 5L                                                                        & 6L                                                        & 3                                                                     & 0                     & 2                     & 5                    &                      &                      \\ \hline
+2.3959    & 6L                                                                        & 2R                                                        & 4                                                                     & 0                     & 2                     & 5                    & 6                    &                      \\ \hline
+2.4258    & 2R                                                                        & 7L                                                        & 3                                                                     & 0                     & 5                     & 6                    &                      &                      \\ \hline
+2.7868    & 7L                                                                        & 8L                                                        & 4                                                                     & 0                     & 5                     & 6                    & 7                    &                      \\ \hline
+2.8358    & 8L                                                                        & 5R                                                        & 5                                                                     & 0                     & 5                     & 6                    & 7                    & 8                    \\ \hline
+2.9184    & 5R                                                                        & 7R                                                        & 4                                                                     & 0                     & 6                     & 7                    & 8                    &                      \\ \hline
+3.3301    & 7R                                                                        & 9R                                                        & 3                                                                     & 0                     & 6                     & 8                    &                      &                      \\ \hline
+3.9464    & 9R                                                                        & 6R                                                        & 4                                                                     & 0                     & 6                     & 8                    & 9                    &                      \\ \hline
+4.767     & 6R                                                                        & 3L                                                        & 3                                                                     & 0                     & 8                     & 9                    &                      &                      \\ \hline
+4.8425    & 3L                                                                        & 8R                                                        & 4                                                                     & 0                     & 3                     & 8                    & 9                    &                      \\ \hline
+4.9072    & 8R                                                                        & 4L                                                        & 3                                                                     & 0                     & 3                     & 9                    &                      &                      \\ \hline
+5.3804    & 4L                                                                        & 9R                                                        & 4                                                                     & 0                     & 3                     & 4                    & 9                    &                      \\ \hline
+5.9157    & 9R                                                                        & 1L                                                        & 3                                                                     & 0                     & 3                     & 4                    &                      &                      \\ \hline
 \end{tabular}
 
 \label{my-label}
@@ -423,7 +424,7 @@ above is thus given by the sixth line of the table.
 
 %The optimization algorithm that used by LiCO protocol based on the perimeter coverage levels of the left and right points of the segments and worked to minimize the number of sensor nodes for each left or right point of the segments within each sensor node. The algorithm minimize the perimeter coverage level of the left and right points of the segments, while, it assures that every perimeter coverage level of the left and right points of the segments greater than or equal to 1.
 
-In LiCO  protocol, scheduling of sensor  nodes activities is formulated  with an
+In LiCO  protocol, scheduling of sensor  nodes' activities is formulated  with an
 integer program  based on  coverage intervals. The  formulation of  the coverage
 optimization problem is  detailed in~section~\ref{cp}.  Note that  when a sensor
 node  has a  part of  its sensing  range outside  the WSN  sensing field,  as in
@@ -445,43 +446,67 @@ optimization algorithm.
 %\label{ex5pcm}
 %\end{figure} 
 
-% MICHEL TO BE CONTINUED FROM HERE
-
 \subsection{The Main Idea}
-\noindent The area  of  interest can  be  divided into smaller areas called subregions and
-then our protocol will be implemented in each subregion simultaneously. LiCO protocol works into periods fashion as shown in figure~\ref{fig2}.
-\begin{figure}[ht!]
+
+\noindent The  WSN area of  interest is, in a  first step, divided  into regular
+homogeneous subregions  using a divide-and-conquer  algorithm. In a  second step
+our  protocol  will  be  executed  in   a  distributed  way  in  each  subregion
+simultaneously to schedule nodes' activities for one sensing period.
+
+As  shown in  figure~\ref{fig2}, node  activity  scheduling is  produced by  our
+protocol in a periodic manner. Each period is divided into 4 stages: Information
+(INFO)  Exchange,  Leader Election,  Decision  (the  result of  an  optimization
+problem),  and  Sensing.   For  each  period there  is  exactly  one  set  cover
+responsible for  the sensing task.  Protocols  based on a periodic  scheme, like
+LiCO, are more  robust against an unexpected  node failure. On the  one hand, if
+node failure is discovered before  taking the decision, the corresponding sensor
+node will  not be considered  by the optimization  algorithm, and, on  the other
+hand, if the sensor failure happens 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, and Decision)
+are energy consuming, even for nodes that will not join the set cover to monitor
+the area.
+
+\begin{figure}[t!]
 \centering
-\includegraphics[width=85mm]{Model.pdf}  
+\includegraphics[width=80mm]{Model.pdf}  
 \caption{LiCO protocol}
 \label{fig2}
 \end{figure} 
 
-Each period is divided into 4 stages: Information (INFO) Exchange, Leader  Election, Optimization Decision,  and  Sensing.  For  each  period there  is exactly one set cover responsible for the sensing task. LiCO is more powerful against an unexpected node failure because it works in periods. On the one hand, if the node failure is discovered before taking the decision of the optimization algorithm, the sensor node would not involved to current stage, and, on the other hand, if the sensor failure takes place 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, and  Decision) are energy  consuming for  some sensor nodes,  even when they do not join the network to monitor the area. 
-
-We define two types of packets to be used by LiCO protocol.
+We define two types of packets to be used by LiCO protocol:
 %\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 transmit to  them their respective status (stay Active  or go Sleep) during
+  sensing phase.
 \end{itemize}
 %\end{enumerate}
 
-There are five status for each sensor node in the network :
+Five status are possible for a sensor node in the network:
 %\begin{enumerate}[(a)] 
 \begin{itemize} 
-\item LISTENING: Sensor is waiting for a decision (to be active or not)
-\item COMPUTATION: Sensor applies the optimization process as leader
-\item ACTIVE: Sensor is active
-\item SLEEP: Sensor is turned off
-\item COMMUNICATION: Sensor is transmitting or receiving packet
+\item LISTENING: waits for a decision (to be active or not);
+\item COMPUTATION: executes the optimization algorithm as leader to
+  determine the activities scheduling;
+\item ACTIVE: node is sensing;
+\item SLEEP: node is turned off;
+\item COMMUNICATION: transmits or recevives packets.
 \end{itemize}
 %\end{enumerate}
 %Below, we describe each phase in more details.
 
 \subsection{LiCO Protocol Algorithm}
-The pseudo-code for LiCO Protocol is illustrated as follows:
 
+\noindent The  pseudocode implementing the  protocol on  a node is  given below.
+More  precisely,  Algorithm~\ref{alg:LiCO}  gives  a brief  description  of  the
+protocol applied by a sensor node $s_k$ where $k$ is the node index in the WSN.
 
 \begin{algorithm}[h!]                
  % \KwIn{all the parameters related to information exchange}
@@ -491,8 +516,8 @@ The pseudo-code for LiCO Protocol is illustrated as follows:
   
   \If{ $RE_k \geq E_{th}$ }{
       \emph{$s_k.status$ = COMMUNICATION}\;
-      \emph{Send $INFO()$ packet to other nodes in the subregion}\;
-      \emph{Wait $INFO()$ packet from other nodes in the subregion}\; 
+      \emph{Send $INFO()$ packet to other nodes in subregion}\;
+      \emph{Wait $INFO()$ packet from other nodes in subregion}\; 
       \emph{Update K.CurrentSize}\;
       \emph{LeaderID = Leader election}\;
       \If{$ s_k.ID = LeaderID $}{
@@ -503,12 +528,12 @@ The pseudo-code for LiCO Protocol is illustrated as follows:
          % \emph{ Determine the segment points using perimeter coverage model}\;
       }
       
-      \If{$ (s_k.ID $ is the same Previous Leader) AND (K.CurrentSize = K.PreviousSize)}{
+      \If{$ (s_k.ID $ is the same Previous Leader) And (K.CurrentSize = K.PreviousSize)}{
       
       	 \emph{ Use the same previous cover set for current sensing stage}\;
       }
       \Else{
-            \emph{ Update $a^j_{ik}$ and prepare data to Algorithm}\;
+            \emph{Update $a^j_{ik}$; prepare data for IP~Algorithm}\;
             \emph{$\left\{\left(X_{1},\dots,X_{l},\dots,X_{K}\right)\right\}$ = Execute Integer Program Algorithm($K$)}\;
             \emph{K.PreviousSize = K.CurrentSize}\;
            }
@@ -531,43 +556,77 @@ The pseudo-code for LiCO Protocol is illustrated as follows:
 
 \end{algorithm}
 
-\noindent Algorithm 1 gives a brief description of the protocol applied by each sensor node (denoted by $s_k$ for a sensor node indexed by $k$). In this algorithm, the K.CurrentSize and K.PreviousSize refer to the current size and the previous size of sensor nodes still alive in the subregion respectively.
-Initially, the sensor node checks its remaining energy $RE_k$, which must be greater than a threshold $E_{th}$ in order to participate in the current period. Each sensor node determines its position and its subregion based Embedded GPS  or Location Discovery Algorithm. After that, all the sensors collect position coordinates, remaining energy, sensor node id, and the number of its one-hop live neighbors during the information exchange. The sensors inside a same region cooperate to elect a leader. The selection criteria for the leader in order  of priority  are: larger number of neighbors,  larger remaining  energy, and  then in  case of equality, larger index. Thereafter the leader collects information to formulate and solve the integer program which allows to construct the set of active sensors in the sensing stage.  
-
+In this  algorithm, K.CurrentSize and  K.PreviousSize refer to the  current size
+and the  previous size of  the subnetwork  in the subregion  respectively.  That
+means the  number of sensor nodes  which are still alive.  Initially, the sensor
+node checks its remaining energy $RE_k$,  which must be greater than a threshold
+$E_{th}$  in order  to  participate  in the  current  period.  Each sensor  node
+determines its  position and its subregion  using an embedded GPS  or a location
+discovery algorithm. After  that, all the sensors  collect position coordinates,
+remaining energy, sensor  node ID, and the number of  its one-hop live neighbors
+during the information  exchange. The sensors inside a same  region cooperate to
+elect a  leader. The selection  criteria for the  leader, in order  of priority,
+are: larger  number of neighbors, larger  remaining energy, and then  in case of
+equality,  larger  index.   Once  chosen, the  leader  collects  information  to
+formulate and  solve the integer  program which allows  to construct the  set of
+active sensors in the sensing stage.
 
 %After the cooperation among the sensor nodes in the same subregion, the leader will be elected in distributed way, where each sensor node and based on it's information decide who is the leader. The selection criteria for the leader in order  of priority  are: larger number of neighbors,  larger remaining  energy, and  then in  case of equality, larger index. Thereafter,  if the sensor node is leader, it will execute the perimeter-coverage model for each sensor in the subregion in order to determine the segment points which would be used in the next stage by the optimization algorithm of the LiCO protocol. Every sensor node is selected as a leader, it is executed the perimeter coverage model only one time during it's life in the network.
 
 % The leader has the responsibility of applying the integer program algorithm (see section~\ref{cp}), which provides a set of sensors planned to be active in the sensing stage.  As leader, it will send an Active-Sleep packet to each sensor in the same subregion to inform 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 stage.
 
-
 \section{Lifetime Coverage problem formulation}
 \label{cp}
-In this section, the coverage model is mathematically formulated.
-For convenience, the notations are described first. 
-%Then the lifetime problem of sensor network is formulated. 
-\noindent $S :$ the set of all sensors in the network.\\
-\noindent $A :$ the set of alive sensors within $S$.\\
-%\noindent $I :$ the set of segment points.\\
-\noindent $I_j :$ the set of coverage intervals (CI)  for sensor $j$.\\
-\noindent $I_j$ refers to the set of intervals which have been defined for each sensor $j$ in section~\ref{sec:The LiCO Protocol Description}.
-\noindent For a coverage interval  $i$,  let  $a^j_{ik}$ denote the indicator function of whether the sensor $k$ is involved in the coverage interval $i$ of sensor $j$, that is:
 
+\noindent In this  section, the coverage model is  mathematically formulated. We
+start  with a  description of  the notations  that will  be used  throughout the
+section.
+
+First, we have the following sets:
+\begin{itemize}
+\item $S$ represents the set of WSN sensor nodes;
+\item $A \subseteq S $ is the subset of alive sensors;
+\item  $I_j$  designates  the  set  of  coverage  intervals  (CI)  obtained  for
+  sensor~$j$.
+\end{itemize}
+$I_j$ refers to the set of  coverage intervals which have been defined according
+to the  method introduced in  subsection~\ref{CI}. For a coverage  interval $i$,
+let $a^j_{ik}$ denote  the indicator function of whether  sensor~$k$ is involved
+in coverage interval~$i$ of sensor~$j$, that is:
 \begin{equation}
 a^j_{ik} = \left \{ 
 \begin{array}{lll}
-  1 & \mbox{if the sensor $k$ is involved in the } \\
+  1 & \mbox{if sensor $k$ is involved in the } \\
   	&	\mbox{coverage interval $i$ of sensor $j$}, \\
-  0 & \mbox{Otherwise.}\\
+  0 & \mbox{otherwise.}\\
 \end{array} \right.
 %\label{eq12} 
 \notag
 \end{equation}
-Note that $a^k_{ik}=1$ by definition of the interval.\\
+Note that $a^k_{ik}=1$ by definition of the interval.
 %, where the objective is to find the maximum number of non-disjoint sets of sensor nodes such that each set cover can assure the coverage for the whole region so as to extend the network lifetime in WSN. Our model uses the PCL~\cite{huang2005coverage} in order to optimize the lifetime coverage in each subregion.
-%We defined some parameters, which are related to our optimization model. In our model,  we  consider binary variables $X_{k}$, which determine the activation of sensor $k$ in the sensing round $k$. .   
-\noindent We  consider binary variables $X_{k}$ ($X_k=1$ if the sensor $k$ is active or 0 otherwise), which determine the activation of sensor $k$ in the sensing phase. We define the integer variable $M^j_i$ which measures the undercoverage for the coverage interval $i$ for sensor $j$. In the same way, we define the integer variable $V^j_i$, which measures the overcoverage for the coverage interval $i$ for sensor $j$. If we decide to sustain a level of coverage equal to $l$ all along the perimeter of the sensor $j$, we have to ensure that at least $l$ sensors involved in each coverage interval $i$ ($i \in I_j$) of sensor $j$ are active. According to the previous notations, the number of active sensors in the coverage interval $i$ of sensor $j$ is given by $\sum_{k \in K} a^j_{ik} X_k$. To extend the network lifetime, the objective is to active a minimal number of sensors in each period to ensure the desired coverage level. As the number of alive sensors decreases, it becomes impossible to satisfy the level of coverage for all covergae intervals. We uses variables $M^j_i$ and $V^j_i$ as a measure of the deviation between the desired number of active sensors in a coverage interval and the effective number of active sensors. And we try to minimize these deviations, first to force the activation of a minimal number of sensors to ensure the desired coverage level, and if the desired level can not be completely  satisfied, to reach a coverage level as close as possible that the desired one.
-
-
+%We defined some parameters, which are related to our optimization model. In our model,  we  consider binary variables $X_{k}$, which determine the activation of sensor $k$ in the sensing round $k$. .
+Second,  we define  several binary  and integer  variables.  Hence,  each binary
+variable $X_{k}$  determines the activation of  sensor $k$ in the  sensing phase
+($X_k=1$ if  the sensor $k$  is active or 0  otherwise).  $M^j_i$ is  an integer
+variable  which  measures  the  undercoverage  for  the  coverage  interval  $i$
+corresponding to  sensor~$j$. In  the same  way, the  overcoverage for  the same
+coverage interval is given by the variable $V^j_i$.
+
+If we decide to sustain a level of coverage equal to $l$ all along the perimeter
+of sensor  $j$, we have  to ensure  that at least  $l$ sensors involved  in each
+coverage  interval $i  \in I_j$  of  sensor $j$  are active.   According to  the
+previous notations, the number of active sensors in the coverage interval $i$ of
+sensor $j$  is given by  $\sum_{k \in A} a^j_{ik}  X_k$.  To extend  the network
+lifetime,  the objective  is to  activate a  minimal number  of sensors  in each
+period to  ensure the  desired coverage  level. As the  number of  alive sensors
+decreases, it becomes impossible to reach  the desired level of coverage for all
+coverage intervals. Therefore we uses variables $M^j_i$ and $V^j_i$ as a measure
+of the  deviation between  the desired  number of active  sensors in  a coverage
+interval and  the effective  number. And  we try  to minimize  these deviations,
+first to  force the  activation of  a minimal  number of  sensors to  ensure the
+desired coverage level, and if the desired level cannot be completely satisfied,
+to reach a coverage level as close as possible to the desired one.
 
 %A system of linear constraints is imposed to attempt to keep the coverage level in each coverage interval to within specified PCL. Since it is physically impossible to satisfy all constraints simultaneously, each constraint uses a variable to either record when the coverage level is achieved, or to record the deviation from the desired coverage level. These additional variables are embedded into an objective function to be minimized. 
 
@@ -589,14 +648,9 @@ Note that $a^k_{ik}=1$ by definition of the interval.\\
 
 %\noindent $V^j_i (overcoverage): $ integer value $\in  \mathbb{N}$ for segment point $i$ of sensor $j$.
 
-
-
-
-
-\noindent Our coverage optimization problem can be mathematically formulated as follows: \\
+Our coverage optimization problem can then be mathematically expressed as follows: 
 %Objective:
-
-\begin{equation} \label{eq:ip2r}
+\begin{equation} %\label{eq:ip2r}
 \left \{
 \begin{array}{ll}
 \min \sum_{j \in S} \sum_{i \in I_j} (\alpha^j_i ~ M^j_i + \beta^j_i ~ V^j_i )&\\
@@ -610,27 +664,35 @@ Note that $a^k_{ik}=1$ by definition of the interval.\\
 X_{k} \in \{0,1\}, \forall k \in A
 \end{array}
 \right.
+\notag
 \end{equation}
-
-
-\noindent $\alpha^j_i$ and $\beta^j_i$ are nonnegative weights selected according to the
-relative importance of satisfying the associated
-level of coverage. For example, weights associated with coverage intervals of a specified part of a region
-may be given a relatively
-larger magnitude than weights associated
-with another region. This kind of integer program is inspired from the model developed for brachytherapy treatment planning for optimizing dose distribution \cite{0031-9155-44-1-012}. The integer program must be solved by the leader in each subregion at the beginning of each sensing phase, whenever the environment has changed (new leader, death of some sensors). Note that the number of constraints in the model is constant (constraints of coverage expressed for all sensors), whereas the number of variables $X_k$ decreases over periods, since we consider only alive sensors (sensors with enough energy to be alive during one sensing phase) in the model. 
-
-
-\section{\uppercase{PERFORMANCE EVALUATION AND ANALYSIS}}  
+$\alpha^j_i$ and $\beta^j_i$  are nonnegative weights selected  according to the
+relative importance of satisfying the associated level of coverage. For example,
+weights associated with  coverage intervals of a specified part  of a region may
+be  given a  relatively larger  magnitude than  weights associated  with another
+region. This  kind of integer program  is inspired from the  model developed for
+brachytherapy    treatment   planning    for   optimizing    dose   distribution
+\cite{0031-9155-44-1-012}. The integer  program must be solved by  the leader in
+each subregion at the beginning of  each sensing phase, whenever the environment
+has  changed (new  leader,  death of  some  sensors). Note  that  the number  of
+constraints in the model is constant  (constraints of coverage expressed for all
+sensors), whereas the number of variables $X_k$ decreases over periods, since we
+consider only alive  sensors (sensors with enough energy to  be alive during one
+sensing phase) in the model.
+
+\section{Performance Evaluation and Analysis}  
 \label{sec:Simulation Results and Analysis}
 %\noindent \subsection{Simulation Framework}
 
 \subsection{Simulation Settings}
 %\label{sub1}
-In this section, we focus on the performance of LiCO protocol, which is distributed in each sensor node in the sixteen subregions of WSN. We use the same energy consumption model which is used in~\cite{Idrees2}. Table~\ref{table3} gives the chosen parameters setting.
+
+The WSN  area of interest is  supposed to be divided  into 16~regular subregions
+and we use the same energy consumption than in our previous work~\cite{Idrees2}.
+Table~\ref{table3} gives the chosen parameters settings.
 
 \begin{table}[ht]
-\caption{Relevant parameters for network initializing.}
+\caption{Relevant parameters for network initialization.}
 % title of Table
 \centering
 % used for centering table
@@ -641,14 +703,14 @@ Parameter & Value  \\ [0.5ex]
    
 \hline
 % inserts single horizontal line
-Sensing  Field  & $(50 \times 25)~m^2 $   \\
+Sensing field & $(50 \times 25)~m^2 $   \\
 
-Nodes Number &  100, 150, 200, 250 and 300~nodes   \\
+WSN size &  100, 150, 200, 250, and 300~nodes   \\
 %\hline
-Initial Energy  & 500-700~joules  \\  
+Initial energy  & in range 500-700~Joules  \\  
 %\hline
-Sensing Period & 60 Minutes \\
-$E_{th}$ & 36 Joules\\
+Sensing period & duration of 60 minutes \\
+$E_{th}$ & 36~Joules\\
 $R_s$ & 5~m   \\     
 %\hline
 $\alpha^j_i$ & 0.6   \\
@@ -660,51 +722,54 @@ $\beta^j_i$ & 0.4
 \label{table3}
 % is used to refer this table in the text
 \end{table}
-Simulations with five  different node densities going from  100 to 250~nodes were
-performed  considering  each  time  25~randomly generated  networks,  to  obtain
-experimental results  which are relevant. All simulations are repeated 25 times and the results are averaged. 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.
-
-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  the
-threshold  $E_{th}=36$~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  corresponds  to the  energy  needed by  the  sensing  phase, obtained  by
-multiplying the energy consumed in active  state (9.72 mW) by the time in seconds
-for one period (3600 seconds), and  adding the energy for the pre-sensing phases.
-According to  the interval of initial energy,  a sensor may be  active during at
-most 20 rounds.
-
-The values of $\alpha^j_i$ and $\beta^j_i$ have been chosen in a way that ensuring a good network coverage and for a longer time during the lifetime of the WSN.  We have given a higher priority for the undercoverage ( by setting the $\alpha^j_i$ with a larger value than $\beta^j_i$) so as to prevent the non-coverage for the interval i of the sensor j. On the other hand, we have given a little bit lower value for $\beta^j_i$ so as to minimize the number of active sensor nodes that contribute in covering the interval i in sensor j.
-
-In the simulations,  we introduce the following performance  metrics to evaluate
-the efficiency of our approach:
+To  obtain  experimental  results  which are  relevant,  simulations  with  five
+different node densities going from  100 to 300~nodes were performed considering
+each time 25~randomly  generated networks. 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. 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  the threshold $E_{th}=36$~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 corresponds to the energy needed by the sensing phase,
+obtained by multiplying  the energy consumed in active state  (9.72 mW) with the
+time in  seconds for one  period (3600 seconds), and  adding the energy  for the
+pre-sensing phases.  According  to the interval of initial energy,  a sensor may
+be active during at most 20 periods.
+
+The values  of $\alpha^j_i$ and  $\beta^j_i$ have been  chosen to ensure  a good
+network coverage and a longer WSN lifetime.  We have given a higher priority for
+the  undercoverage  (by  setting  the  $\alpha^j_i$ with  a  larger  value  than
+$\beta^j_i$) so as to prevent the non-coverage  for the interval i of the sensor
+j. On the other hand, we have given  a little bit lower value for $\beta^j_i$ so
+as to  minimize the number of  active sensor nodes which  contribute in covering
+the interval.
+
+We introduce the following performance metrics to evaluate the efficiency of our
+approach.
 
 %\begin{enumerate}[i)]
 \begin{itemize}
-\item {{\bf Network Lifetime}:} we define the network lifetime as the time until
-  the  coverage  ratio  drops  below  a  predefined  threshold.   We  denote  by
-  $Lifetime_{95}$ (respectively $Lifetime_{50}$) the amount of time during which
-  the  network can  satisfy an  area coverage  greater than  $95\%$ (respectively
-  $50\%$). We assume that the sensor  network can fulfill its task until all its
-  nodes have  been drained of their  energy or it  becomes disconnected. Network
-  connectivity  is crucial because  an active  sensor node  without connectivity
-  towards a base  station cannot transmit any information  regarding an observed
-  event in the area that it monitors.
-  
-    
-\item {{\bf Coverage Ratio (CR)}:} it measures how well the WSN is able to 
-  observe the area of interest. In our case, we discretized the sensor field
-  as a regular grid, which yields the following equation to compute the
-  coverage ratio: 
-\begin{equation*}
-\scriptsize
-\mbox{CR}(\%) = \frac{\mbox{$n$}}{\mbox{$N$}} \times 100.
-\end{equation*}
-where  $n$ is  the number  of covered  grid points  by active  sensors  of every
-subregions during  the current  sensing phase  and $N$ is  total number  of grid
-points in  the sensing field. In  our simulations, we have  a layout of  $N = 51
-\times 26 = 1326$ grid points.
+\item {\bf Network Lifetime}: the lifetime  is defined as the time elapsed until
+  the  coverage  ratio  falls  below a  fixed  threshold.   $Lifetime_{95}$  and
+  $Lifetime_{50}$  denote, respectively,  the  amount of  time  during which  is
+  guaranteed a  level of coverage  greater than $95\%$  and $50\%$. The  WSN can
+  fulfill the expected  monitoring task until all its nodes  have depleted their
+  energy or if the network is not more connected. This last condition is crucial
+  because without  network connectivity a  sensor may not be  able to send  to a
+  base station an event it has sensed.
+\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:
+  \begin{equation*}
+    \scriptsize
+    \mbox{CR}(\%) = \frac{\mbox{$n$}}{\mbox{$N$}} \times 100.
+  \end{equation*}
+  where $n$  is the  number of covered  grid points by  active sensors  of every
+  subregions during  the current sensing phase  and $N$ is total  number of grid
+  points  in the  sensing field.  In our  simulations we  have set  a layout  of
+  $N~=~51~\times~26~=~1326$~grid points.
 
+  % MICHEL TO BE CONTINUED FROM HERE
 
 \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
@@ -841,7 +906,7 @@ We denote by Protocol/50, Protocol/80, Protocol/85, Protocol/90, and Protocol/95
 \section{\uppercase{Conclusion and Future Works}}
 \label{sec:Conclusion and Future Works}
 In this paper we have studied the problem of lifetime coverage optimization in
-WSNs. We designed a protocol LiCO that schedules node activities (wakeup and sleep) with the objective of maintaining a good coverage ratio while maximizing the network lifetime. This protocol is applied on each subregion of the area of interest. It works in periods and is based on the resolution of an integer program to select the subset of sensors operating in active mode for each period. Our work is original in so far as it proposes for the first time an integer program scheduling the activation of sensors based on their perimeter coverage level instead of using a set of targets/points to be covered.
+WSNs. We designed a protocol LiCO that schedules node' activities (wakeup and sleep) with the objective of maintaining a good coverage ratio while maximizing the network lifetime. This protocol is applied on each subregion of the area of interest. It works in periods and is based on the resolution of an integer program to select the subset of sensors operating in active mode for each period. Our work is original in so far as it proposes for the first time an integer program scheduling the activation of sensors based on their perimeter coverage level instead of using a set of targets/points to be covered.
 
  
 
