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\chapter{Multiround Distributed Lifetime Coverage Optimization Protocol}
\label{ch5}


\section{Introduction}
\label{ch5:sec:01}

%The fast  developments of low-cost  sensor devices  and  wireless communications have allowed the emergence of WSNs. A WSN includes a large number of small, limited-power sensors that  can sense, process, and transmit data over a wireless  communication. They communicate  with each other by using multi-hop wireless communications and cooperate together  to monitor the area of interest, so that  each measured data can be  reported to a monitoring  center called sink for further  analysis~\cite{ref222}.  There  are several fields  of application covering  a wide  spectrum for a  WSN, including health, home, environmental, military, and industrial applications~\cite{ref19}.

%On the one hand sensor nodes run on batteries with limited capacities, and it is often  costly  or  simply  impossible  to  replace  and/or  recharge  batteries, especially in remote and hostile environments. Obviously, to achieve a long life of the  network it is important  to conserve battery  power. Therefore, lifetime optimization is one of the most critical issues in wireless sensor networks. On the other hand we must guarantee  coverage over the area of interest. To fulfill these two objectives, the main idea  is to take advantage of overlapping sensing regions to turn-off redundant sensor nodes  and thus save energy. In this paper, we concentrate  on the area coverage  problem, with the  objective of maximizing the network lifetime by using an optimized multiround scheduling.
We study the problem of designing an energy-efficient optimization algorithm that divides the sensor nodes in a WSN into multiple cover sets such that the area of interest is monitored as long as possible. Providing multiple cover sets can be used to improve the energy efficiency of WSNs. Therefore, in order to increase the longevity of the WSN and conserve the energy, it can be useful to provide multiple cover sets in one time step and schedule them for multiple rounds, so that the battery life of a sensor is not wasted due to the repeated execution of the presensing phases of MuDiLCO protocol.

The MuDiLCO protocol (for Multiround Distributed Lifetime Coverage Optimization protocol) presented in this chapter is an extension of the approach introduced in chapter 4. 
%Simulation results have shown that it was more interesting to divide the area into several subregions, given the computation complexity. 
Compared to DiLCO protocol in chapter 4, in this one we study the possibility of dividing the sensing phase into multiple rounds. In fact, in this chapter we make a multiround optimization while it was a single round optimization in our protocol in chapter 4.


The remainder of this chapter continues with section \ref{ch5:sec:02} where a detailed description of MuDiLCO Protocol is given. The next section describes the primary points based multiround coverage problem formulation  which is used to schedule the activation of sensors in multiple cover sets. Section \ref{ch5:sec:04} shows the simulation
results. The chapter ends with a conclusion and some suggestions for further works.


 


\section{Description of the MuDiLCO Protocol }
\label{ch5:sec:02}
%\noindent In this section, we introduce the MuDiLCO protocol which is distributed on each subregion in the area of interest. It 
Like DiLCO, the MuDiLCO protocol is based on two energy-efficient
mechanisms: subdividing the area of interest into several subregions (like a cluster architecture) using the divide and conquer method, where the sensor nodes cooperate within each subregion as independent group in order to achieve a network leader election. Sensor activity scheduling is used to maintain the coverage and to prolong the network lifetime, it is applied periodically. MuDiLCO uses the same assumptions, primary point coverage and network models, than DiLCO, given in section \ref{ch4:sec:02:01} and \ref{ch4:sec:02:02}, respectively.

 
%\subsection{Background Idea and Algorithm}
%\label{ch5: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. 
%The information exchange among wireless sensor nodes is described in chapter 4, section \ref{ch4:sec:02:03:01}. The leader election in each subregion is explained in chapter 4, section \ref{ch4:sec:02:03:02}, 
The difference with MuDiLCO in that the elected leader in each subregion is for each period. In the decision phase, each leader 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 leader will send an ActiveSleep  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. Each sensing phase is itself divided into $T$ rounds and for each round a set of sensors (a cover set) is responsible for the sensing task. 
%Each sensor node in the subregion will receive an ActiveSleep packet from leader, 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 ActiveSleep packet is obtained. 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/ch5/GeneralModel.jpg} % 70mm  Modelgeneral.pdf
\caption{MuDiLCO protocol.}
\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 during this phase. 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.

 

\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{ $RE_j \geq E_{th}$ }{
      \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{Primary Points based Multiround Coverage Problem Formulation}
\label{ch5:sec:03}


According to Algorithm~\ref{alg:MuDiLCO}, the integer program is based on the model
proposed by  \cite{ref156} with some modifications, where  the objective of our model is
to find  a maximum number of non-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{eq133} 
\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{eq1114} 
\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_{th}} \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 $E_{th}$ 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 $W_{\theta}$.


 
 

\section{Experimental Study and Analysis}
\label{ch5:sec:04}

\subsection{Simulation Setup}
\label{ch5:sec:04:01}
We conducted  a  series of  simulations  to  evaluate  the efficiency  and  the
relevance  of our  approach,  using  the  discrete   event  simulator  OMNeT++
\cite{ref158}. We performed the optimization in the same manner than in chapter 4, considering the same energy model. The simulation  parameters are summarized in Table~\ref{tablech4}.  Each experiment for  a network  is  run over  25~different random topologies and  the results presented hereafter are  the average of these 25 runs.
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.

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 three other methods. DESK \cite{DESK}, GAF~\cite{GAF}, and DiLCO~\cite{Idrees2}, where MuDiLCO-1 is the same of DiLCO.
%Some preliminary experiments were performed in chapter 4 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 set the number of subregions to 16 rather than 32 as explained in section \ref{ch4:sec:04:05}. 
We use the modeling language and the optimization solver which are mentioned in section \ref{ch4:sec:04:02}. 
%In addition, the energy consumption model is presented in chapter 4, section \ref{ch4:sec:04:03}. 

\subsection{Metrics}
\label{ch5:sec:04:02}
To evaluate our approach we consider the following performance metrics
\begin{frame}{}
\begin{enumerate}[i)]
  
\item {{\bf Coverage Ratio (CR)}:} 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)}:}  The Active Sensors
  Ratio for round t is defined as follows:
\begin{equation*}
\scriptsize  \mbox{$ASR^t$}(\%) = \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  Energy Consumption  (EC)}:}  EC  can  be computed  as follows:

  \begin{equation*}
    \scriptsize
    \mbox{EC} = \frac{\sum\limits_{m=1}^{M} \left[ \left( E^{\mbox{com}}_m+E^{\mbox{list}}_m+E^{\mbox{comp}}_m \right) +\sum\limits_{t=1}^{T} \left( E^{a}_t+E^{s}_t \right) \right]}{\sum\limits_{m=1}^{M} T},
  \end{equation*}

The energy factors of above equation are described in section \ref{ch4:sec:04:04}. $E^a_t$ and $E^s_t$
indicate the energy consumed by the whole network in round $t$ of the sensing phase.

%where  $M$ is  the number  of periods  and  $T$ 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 consumed by the whole network in round $t$.

 %$\right\}$
 
\item {{\bf Execution Time}:} 
\item {{\bf Stopped simulation runs}:} \makebox(0,0){\put(0,2.2\normalbaselineskip){%
               $\left.\rule{0pt}{2.3\normalbaselineskip}\right\}$ Described in section \ref{ch4:sec:04:04}.}}
\item {{\bf Network Lifetime}:} 

\end{enumerate}

\end{frame}

\subsection{Results Analysis and Comparison }
\label{ch5:sec:04:03}


\begin{enumerate}[i)]

\item {{\bf Coverage Ratio}}
%\subsection{Coverage ratio} 
%\label{ch5: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 fewer dead nodes, at most for several
rounds, and thus should extend the network lifetime.

\begin{figure}[h!]
\centering
 \includegraphics[scale=0.8] {Figures/ch5/R1/CR.pdf}   
\caption{Average coverage ratio for 150 deployed nodes}
\label{fig3}
\end{figure} 


\item {{\bf Active sensors ratio}}
%\subsection{Active sensors ratio} 
%\label{ch5: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 mode, whereas
MuDiLCO clearly  outperforms them  with only 23.7\%  of active nodes.  After the
thirty-sixth 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 fewer active nodes since they have activated many nodes  in the beginning. Anyway, MuDiLCO  activates the available nodes in a more efficient manner.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.8]{Figures/ch5/R1/ASR.pdf}  
\caption{Active sensors ratio for 150 deployed nodes}
\label{fig4}
\end{figure} 

\item {{\bf Stopped simulation runs}}
%\subsection{Stopped simulation runs}
%\label{ch5: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}[t]
\centering
\includegraphics[scale=0.8]{Figures/ch5/R1/SR.pdf} 
\caption{Cumulative percentage of stopped simulation runs for 150 deployed nodes }
\label{fig6}
\end{figure} 

 
\item {{\bf Energy consumption}} \label{subsec:EC} 
%\subsection{Energy consumption} 
%\label{ch5: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{multicols}{1}
\centering
\includegraphics[scale=0.8]{Figures/ch5/R1/EC95.pdf}\\~ ~ ~ ~ ~(a) \\
%\vfill
\includegraphics[scale=0.8]{Figures/ch5/R1/EC50.pdf}\\~ ~ ~ ~ ~(b)

%\end{multicols} 
\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, the larger the time computation to solve the optimization problem. To improve the performances of MuDiLCO-7, we  should increase the  number of subregions  in order to  have fewer sensors to consider in the integer program.
\\ \\ \\


 \item {{\bf Execution time}}
%\subsection{Execution time}
%\label{ch5: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 4, section \ref{ch4: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}[h!]
\centering
\includegraphics[scale=0.8]{Figures/ch5/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. \\


\item {{\bf Network lifetime}}
%\subsection{Network lifetime}
%\label{ch5: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. \\ \\ \\ 

\begin{figure}[h!]
\centering
% \begin{multicols}{0}
\centering
\includegraphics[scale=0.8]{Figures/ch5/R1/LT95.pdf}\\~ ~ ~ ~ ~(a) \\
%\hfill 
\includegraphics[scale=0.8]{Figures/ch5/R1/LT50.pdf}\\~ ~ ~ ~ ~(b)

%\end{multicols} 
\caption{Network lifetime for (a) $Lifetime_{95}$ and (b) $Lifetime_{50}$}
  \label{fig8}
\end{figure}


\end{enumerate} 
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 \ref{subsec:EC} devoted  to the
energy consumption,  since network lifetime and energy  consumption are directly linked.

\section{Conclusion}
\label{ch5:sec:05}

%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.  
In this chapter, we have presented a protocol, called MuDiLCO (Multiround  Distributed Lifetime Coverage Optimization) that 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 into 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. Compared with DiLCO, it is clear that MuDiLCO improves the network lifetime especially for the dense network, but it is  less robust than DiLCO under sensor nodes failures. Therefore, choosing the number of rounds $T$ depends on the type of application the WSN is deployed for.