Modifications made up to subsection 5.2.2

diff --git a/Example.aux b/Example.aux
index af343110308c7267b5f7cd78bdbeef1d3a53cc8d..41dd5f6bb940a41ecff40c3c711b334def264bc4 100644 (file)
--- a/Example.aux
+++ b/Example.aux
@@ -16,6 +16,7 @@
 \citation{Zhang05}
 \citation{idrees2014coverage}
 \citation{pedraza2006}
 \citation{Zhang05}
 \citation{idrees2014coverage}
 \citation{pedraza2006}

+\newlabel{main_idea}{{3.2}{3}}

 \providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}}
 \newlabel{fig2}{{1}{3}}
 \citation{varga}
 \providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}}
 \newlabel{fig2}{{1}{3}}
 \citation{varga}

diff --git a/Example.tex b/Example.tex
index 5a2a5cb009378db3ba9788d210e8f58c29bfdc99..8ed81a642e304cb0eef1c8bad61eb1ce8328095b 100644 (file)
--- a/Example.tex
+++ b/Example.tex
@@ -295,6 +295,7 @@ $X_{13}=( p_x + R_s * (0), p_y + R_s * (\frac{-\sqrt{2}}{2})) $.
 \fi
 
 \subsection{The main idea}
 \fi
 
 \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
 
 \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


@@ -390,15 +391,15 @@ 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
 
 
 \fi
 


@@ -603,13 +604,13 @@ 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
 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
+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
 node typically  consists of  four units: a  MicroController Unit, an  Atmels AVR
 ATmega103L in  case of Medusa II,  to perform the  computations; a communication

-(adio) 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
+(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]
 second) by a node for each of its possible status.
 
 \begin{table}[ht]


@@ -643,27 +644,25 @@ Computation & ON & ON & ON & 26.83 \\
 % is used to refer this table in the text
 \end{table}
 
 % is used to refer this table in the text
 \end{table}
 

-% MICHEL - TO BE CONTINUED
-
-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.\\
+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:
 
 In the simulations,  we introduce the following performance  metrics to evaluate
 the efficiency of our approach:


@@ -671,13 +670,18 @@ 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.


@@ -696,19 +700,23 @@ 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 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 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: \\
+\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
  \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},
+\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*}
 
 %\begin{equation*}
 \end{equation*}
 
 %\begin{equation*}


@@ -716,15 +724,16 @@ transmit information on an event in the area that it monitors.
 %\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*}
 
 %\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.
+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).

 
 \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,


@@ -742,45 +751,73 @@ 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 for different subregions}

 \subsection{Performance Analysis}
 \label{sub1}
 
 \subsection{Performance Analysis}
 \label{sub1}
 

-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).
+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.

 
 

-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.
+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.

 
 

-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. 
+\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} 
 \caption{The Coverage Ratio}
 \label{fig3}
 \end{figure} 
 
 \centering
  \includegraphics[scale=0.45] {R/CR.pdf} 
 \caption{The Coverage Ratio}
 \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}

 
 

+% MICHEL - TO BE CONTINUED

 
 

-\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.
+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 $Lifetime50$. 
 % 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.
+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} 


@@ -788,13 +825,15 @@ We denote by $DiLCO-/50$ (respectively  $DiLCO-/95$) as the amount of energy con
 \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 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.

 
 
 %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}
 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}.
 \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}.