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\chapter{Perimeter-based Coverage Optimization to Improve Lifetime in Wireless Sensor Networks}
\label{ch6}


\section{summary}
\label{ch6:sec:01}

The most important problem in a Wireless Sensor Network (WSN) is to optimize the
use of its limited energy provision, so that it can fulfill its monitoring task
as long as  possible. Among  known  available approaches  that can  be used  to
improve  power  management,  lifetime coverage  optimization  provides  activity
scheduling which ensures sensing coverage while minimizing the energy cost. In
this paper,  we propose such an approach called Perimeter-based Coverage Optimization
protocol (PeCO). It is a  hybrid of centralized and distributed methods: the
region of interest is first subdivided into subregions and our protocol is then
distributed among sensor nodes in each  subregion.
The novelty of our approach lies essentially in the formulation of a new
mathematical optimization  model based on the  perimeter coverage level  to schedule
sensors' activities.  Extensive simulation experiments have been performed using
OMNeT++, the  discrete event simulator, to  demonstrate that PeCO  can
offer longer lifetime coverage for WSNs in comparison with some other protocols.

\section{THE PeCO PROTOCOL DESCRIPTION}
\label{ch6:sec:02}

\noindent  In  this  section,  we  describe in  details  our  Lifetime  Coverage
Optimization protocol.  First we present the  assumptions we made and the models
we considered (in particular the perimeter coverage one), second we describe the
background idea of our protocol, and third  we give the outline of the algorithm
executed by each node.



\subsection{Assumptions and Models}
\label{ch6:sec:02:01}
PeCO protocol uses the same assumptions and network model that presented in chapter 4, section \ref{ch4:sec:02:01}.

The PeCO protocol  uses the  same perimeter-coverage  model as  Huang and
Tseng in~\cite{ref133}. It  can be expressed as follows:  a sensor is
said to be perimeter  covered if all the points on its  perimeter are covered by
at least  one sensor  other than  itself.  They  proved that  a network  area is
$k$-covered if and only if each sensor in the network is $k$-perimeter-covered (perimeter covered by at least $k$ sensors).
  
Figure~\ref{pcm2sensors}(a)  shows  the coverage  of  sensor  node~$0$. On  this
figure, we can  see that sensor~$0$ has  nine neighbors and we  have reported on
its  perimeter (the  perimeter  of the  disk  covered by  the  sensor) for  each
neighbor  the  two  points  resulting  from  intersection  of  the  two  sensing
areas. These points are denoted for  neighbor~$i$ by $iL$ and $iR$, respectively
for  left and  right from  neighbor  point of  view.  The  resulting couples  of
intersection points subdivide the perimeter of sensor~$0$ into portions called
arcs.

\begin{figure}[ht!]
  \centering
  \begin{tabular}{@{}cr@{}}
    \includegraphics[width=95mm]{Figures/ch6/pcm.jpg} & \raisebox{3.25cm}{(a)} \\
    \includegraphics[width=95mm]{Figures/ch6/twosensors.jpg} & \raisebox{2.75cm}{(b)}
  \end{tabular}
  \caption{(a) Perimeter  coverage of sensor node  0 and (b) finding  the arc of
    $u$'s perimeter covered by $v$.}
  \label{pcm2sensors}
\end{figure} 

Figure~\ref{pcm2sensors}(b) describes the geometric information used to find the
locations of the  left and right points of  an arc on the perimeter  of a sensor
node~$u$ covered by a sensor node~$v$. Node~$v$ is supposed to be located on the
west  side of  sensor~$u$,  with  the following  respective  coordinates in  the
sensing area~: $(v_x,v_y)$ and $(u_x,u_y)$. From the previous coordinates we can
compute the euclidean distance between nodes~$u$ and $v$: $Dist(u,v)=\sqrt{\vert
  u_x  - v_x  \vert^2 +  \vert u_y-v_y  \vert^2}$, while  the angle~$\alpha$  is
obtained through  the formula: $$\alpha =  \arccos \left(\dfrac{Dist(u,v)}{2R_s}
\right).$$ The arc on the perimeter of~$u$ defined by the angular interval $[\pi
  - \alpha,\pi + \alpha]$ is said to be perimeter-covered by sensor~$v$.

Every couple of intersection points is placed on the angular interval $[0,2\pi]$
in  a  counterclockwise manner,  leading  to  a  partitioning of  the  interval.
Figure~\ref{pcm2sensors}(a)  illustrates  the arcs  for  the  nine neighbors  of
sensor $0$ and  Figure~\ref{expcm} gives the position of  the corresponding arcs
in  the interval  $[0,2\pi]$. More  precisely, we  can see  that the  points are
ordered according  to the  measures of  the angles  defined by  their respective
positions. The intersection points are  then visited one after another, starting
from the first  intersection point  after  point~zero,  and  the maximum  level  of
coverage is determined  for each interval defined by two  successive points. The
maximum  level of  coverage is  equal to  the number  of overlapping  arcs.  For
example, 
between~$5L$  and~$6L$ the maximum  level of  coverage is equal  to $3$
(the value is highlighted in yellow  at the bottom of Figure~\ref{expcm}), which
means that at most 2~neighbors can cover  the perimeter in addition to node $0$. 
Table~\ref{my-label} summarizes for each coverage  interval the maximum level of
coverage and  the sensor  nodes covering the  perimeter.  The  example discussed
above is thus given by the sixth line of the table.


\begin{figure*}[t!]
\centering
\includegraphics[width=150.5mm]{Figures/ch6/expcm2.jpg}  
\caption{Maximum coverage levels for perimeter of sensor node $0$.}
\label{expcm}
\end{figure*} 


 \begin{table}[h!]
 \caption{Coverage intervals and contributing sensors for sensor node 0.}
 \centering
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\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 coverage interval\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{table}


In the PeCO  protocol, the scheduling of the 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{ch6:sec:03}.  Note that  when a sensor
node  has a  part of  its sensing  range outside  the WSN  sensing field,  as in
Figure~\ref{ex4pcm}, the maximum coverage level for  this arc is set to $\infty$
and  the  corresponding  interval  will  not   be  taken  into  account  by  the
optimization algorithm.


\begin{figure}[h!]
\centering
\includegraphics[width=95.5mm]{Figures/ch6/ex4pcm.jpg}  
\caption{Sensing range outside the WSN's area of interest.}
\label{ex4pcm}
\end{figure} 





\subsection{The Main Idea}
\label{ch6:sec:02:02}

\noindent The  WSN area of interest is, in a  first step, divided  into regular
homogeneous subregions  using a divide-and-conquer  algorithm. In a  second step
our  protocol  will  be  executed  in a distributed way in each subregion
simultaneously to schedule nodes' activities for one sensing period.

As  shown in  Figure~\ref{fig2}, node  activity  scheduling is  produced by  our
protocol in a periodic manner. Each period is divided into 4 stages: Information
(INFO)  Exchange,  Leader Election,  Decision  (the  result of  an  optimization
problem),  and  Sensing.   For  each  period there  is  exactly  one  set  cover
responsible for  the sensing task.  Protocols  based on a periodic  scheme, like
PeCO, are more  robust against an unexpected  node failure. On the  one hand, if
a node failure is discovered before  taking the decision, the corresponding sensor
node will  not be considered  by the optimization  algorithm. On  the other
hand, if the sensor failure happens after  the decision, the sensing task of the
network will be temporarily affected: only  during the period of sensing until a
new period starts, since a new set cover will take charge of the sensing task in
the next period. The energy consumption and some other constraints can easily be
taken  into  account since  the  sensors  can  update  and then  exchange  their
information (including their  residual energy) at the beginning  of each period.
However, the pre-sensing  phases (INFO Exchange, Leader  Election, and Decision)
are energy consuming, even for nodes that will not join the set cover to monitor
the area.

\begin{figure}[t!]
\centering
\includegraphics[width=95.5mm]{Figures/ch6/Model.pdf}  
\caption{PeCO protocol.}
\label{fig2}
\end{figure} 




\subsection{PeCO Protocol Algorithm}
\label{ch6:sec:02:03}


\noindent The  pseudocode implementing the protocol on a node is  given below.
More  precisely,  Algorithm~\ref{alg:PeCO}  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}
%  \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{ $RE_k \geq E_{th}$ }{
      \emph{$s_k.status$ = COMMUNICATION}\;
      \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 $}{
         \emph{$s_k.status$ = COMPUTATION}\;
         
      \If{$ s_k.ID $ is Not previously selected as a Leader }{
          \emph{ Execute the 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)}{
      
         \emph{ Use the same previous cover set for current sensing stage}\;
      }
      \Else{
            \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{$s_k.status$ = COMMUNICATION}\;
        \emph{Send $ActiveSleep()$ to each node $l$ in subregion}\;
        \emph{Update $RE_k $}\;
      }   
      \Else{
        \emph{$s_k.status$ = LISTENING}\;
        \emph{Wait $ActiveSleep()$ packet from the Leader}\;
        \emph{Update $RE_k $}\;
      }  
  }
  \Else { Exclude $s_k$ from entering in the current sensing stage}
\caption{PeCO($s_k$)}
\label{alg:PeCO}
\end{algorithm}

In this  algorithm, K.CurrentSize and K.PreviousSize  respectively represent the
current number and  the previous number of living nodes in  the subnetwork of the
subregion.  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 their  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 numbers 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.



\section{Perimeter-based Coverage Problem Formulation}
\label{ch6:sec:03}


\noindent In this  section, the coverage model is  mathematically formulated. We
start  with a  description of  the notations  that will  be used  throughout the
section.

First, we have the following sets:
\begin{itemize}
\item $S$ represents the set of WSN sensor nodes;
\item $A \subseteq S $ is the subset of alive sensors;
\item  $I_j$  designates  the  set  of  coverage  intervals  (CI)  obtained  for
  sensor~$j$.
\end{itemize}
$I_j$ refers to the set of  coverage intervals which have been defined according
to the  method introduced in  subsection~\ref{ch6:sec:02:01}. For a coverage  interval $i$,
let $a^j_{ik}$ denotes  the indicator function of whether  sensor~$k$ is involved
in coverage interval~$i$ of sensor~$j$, that is:
\begin{equation}
a^j_{ik} = \left \{ 
\begin{array}{lll}
  1 & \mbox{if sensor $k$ is involved in the } \\
        &       \mbox{coverage interval $i$ of sensor $j$}, \\
  0 & \mbox{otherwise.}\\
\end{array} \right.
%\label{eq12} 
\notag
\end{equation}
Note that $a^k_{ik}=1$ by definition of the interval.

Second,  we define  several binary  and integer  variables.  Hence,  each binary
variable $X_{k}$  determines the activation of  sensor $k$ in the  sensing phase
($X_k=1$ if  the sensor $k$  is active or 0  otherwise).  $M^j_i$ is  an integer
variable  which  measures  the  undercoverage  for  the  coverage  interval  $i$
corresponding to  sensor~$j$. In  the same  way, the  overcoverage for  the same
coverage interval is given by the variable $V^j_i$.

If we decide to sustain a level of coverage equal to $l$ all along the perimeter
of sensor  $j$, we have  to ensure  that at least  $l$ sensors involved  in each
coverage  interval $i  \in I_j$  of  sensor $j$  are active.   According to  the
previous notations, the number of active sensors in the coverage interval $i$ of
sensor $j$  is given by  $\sum_{k \in A} a^j_{ik}  X_k$.  To extend  the network
lifetime,  the objective  is to  activate a  minimal number  of sensors  in each
period to  ensure the  desired coverage  level. As the  number of  alive sensors
decreases, it becomes impossible to reach  the desired level of coverage for all
coverage intervals. Therefore we use variables  $M^j_i$ and $V^j_i$ as a measure
of the  deviation between  the desired  number of active  sensors in  a coverage
interval and  the effective  number. And  we try  to minimize  these deviations,
first to  force the  activation of  a minimal  number of  sensors to  ensure the
desired coverage level, and if the desired level cannot be completely satisfied,
to reach a coverage level as close as possible to the desired one.


Our coverage optimization problem can then be mathematically expressed as follows: 
%Objective:
\begin{equation} %\label{eq:ip2r}
\left \{
\begin{array}{ll}
\min \sum_{j \in S} \sum_{i \in I_j} (\alpha^j_i ~ M^j_i + \beta^j_i ~ V^j_i )&\\
\textrm{subject to :}&\\
\sum_{k \in A} ( a^j_{ik} ~ X_{k}) + M^j_i  \geq l \quad \forall i \in I_j, \forall j \in S\\
%\label{c1} 
\sum_{k \in A} ( a^j_{ik} ~ X_{k}) - V^j_i  \leq l \quad \forall i \in I_j, \forall j \in S\\
% \label{c2}
% \Theta_{p}\in \mathbb{N}, &\forall p \in P\\
% U_{p} \in \{0,1\}, &\forall p \in P\\
X_{k} \in \{0,1\}, \forall k \in A
\end{array}
\right.
\notag
\end{equation}
$\alpha^j_i$ and $\beta^j_i$  are nonnegative weights selected  according to the
relative importance of satisfying the associated level of coverage. For example,
weights associated with  coverage intervals of a specified part  of a region may
be  given by a  relatively larger  magnitude than  weights associated  with another
region. This  kind of integer program  is inspired from the  model developed for
brachytherapy treatment planning  for optimizing dose  distribution
\cite{0031-9155-44-1-012}. The integer  program must be solved by  the leader in
each subregion at the beginning of  each sensing phase, whenever the environment
has  changed (new  leader,  death of  some  sensors). Note  that  the number  of
constraints in the model is constant  (constraints of coverage expressed for all
sensors), whereas the number of variables $X_k$ decreases over periods, since we
consider only alive  sensors (sensors with enough energy to  be alive during one
sensing phase) in the model.

\section{Performance Evaluation and Analysis}
\label{ch6:sec:04}

\subsection{Simulation Settings}
\label{ch6:sec:04:01}

The WSN  area of interest is  supposed to be divided  into 16~regular subregions. %and we use the same energy consumption than in our previous work~\cite{Idrees2}.
Table~\ref{table3} gives the chosen parameters settings.

\begin{table}[ht]
\caption{Relevant parameters for network initialization.}
% title of Table
\centering
% used for centering table
\begin{tabular}{c|c}
% centered columns (4 columns)
\hline
Parameter & Value  \\ [0.5ex]
   
\hline
% inserts single horizontal line
Sensing field & $(50 \times 25)~m^2 $   \\

WSN size &  100, 150, 200, 250, and 300~nodes   \\
%\hline
Initial energy  & in range 500-700~Joules  \\  
%\hline
Sensing period & duration of 60 minutes \\
$E_{th}$ & 36~Joules\\
$R_s$ & 5~m   \\     
%\hline
$\alpha^j_i$ & 0.6   \\
% [1ex] adds vertical space
%\hline
$\beta^j_i$ & 0.4
%inserts single line
\end{tabular}
\label{table3}
% is used to refer this table in the text
\end{table}


To obtain experimental results which are relevant,  simulations  with  five
different node densities going from  100 to 300~nodes were performed considering
each time 25~randomly  generated networks. The nodes are deployed  on a field of
interest of $(50 \times 25)~m^2 $ in such a way that they cover the field with a
high coverage ratio. Each node has an  initial energy level, in Joules, which is
randomly drawn in the interval $[500-700]$. If its energy provision reaches a
value below  the threshold $E_{th}=36$~Joules,  the minimum energy needed  for a
node  to stay  active during  one period,  it will no more  participate in the
coverage task. This value corresponds to the energy needed by the sensing phase,
obtained by multiplying the energy consumed in active state (9.72 mW) with the
time in seconds for one  period (3600 seconds), and  adding the energy  for the
pre-sensing phases. According  to the interval of initial energy,  a sensor may
be active during at most 20 periods.


The values  of $\alpha^j_i$ and  $\beta^j_i$ have been  chosen to ensure  a good
network coverage and a longer WSN lifetime.  We have given a higher priority to
the  undercoverage  (by  setting  the  $\alpha^j_i$ with  a  larger  value  than
$\beta^j_i$)  so as  to prevent  the non-coverage  for the  interval~$i$ of  the
sensor~$j$.  On the  other hand,  we have assigned to
$\beta^j_i$ a value which is slightly lower so as to minimize the number of active sensor nodes which contribute
in covering the interval.

We applied the performance metrics, which are described in chapter 4, section \ref{ch4:sec:04:04} in order to evaluate the efficiency of our approach. We used the modeling language and the optimization solver which are mentioned in chapter 4, section \ref{ch4:sec:04:02}. In addition, we employed an energy consumption model, which is presented in chapter 4, section \ref{ch4:sec:04:03}.


\subsection{Simulation Results}
\label{ch6:sec:04:02}

In  order  to  assess and  analyze  the  performance  of  our protocol  we  have implemented PeCO protocol in  OMNeT++~\cite{ref158} simulator.  Besides PeCO, three other protocols,  described in  the next paragraph,  will  be  evaluated for comparison purposes. 
%The simulations were run  on a laptop DELL with an Intel Core~i3~2370~M (2.4~GHz) processor (2  cores) whose MIPS  (Million Instructions Per Second) rate  is equal to 35330. To  be consistent with the use  of a sensor node based on  Atmels AVR ATmega103L microcontroller (6~MHz) having  a MIPS rate equal to 6, the original execution time  on the laptop is  multiplied by 2944.2 $\left(\frac{35330}{2} \times  \frac{1}{6} \right)$.  The modeling  language for Mathematical Programming (AMPL)~\cite{AMPL} is  employed to generate the integer program instance  in a  standard format, which  is then read  and solved  by the optimization solver  GLPK (GNU  linear Programming Kit  available in  the public domain) \cite{glpk} through a Branch-and-Bound method.
As said previously, the PeCO is  compared with three other approaches. The first one,  called  DESK,  is  a  fully distributed  coverage  algorithm  proposed  by \cite{DESK}. The second one,  called GAF~\cite{GAF}, consists in dividing  the monitoring  area into  fixed  squares. Then,  during the  decision phase, in each square, one sensor is  chosen to remain active during the sensing phase. The last  one, the DiLCO protocol~\cite{Idrees2}, is  an improved version of a research work we presented in~\cite{ref159}. Let us notice that PeCO and  DiLCO protocols are  based on the  same framework. In  particular, the choice for the simulations of a partitioning in 16~subregions was chosen because it corresponds to the configuration producing the better results for DiLCO. The protocols are distinguished from one another by the formulation  of the integer program providing the set of sensors which have to be activated in each sensing phase. DiLCO protocol tries to satisfy the coverage of a set of primary points, whereas PeCO protocol objective is to reach a desired level of coverage for each sensor perimeter. In our experimentations, we chose a level of coverage equal to one ($l=1$).



\subsubsection{Coverage Ratio}
\label{ch6:sec:04:02:01}

Figure~\ref{fig333}  shows the  average coverage  ratio for  200 deployed  nodes
obtained with the  four protocols. DESK, GAF, and DiLCO  provide a slightly better
coverage ratio with respectively 99.99\%,  99.91\%, and 99.02\%, compared to the 98.76\%
produced by  PeCO for the  first periods. This  is due to  the fact that  at the
beginning the DiLCO protocol  puts to  sleep status  more redundant  sensors (which
slightly decreases the coverage ratio), while the three other protocols activate
more sensor  nodes. Later, when the  number of periods is  beyond~70, it clearly
appears that  PeCO provides a better  coverage ratio and keeps  a coverage ratio
greater  than 50\%  for  longer periods  (15  more compared  to  DiLCO, 40  more
compared to DESK). The energy saved by  PeCO in the early periods allows later a
substantial increase of the coverage performance.

\parskip 0pt    
\begin{figure}[h!]
\centering
 \includegraphics[scale=0.5] {Figures/ch6/R/CR.eps} 
\caption{Coverage ratio for 200 deployed nodes.}
\label{fig333}
\end{figure} 



\subsubsection{Active Sensors Ratio}
\label{ch6:sec:04:02:02}

Having the less active sensor nodes in  each period is essential to minimize the
energy consumption  and thus to  maximize the network  lifetime.  Figure~\ref{fig444}
shows the  average active nodes ratio  for 200 deployed nodes.   We observe that
DESK and  GAF have 30.36  \% and  34.96 \% active  nodes for the  first fourteen
rounds and  DiLCO and PeCO  protocols compete perfectly  with only 17.92  \% and
20.16 \% active  nodes during the same  time interval. As the  number of periods
increases, PeCO protocol  has a lower number of active  nodes in comparison with
the three other approaches, while keeping a greater coverage ratio as shown in
Figure \ref{fig333}.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.5]{Figures/ch6/R/ASR.eps}  
\caption{Active sensors ratio for 200 deployed nodes.}
\label{fig444}
\end{figure} 

\subsubsection{The Energy Consumption}
\label{ch6:sec:04:02:03}

We studied the effect of the energy  consumed by the WSN during the communication,
computation, listening, active, and sleep status for different network densities
and  compared  it for  the  four  approaches.  Figures~\ref{fig3EC}(a)  and  (b)
illustrate  the  energy   consumption  for  different  network   sizes  and  for
$Lifetime95$ and  $Lifetime50$. The results show  that our PeCO protocol  is the
most competitive  from the energy  consumption point of  view. As shown  in both
figures, PeCO consumes much less energy than the three other methods.  One might
think that the  resolution of the integer  program is too costly  in energy, but
the  results show  that it  is very  beneficial to  lose a  bit of  time in  the
selection of  sensors to  activate.  Indeed the  optimization program  allows to
reduce significantly the number of active  sensors and so the energy consumption
while keeping a good coverage level.

\begin{figure}[h!]
  \centering
  \begin{tabular}{@{}cr@{}}
    \includegraphics[scale=0.475]{Figures/ch6/R/EC95.eps} & \raisebox{2.75cm}{(a)} \\
    \includegraphics[scale=0.475]{Figures/ch6/R/EC50.eps} & \raisebox{2.75cm}{(b)}
  \end{tabular}
  \caption{Energy consumption per period for (a)~$Lifetime_{95}$ and (b)~$Lifetime_{50}$.}
  \label{fig3EC}
\end{figure} 



\subsubsection{The Network Lifetime}
\label{ch6:sec:04:02:04}

We observe the superiority of PeCO and DiLCO protocols in comparison with the
two    other   approaches    in    prolonging   the    network   lifetime.    In
Figures~\ref{fig3LT}(a)  and (b),  $Lifetime95$ and  $Lifetime50$ are  shown for
different  network  sizes.   As  highlighted  by  these  figures,  the  lifetime
increases with the size  of the network, and it is clearly   largest for DiLCO
and PeCO  protocols.  For instance,  for a  network of 300~sensors  and coverage
ratio greater than 50\%, we can  see on Figure~\ref{fig3LT}(b) that the lifetime
is about twice longer with  PeCO compared to DESK protocol.  The performance
difference    is    more    obvious   in    Figure~\ref{fig3LT}(b)    than    in
Figure~\ref{fig3LT}(a) because the gain induced  by our protocols increases with
 time, and the lifetime with a coverage  of 50\% is far  longer than with
95\%.

\begin{figure}[h!]
  \centering
  \begin{tabular}{@{}cr@{}}
    \includegraphics[scale=0.475]{Figures/ch6/R/LT95.eps} & \raisebox{2.75cm}{(a)} \\  
    \includegraphics[scale=0.475]{Figures/ch6/R/LT50.eps} & \raisebox{2.75cm}{(b)}
  \end{tabular}
  \caption{Network Lifetime for (a)~$Lifetime_{95}$ and (b)~$Lifetime_{50}$.}
  \label{fig3LT}
\end{figure} 

Figure~\ref{figLTALL}  compares  the  lifetime  coverage of  our  protocols  for
different coverage  ratios. We denote by  Protocol/50, Protocol/80, Protocol/85,
Protocol/90, and  Protocol/95 the amount  of time  during which the  network can
satisfy an area coverage greater than $50\%$, $80\%$, $85\%$, $90\%$, and $95\%$
respectively, where the term Protocol refers to  DiLCO  or PeCO.  Indeed there  are applications
that do not require a 100\% coverage of  the area to be monitored. PeCO might be
an interesting  method since  it achieves  a good balance  between a  high level
coverage ratio and network lifetime. PeCO always outperforms DiLCO for the three
lower  coverage  ratios,  moreover  the   improvements  grow  with  the  network
size. DiLCO is better  for coverage ratios near 100\%, but in  that case PeCO is
not ineffective for the smallest network sizes.

\begin{figure}[h!]
\centering \includegraphics[scale=0.5]{Figures/ch6/R/LTa.eps}
\caption{Network lifetime for different coverage ratios.}
\label{figLTALL}
\end{figure} 



\section{Conclusion}
\label{ch6:sec:04}

In this chapter, we have studied the problem of  Perimeter-based Coverage Optimization in
WSNs. We have designed  a new protocol, called Perimeter-based  Coverage Optimization, which
schedules nodes'  activities (wake up  and sleep  stages) with the  objective of
maintaining a  good coverage ratio  while maximizing the network  lifetime. This
protocol is  applied in a distributed  way in regular subregions  obtained after
partitioning the area of interest in a preliminary step. It works in periods and
is based on the resolution of an integer program to select the subset of sensors
operating in active status 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. We  have carried out  several simulations  to  evaluate the  proposed protocol. The simulation  results  show   that  PeCO  is  more   energy-efficient  than  other approaches, with respect to lifetime,  coverage ratio, active sensors ratio, and
energy consumption.

We plan to extend our framework so that the schedules are planned for multiple
sensing periods.
%in order to compute all active sensor schedules in only one step for many periods;
We also want  to improve our integer program to  take into account heterogeneous
sensors  from both  energy  and node  characteristics point of views.
%the third, we are investigating new optimization model based on the sensing range so as to maximize the lifetime coverage in WSN;
Finally,  it   would  be   interesting  to  implement   our  protocol   using  a
sensor-testbed to evaluate it in real world applications.