Michel : Section 3 terminee

diff --git a/LiCO_Journal.tex b/LiCO_Journal.tex
index ead2af07748fa1d9b42652edc84efffd6d3f2b35..c8e2eb501a2fd70c6b02f1cc4d7fb92315b95289 100644 (file)
--- 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
 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
 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.
 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
 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,
 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
   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
   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
 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
 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


@@ -396,25 +396,25 @@ 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
  \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}
 \end{tabular}
 
 \label{my-label}


@@ -423,7 +423,7 @@ 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.
 
 
 %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
 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 +445,67 @@ optimization algorithm.
 %\label{ex5pcm}
 %\end{figure} 
 
 %\label{ex5pcm}
 %\end{figure} 
 

-% MICHEL TO BE CONTINUED FROM HERE
-

 \subsection{The Main Idea}
 \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~\label{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
 \centering

-\includegraphics[width=85mm]{Model.pdf}  
+\includegraphics[width=80mm]{Model.pdf}  

 \caption{LiCO protocol}
 \label{fig2}
 \end{figure} 
 
 \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} 
 %\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}
 
 \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} 
 %\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}
 \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:

 
 

+The  pseudocode  implementing the  protocol  on  a  node  is given  below.  More
+precisely, Algorithm~\label{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}
 
 \begin{algorithm}[h!]                
  % \KwIn{all the parameters related to information exchange}


@@ -491,8 +515,8 @@ The pseudo-code for LiCO Protocol is illustrated as follows:
   
   \If{ $RE_k \geq E_{th}$ }{
       \emph{$s_k.status$ = COMMUNICATION}\;
   
   \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 $}{
       \emph{Update K.CurrentSize}\;
       \emph{LeaderID = Leader election}\;
       \If{$ s_k.ID = LeaderID $}{


@@ -503,12 +527,12 @@ The pseudo-code for LiCO Protocol is illustrated as follows:
          % \emph{ Determine the segment points using perimeter coverage model}\;
       }
       
          % \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{ 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}\;
            }
             \emph{$\left\{\left(X_{1},\dots,X_{l},\dots,X_{K}\right)\right\}$ = Execute Integer Program Algorithm($K$)}\;
             \emph{K.PreviousSize = K.CurrentSize}\;
            }


@@ -531,16 +555,29 @@ The pseudo-code for LiCO Protocol is illustrated as follows:
 
 \end{algorithm}
 
 
 \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.  
-
+\noindent  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.
 
 
 %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.
 

+% MICHEL TO BE CONTINUED

 
 \section{Lifetime Coverage problem formulation}
 
 \section{Lifetime Coverage problem formulation}

+

 \label{cp}
 In this section, the coverage model is mathematically formulated.
 For convenience, the notations are described first. 
 \label{cp}
 In this section, the coverage model is mathematically formulated.
 For convenience, the notations are described first. 


@@ -551,7 +588,6 @@ For convenience, the notations are described first.
 \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 $I_j :$ 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:

-

 \begin{equation}
 a^j_{ik} = \left \{ 
 \begin{array}{lll}
 \begin{equation}
 a^j_{ik} = \left \{ 
 \begin{array}{lll}


@@ -593,9 +629,8 @@ Note that $a^k_{ik}=1$ by definition of the interval.\\
 
 
 
 
 
 

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

 %Objective:
 %Objective:

-

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


@@ -841,7 +876,7 @@ 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
 \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.