Update on the general introduction

[ThesisAli.git] / CHAPITRE_04.tex

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\chapter{Multiround Distributed Lifetime Coverage Optimization Protocol in Wireless Sensor Networks}
\label{ch4}

\section{Summary}
\label{ch4:sec:01}
Coverage and lifetime are two paramount problems in Wireless  Sensor Networks (WSNs). In this paper, a method called Multiround Distributed Lifetime Coverage
Optimization  protocol (MuDiLCO)  is proposed  to maintain  the coverage  and to improve the lifetime in wireless sensor  networks. The area of interest is first
divided  into subregions and  then the  MuDiLCO protocol  is distributed  on the sensor nodes in each subregion. The proposed MuDiLCO protocol works in periods
during which sets of sensor nodes are scheduled to remain active for a number of rounds  during the  sensing phase,  to ensure coverage  so as  to maximize the
lifetime of  WSN. The decision process is  carried out by a  leader node, which solves an  integer program to  produce the best  representative sets to  be used
during the rounds  of the sensing phase. Compared  with some existing protocols, simulation  results based  on  multiple criteria  (energy consumption,  coverage
ratio, and  so on) show that  the proposed protocol can  prolong efficiently the network lifetime and improve the coverage performance.

\section{MuDiLCO protocol description}
\label{ch4:sec:02}
\noindent In this section, we introduce the MuDiLCO protocol which is distributed on each subregion in the area of interest. It is based on two energy-efficient
mechanisms: subdividing the area of interest into several subregions (like cluster architecture) using divide and conquer method, where the sensor nodes cooperate within each subregion as independent group in order to achieve a network leader election; and sensor activity scheduling for maintaining the coverage and prolonging the network lifetime, which are applied periodically. MuDiLCO protocol uses the same assumptions and network model that presented in chapter 3, section \ref{ch3:sec:02:01} and it has been used the primary point coverage model which is described in the same chapter, section \ref{ch3:sec:02:02}.

 
\subsection{Background Idea}
\label{ch4:sec:02:02}
The area of  interest can be divided using  the divide-and-conquer strategy into
smaller  areas,  called  subregions,  and  then our MuDiLCO  protocol will be
implemented in each subregion in a distributed way.

As  can be seen  in Figure~\ref{fig2},  our protocol  works in  periods fashion,
where  each is  divided  into 4  phases: Information~Exchange,  Leader~Election,
Decision, and Sensing.  Each sensing phase may be itself divided into $T$ rounds
and for each round a set of sensors (a cover set) is responsible for the sensing
task. In  this way  a multiround optimization  process is performed  during each
period  after  Information~Exchange  and  Leader~Election phases,  in  order  to
produce $T$ cover sets that will take the mission of sensing for $T$ rounds.
\begin{figure}[ht!]
\centering \includegraphics[width=160mm]{Figures/ch4/GeneralModel.jpg} % 70mm  Modelgeneral.pdf
\caption{The MuDiLCO protocol scheme executed on each node}
\label{fig2}
\end{figure} 


This protocol minimizes the impact of unexpected node failure (not due to batteries
running out of energy), because it works in periods. 

 On the one hand, if a node failure is detected before  making the
decision, the node will not participate to this phase, and, on the other hand,
if the node failure occurs after the decision, the sensing  task of the network
will be temporarily affected:  only during  the period of sensing until a new
period starts.

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 (Information  Exchange, Leader  Election, and  Decision) are
energy  consuming for some  nodes, even  when they  do not  join the  network to
monitor the area.


These phases can be described in more details as follow:
\subsection{Information Exchange Phase}
\label{ch4:sec:02:02:01}
The information exchange among the wireless sensor nodes is similar to that one which is described in chapter 3, sections \ref{ch3:sec:02:03:01}.

\subsection{Leader Election phase}
\label{ch4:sec:02:02:02}
The leader election in each subregion is similar to that one which is described in chapter 3, sections\ref{ch3:sec:02:03:02}, but the difference in that the elected leader in each subregion is for each period. 

\subsection{Decision phase}
\label{ch4:sec:02:02:03}
Each  WSNL will solve  an integer  program to  select which  cover sets  will be
activated in  the following  sensing phase  to cover the  subregion to  which it
belongs.  The integer  program will produce $T$ cover sets,  one for each round.
The WSNL will send an Active-Sleep  packet to each sensor in the subregion based
on the algorithm's results, indicating if  the sensor should be active or not in
each round  of the  sensing phase.  The  integer program  is based on  the model
proposed by  \cite{ref156} with some modifications, where  the objective is
to find  a maximum  number of disjoint  cover sets.   To fulfill this  goal, the
authors proposed an integer  program which forces undercoverage and overcoverage
of  targets to  become minimal  at  the same  time.  They  use binary  variables
$x_{jl}$ to indicate if  sensor $j$ belongs to cover set $l$.   In our model, we
consider binary  variables $X_{t,j}$ to determine the  possibility of activating
sensor $j$ during round $t$ of  a given sensing phase.  We also consider primary
points as targets.  The  set of primary points is denoted by  $P$ and the set of
sensors by  $J$. Only sensors  able to  be alive during  at least one  round are
involved in the integer program.


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

Our coverage optimization problem can then be formulated as follows:
\begin{equation}
 \min \sum_{t=1}^{T} \sum_{p=1}^{P} \left(W_{\theta}* \Theta_{t,p} + W_{U} * U_{t,p}  \right)  \label{eq15} 
\end{equation}

Subject to
\begin{equation}
  \sum_{j=1}^{|J|} \alpha_{j,p} * X_{t,j}   = \Theta_{t,p} - U_{t,p} + 1 \label{eq16} \hspace{6 mm} \forall p \in P, t = 1,\dots,T
\end{equation}

\begin{equation}
  \sum_{t=1}^{T}  X_{t,j}   \leq  \lfloor {RE_{j}/E_{R}} \rfloor \hspace{6 mm} \forall j \in J, t = 1,\dots,T
  \label{eq144} 
\end{equation}

\begin{equation}
X_{t,j} \in \lbrace0,1\rbrace,   \hspace{10 mm} \forall j \in J, t = 1,\dots,T \label{eq17} 
\end{equation}

\begin{equation}
U_{t,p} \in \lbrace0,1\rbrace, \hspace{10 mm}\forall p \in P, t = 1,\dots,T  \label{eq18} 
\end{equation}

\begin{equation}
 \Theta_{t,p} \geq 0 \hspace{10 mm}\forall p \in P, t = 1,\dots,T \label{eq178}
\end{equation}



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

The first group  of constraints indicates that some primary  point $p$ should be
covered by at least  one sensor and, if it is not  always the case, overcoverage
and undercoverage  variables help balancing the restriction  equations by taking
positive values. The constraint  given by equation~(\ref{eq144}) guarantees that
the sensor has enough energy ($RE_j$  corresponds to its remaining energy) to be
alive during  the selected rounds knowing  that $E_{R}$ is the  amount of energy
required to be alive during one round.

There  are two main  objectives.  First,  we limit  the overcoverage  of primary
points in order to activate a  minimum number of sensors.  Second we prevent the
absence  of  monitoring  on  some  parts  of the  subregion  by  minimizing  the
undercoverage.  The weights  $W_\theta$ and $W_U$ must be  properly chosen so as
to guarantee that the maximum number of points are covered during each round. 
%% MS W_theta is smaller than W_u => problem with the following sentence
In our simulations priority is given  to the coverage by choosing $W_{U}$ very
large compared to $W_{\theta}$.

\subsection{Sensing phase}
\label{ch4:sec:02:02:04}
The sensing phase consists of $T$ rounds. Each sensor node in the subregion will
receive an Active-Sleep packet from WSNL, informing it to stay awake or to go to
sleep for  each round of the sensing  phase.  Algorithm~\ref{alg:MuDiLCO}, which
will be  executed by each node  at the beginning  of a period, explains  how the
Active-Sleep packet is obtained.

\begin{algorithm}[h!]                
 % \KwIn{all the parameters related to information exchange}
%  \KwOut{$winer-node$ (: the id of the winner sensor node, which is the leader of current round)}
  \BlankLine
  %\emph{Initialize the sensor node and determine it's position and subregion} \; 
  
  \If{ $RE_j \geq E_{R}$ }{
      \emph{$s_j.status$ = COMMUNICATION}\;
      \emph{Send $INFO()$ packet to other nodes in the subregion}\;
      \emph{Wait $INFO()$ packet from other nodes in the subregion}\; 
      %\emph{UPDATE $RE_j$ for every sent or received INFO Packet}\;
      %\emph{ Collect information and construct the list L for all nodes in the subregion}\;
      
      %\If{ the received INFO Packet = No. of nodes in it's subregion -1  }{
      \emph{LeaderID = Leader election}\;
      \If{$ s_j.ID = LeaderID $}{
        \emph{$s_j.status$ = COMPUTATION}\;
        \emph{$\left\{\left(X_{1,k},\dots,X_{T,k}\right)\right\}_{k \in J}$ =
          Execute Integer Program Algorithm($T,J$)}\;
        \emph{$s_j.status$ = COMMUNICATION}\;
        \emph{Send $ActiveSleep()$ to each node $k$ in subregion a packet \\
          with vector of activity scheduling $(X_{1,k},\dots,X_{T,k})$}\;
        \emph{Update $RE_j $}\;
      }   
      \Else{
        \emph{$s_j.status$ = LISTENING}\;
        \emph{Wait $ActiveSleep()$ packet from the Leader}\;
        % \emph{After receiving Packet, Retrieve the schedule and the $T$ rounds}\;
        \emph{Update $RE_j $}\;
      }  
      %  }
  }
  \Else { Exclude $s_j$ from entering in the current sensing phase}
  
 %   \emph{return X} \;
\caption{MuDiLCO($s_j$)}
\label{alg:MuDiLCO}

\end{algorithm}




\section{Experimental Study and Analysis}
\label{ch4:sec:03}

\subsection{Simulation Setup}
\label{ch4:sec:03:01}
We  conducted  a  series of  simulations  to  evaluate  the efficiency  and  the
relevance  of   our  approach,  using  the  discrete   event  simulator  OMNeT++
\cite{ref158}.     The simulation     parameters    are     summarized     in
Table~\ref{table3}.  Each experiment  for  a network  is  run over  25~different
random topologies and  the results presented hereafter are  the average of these
25 runs.
%Based on the results of our proposed work in~\cite{idrees2014coverage}, we found as the region of interest are divided into larger subregions as the network lifetime increased. In this simulation, the network are divided into 16 subregions. 
We  performed  simulations for  five  different  densities  varying from  50  to
250~nodes deployed  over  a  $50 \times  25~m^2  $  sensing field.  More
precisely, the  deployment is controlled  at a coarse  scale in order  to ensure
that  the deployed  nodes can  cover the  sensing field  with the  given sensing
range.

%%RC these parameters are realistic?
%% maybe we can increase the field and sensing range. 5mfor Rs it seems very small... what do the other good papers consider ?

\begin{table}[ht]
\caption{Relevant parameters for network initializing.}
% title of Table
\centering
% used for centering table
\begin{tabular}{c|c}
% centered columns (4 columns)
      \hline
%inserts double horizontal lines
Parameter & Value  \\ [0.5ex]
   
%Case & Strategy (with Two Leaders) & Strategy (with One Leader) & Simple Heuristic \\ [0.5ex]
% inserts table
%heading
\hline
% inserts single horizontal line
Sensing field size & $(50 \times 25)~m^2 $   \\
% inserting body of the table
%\hline
Network size &  50, 100, 150, 200 and 250~nodes   \\
%\hline
Initial energy  & 500-700~joules  \\  
%\hline
Sensing time for one round & 60 Minutes \\
$E_{R}$ & 36 Joules\\
$R_s$ & 5~m   \\     
%\hline
$W_{\Theta}$ & 1   \\
% [1ex] adds vertical space
%\hline
$W_{U}$ & $|P|^2$
%inserts single line
\end{tabular}
\label{table3}
% is used to refer this table in the text
\end{table}
  
Our protocol  is declined into  four versions: MuDiLCO-1,  MuDiLCO-3, MuDiLCO-5,
and  MuDiLCO-7, corresponding  respectively to  $T=1,3,5,7$ ($T$  the  number of
rounds in one sensing period).  In  the following, we will make comparisons with
two other methods. The first method, called DESK and proposed by \cite{DESK},
is  a   full  distributed  coverage   algorithm.   The  second   method,  called
GAF~\cite{GAF}, consists in dividing  the region into fixed squares.
During the decision  phase, in each square, one sensor is  then chosen to remain
active during the sensing phase time.

Some preliminary experiments were performed in chapter 3 to study the choice of the number of
subregions  which subdivides  the  sensing field,  considering different  network
sizes. They show that as the number of subregions increases, so does the network
lifetime. Moreover,  it makes  the MuDiLCO protocol  more robust  against random
network  disconnection due  to node  failures.  However,  too  many subdivisions
reduce the advantage  of the optimization. In fact, there  is a balance between
the  benefit  from the  optimization  and the  execution  time  needed to  solve
it. Therefore, we have set the number of subregions to 16 rather than 32.

We have used an energy consumption model, which is presented in chapter 1, section \ref{ch1:sec9:subsec2}. The initial energy of each node  is randomly set in the interval $[500;700]$.  A sensor node  will not participate in the  next round if its  remaining energy is less than  $E_{R}=36~\mbox{Joules}$, 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.

\subsection{Metrics}
\label{ch4:sec:03:02}
To evaluate our approach we consider the following performance metrics:

\begin{enumerate}[i]
  
\item {{\bf Coverage Ratio (CR)}:} the coverage ratio measures how much of the area
  of a sensor field is covered. In our case, the sensing field is represented as
  a connected grid  of points and we use  each grid point as a  sample point to
  compute the coverage. The coverage ratio can be calculated by:
\begin{equation*}
\scriptsize
\mbox{CR}(\%) = \frac{\mbox{$n^t$}}{\mbox{$N$}} \times 100,
\end{equation*}
where $n^t$ is  the number of covered  grid points by the active  sensors of all
subregions during round $t$ in the current sensing phase and $N$ is the total number
of grid points  in the sensing field of  the network. In our simulations $N = 51
\times 26 = 1326$ grid points.

\item{{\bf Number  of Active Sensors Ratio  (ASR)}:} it is important  to have as
  few  active  nodes  as  possible  in  each  round, in  order  to  minimize  the
  communication overhead  and maximize the network lifetime.  The Active Sensors
  Ratio is defined as follows:
\begin{equation*}
\scriptsize  \mbox{ASR}(\%) = \frac{\sum\limits_{r=1}^R
  \mbox{$A_r^t$}}{\mbox{$|J|$}} \times 100,
\end{equation*}
where $A_r^t$ is the number of  active sensors in the subregion $r$ during round
$t$ in the  current sensing phase, $|J|$  is the total number of  sensors in the
network, and $R$ is the total number of subregions in the network.

\item {{\bf Network Lifetime}:} is described in chapter 3, section \ref{ch3:sec:04:02}.

\item {{\bf  Energy Consumption  (EC)}:} the average energy consumption  can be
  seen as the total energy consumed by the sensors during the $Lifetime_{95}$ or
  $Lifetime_{50}$  divided  by the  number  of rounds.  EC  can  be computed  as
  follows:

 % New version with global loops on period
  \begin{equation*}
    \scriptsize
    \mbox{EC} = \frac{\sum\limits_{m=1}^{M_L} \left[ \left( E^{\mbox{com}}_m+E^{\mbox{list}}_m+E^{\mbox{comp}}_m \right) +\sum\limits_{t=1}^{T_m} \left( E^{a}_t+E^{s}_t \right) \right]}{\sum\limits_{m=1}^{M_L} T_m},
  \end{equation*}


where  $M_L$ is  the number  of periods  and  $T_m$ the  number of rounds in  a
period~$m$, both  during $Lifetime_{95}$  or $Lifetime_{50}$. The total energy
consumed by the  sensors (EC) comes through taking  into consideration four main
energy  factors.   The  first  one  ,  denoted  $E^{\scriptsize  \mbox{com}}_m$,
represents  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, $E^a_t$ and $E^s_t$
indicate the energy consummed by the whole network in round $t$.


\item {{\bf Execution Time}:} is described in chapter 3, section \ref{ch3:sec:04:02}.
  
\item {{\bf Stopped simulation runs}:} is described in chapter 3, section \ref{ch3:sec:04:02}.

\end{enumerate}



\subsection{Results analysis and Comparison }
\label{ch4:sec:03:02}

\subsection{Coverage ratio} 
\label{ch4:sec:03:02:01}
Figure~\ref{fig3} shows  the average coverage  ratio for 150 deployed  nodes. We
can notice that for the first thirty rounds both DESK and GAF provide a coverage
which is a little bit better than the one of MuDiLCO.  

This is due  to the fact that, in comparison with  MuDiLCO which uses optimization
to put in  SLEEP status redundant sensors, more sensor  nodes remain active with
DESK and GAF.   As a consequence, when the number of  rounds increases, a larger
number of node failures  can be observed in DESK and GAF,  resulting in a faster
decrease of the coverage ratio.   Furthermore, our protocol allows to maintain a
coverage ratio  greater than  50\% for far  more rounds.  Overall,  the proposed
sensor  activity scheduling based  on optimization  in MuDiLCO  maintains higher
coverage ratios of the  area of interest for a larger number  of rounds. It also
means that MuDiLCO saves more energy,  with less dead nodes, at most for several
rounds, and thus should extend the network lifetime.

\begin{figure}[ht!]
\centering
 \includegraphics[scale=0.5] {Figures/ch4/R1/CR.pdf}   
\caption{Average coverage ratio for 150 deployed nodes}
\label{fig3}
\end{figure} 

\subsection{Active sensors ratio} 
\label{ch4:sec:03:02:02}
It is crucial to have as few active nodes as possible in each round, in order to 
minimize    the    communication    overhead    and   maximize    the    network
lifetime. Figure~\ref{fig4}  presents the active  sensor ratio for  150 deployed
nodes all along the network lifetime. It appears that up to round thirteen, DESK
and GAF have  respectively 37.6\% and 44.8\% of nodes  in ACTIVE status, whereas
MuDiLCO clearly  outperforms them  with only 24.8\%  of active nodes.  After the
thirty-fifth round, MuDiLCO exhibits larger numbers of active nodes, which agrees
with  the  dual  observation  of  higher  level  of  coverage  made  previously.
Obviously, in  that case DESK  and GAF have  less active nodes, since  they have
activated many nodes  at the beginning. Anyway, MuDiLCO  activates the available
nodes in a more efficient manner.

\begin{figure}[ht!]
\centering
\includegraphics[scale=0.5]{Figures/ch4/R1/ASR.pdf}  
\caption{Active sensors ratio for 150 deployed nodes}
\label{fig4}
\end{figure} 

\subsection{Stopped simulation runs}
\label{ch4:sec:03:02:03}
Figure~\ref{fig6} reports the cumulative  percentage of stopped simulations runs
per round for 150 deployed nodes. This figure gives the  breakpoint for each method.  DESK stops first,  after approximately 45~rounds, because it consumes the
more energy by  turning on a large number of redundant  nodes during the sensing
phase. GAF  stops secondly for the  same reason than  DESK.  MuDiLCO overcomes
DESK and GAF because the  optimization process distributed on several subregions
leads  to coverage  preservation and  so extends  the network  lifetime.  Let us
emphasize that the  simulation continues as long as a network  in a subregion is
still connected.


\begin{figure}[ht!]
\centering
\includegraphics[scale=0.5]{Figures/ch4/R1/SR.pdf} 
\caption{Cumulative percentage of stopped simulation runs for 150 deployed nodes }
\label{fig6}
\end{figure} 

\subsection{Energy consumption} \label{subsec:EC}
\label{ch4:sec:03:02:04}
We  measure  the  energy  consumed  by the  sensors  during  the  communication,
listening, computation, active, and sleep status for different network densities
and   compare   it   with   the  two   other   methods.    Figures~\ref{fig7}(a)
and~\ref{fig7}(b)  illustrate  the  energy  consumption,  considering  different
network sizes, for $Lifetime_{95}$ and $Lifetime_{50}$.

\begin{figure}[h!]
  \centering
  \begin{tabular}{cl}
    \parbox{9.5cm}{\includegraphics[scale=0.5]{Figures/ch4/R1/EC95.pdf}} & (a) \\
    \verb+ + \\
    \parbox{9.5cm}{\includegraphics[scale=0.5]{Figures/ch4/R1/EC50.pdf}} & (b)
  \end{tabular}
  \caption{Energy consumption for (a) $Lifetime_{95}$ and 
    (b) $Lifetime_{50}$}
  \label{fig7}
\end{figure} 

The  results  show  that  MuDiLCO  is  the  most  competitive  from  the  energy
consumption point of view.  The  other approaches have a high energy consumption
due  to activating a  larger number  of redundant  nodes as  well as  the energy
consumed during  the different  status of the  sensor node. Among  the different
versions of our protocol, the MuDiLCO-7  one consumes more energy than the other
versions. This is  easy to understand since the bigger the  number of rounds and
the number of  sensors involved in the integer program are,  the larger the time
computation to solve the optimization problem is. To improve the performances of
MuDiLCO-7, we  should increase the  number of subregions  in order to  have less
sensors to consider in the integer program.


\subsection{Execution time}
\label{ch4:sec:03:02:05}
We observe  the impact of the  network size and of  the number of  rounds on the
computation  time.   Figure~\ref{fig77} gives  the  average  execution times  in
seconds (needed to solve optimization problem) for different values of $T$. The original execution time is computed as described in chapter 3, section \ref{ch3:sec:04:02}.

%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{fig77} for different network sizes.

\begin{figure}[ht!]
\centering
\includegraphics[scale=0.5]{Figures/ch4/R1/T.pdf}  
\caption{Execution Time (in seconds)}
\label{fig77}
\end{figure} 

As expected,  the execution time increases  with the number of  rounds $T$ taken
into account to schedule the sensing phase. The times obtained for $T=1,3$
or $5$ seem bearable, but for $T=7$ they become quickly unsuitable for a sensor
node, especially when  the sensor network size increases.   Again, we can notice
that if we want  to schedule the nodes activities for a  large number of rounds,
we need to choose a relevant number of subregions in order to avoid a complicated
and cumbersome optimization.  On the one hand, a large value  for $T$ permits to
reduce the  energy-overhead due  to the three  pre-sensing phases, on  the other
hand  a leader  node may  waste a  considerable amount  of energy  to  solve the
optimization problem.

\subsection{Network lifetime}
\label{ch4:sec:03:02:06}
The next  two figures,  Figures~\ref{fig8}(a) and \ref{fig8}(b),  illustrate the
network lifetime  for different network sizes,  respectively for $Lifetime_{95}$
and  $Lifetime_{50}$.  Both  figures show  that the  network  lifetime increases
together with the  number of sensor nodes, whatever the  protocol, thanks to the
node  density  which  results in  more  and  more  redundant  nodes that  can  be
deactivated and thus save energy.  Compared to the other approaches, our MuDiLCO
protocol  maximizes the  lifetime of  the network.   In particular  the  gain in
lifetime for a  coverage over 95\% is greater than 38\%  when switching from GAF
to MuDiLCO-3.  The  slight decrease that can be observed  for MuDiLCO-7 in case
of  $Lifetime_{95}$  with  large  wireless  sensor  networks  results  from  the
difficulty  of the optimization  problem to  be solved  by the  integer program.
This  point was  already noticed  in subsection  \ref{subsec:EC} devoted  to the
energy consumption,  since network lifetime and energy  consumption are directly
linked.

\begin{figure}[t!]
  \centering
  \begin{tabular}{cl}
    \parbox{9.5cm}{\includegraphics[scale=0.5]{Figures/ch4/R1/LT95.pdf}} & (a) \\
    \verb+ + \\
    \parbox{9.5cm}{\includegraphics[scale=0.5]{Figures/ch4/R1/LT50.pdf}} & (b)
  \end{tabular}
  \caption{Network lifetime for (a) $Lifetime_{95}$ and 
    (b) $Lifetime_{50}$}
  \label{fig8}
\end{figure} 


\section{Conclusion}
\label{ch4:sec:04}

We have addressed  the problem of the coverage and of the lifetime optimization in
wireless  sensor networks.  This is  a key  issue as  sensor nodes  have limited
resources in terms of memory, energy, and computational power. To cope with this
problem,  the field  of sensing  is divided  into smaller  subregions  using the
concept  of divide-and-conquer  method, and  then  we propose  a protocol  which
optimizes coverage  and lifetime performances in each  subregion.  Our protocol,
called MuDiLCO (Multiround  Distributed Lifetime Coverage Optimization) combines
two  efficient   techniques:  network   leader  election  and   sensor  activity
scheduling.

The activity  scheduling in each subregion  works in periods,  where each period
consists of four  phases: (i) Information Exchange, (ii)  Leader Election, (iii)
Decision Phase to plan the activity  of the sensors over $T$ rounds, (iv) Sensing
Phase itself divided into T rounds.

Simulations  results show the  relevance of  the proposed  protocol in  terms of
lifetime, coverage  ratio, active  sensors ratio, energy  consumption, execution
time. Indeed,  when dealing with  large wireless sensor networks,  a distributed
approach, like  the one we  propose, allows to  reduce the difficulty of  a single
global optimization problem by partitioning it in many smaller problems, one per
subregion, that can be solved  more easily. Nevertheless, results also show that
it is not possible to plan the activity of sensors over too many rounds, because
the resulting optimization problem leads to too high resolution times and thus to
an excessive energy consumption.