\DeclareGraphicsRule{.ps}{pdf}{.pdf}{`ps2pdf -dEPSCrop -dNOSAFER #1 \noexpand\OutputFile}
\begin{document}
-\title{Lifetime Coverage Optimization Protocol \\
- in Wireless Sensor Networks} %LiCO Protocol
+%\title{Lifetime Coverage Optimization Protocol \\
+% in Wireless Sensor Networks}
+\title{Perimeter-based Coverage Optimization to Improve \\
+ Lifetime in Wireless Sensor Networks}
\author{Ali Kadhum Idrees,~\IEEEmembership{}
Karine Deschinkel,~\IEEEmembership{}
Michel Salomon,~\IEEEmembership{}
and~Rapha\"el Couturier ~\IEEEmembership{}
- \thanks{The authors are with FEMTO-ST Institute, UMR 6174 CNRS, University of Franche-Comt\'e,
+ \thanks{The authors are with FEMTO-ST Institute, UMR 6174 CNRS, University of Franche-Comte,
Belfort, France. Email: ali.idness@edu.univ-fcomte.fr, $\lbrace$karine.deschinkel,
michel.salomon, raphael.couturier$\rbrace$@univ-fcomte.fr}}
\maketitle
\begin{abstract}
-The most important problem in Wireless Sensor Networks (WSNs) 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
+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 a such approach called Lifetime Coverage Optimization
-protocol (LiCO). 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. A sensor node which runs LiCO
-protocol repeats periodically four stages: information exchange, leader
-election, optimization decision, and sensing. More precisely, the scheduling of
-nodes' activities (sleep/wake up duty cycles) is achieved in each subregion by a
-leader selected after cooperation between nodes within the same subregion. The
-novelty of approach lies essentially in the formulation of a new mathematical
-optimization model based on perimeter coverage level to schedule sensors'
-activities. Extensive simulation experiments have been performed using OMNeT++,
-the discrete event simulator, to demonstrate that LiCO is capable to offer
-longer lifetime coverage for WSNs in comparison with some other protocols.
+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.
+% A sensor node which runs LiCO protocol repeats periodically four stages:
+%information exchange, leader election, optimization decision, and sensing.
+%More precisely, the scheduling of nodes' activities (sleep/wake up duty cycles)
+%is achieved in each subregion by a leader selected after cooperation between
+%nodes within the same 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.
\end{abstract}
% Note that keywords are not normally used for peerreview papers.
\begin{IEEEkeywords}
-Wireless Sensor Networks, Area Coverage, Network lifetime, Optimization, Scheduling.
+Wireless Sensor Networks, Area Coverage, Network Lifetime, Optimization, Scheduling.
\end{IEEEkeywords}
\IEEEpeerreviewmaketitle
can send the data it collects in its environment, thanks to its sensor, to the
user by means of sink nodes. The features of a WSN made it suitable for a wide
range of application in areas such as business, environment, health, industry,
-military, and son~\cite{yick2008wireless}. Typically, a sensor node contains
+military, and so on~\cite{yick2008wireless}. Typically, a sensor node contains
three main components~\cite{anastasi2009energy}: a sensing unit able to measure
physical, chemical, or biological phenomena observed in the environment; a
processing unit which will process and store the collected measurements; a radio
communication is supplied by a power supply which is a battery. This battery has
a limited energy provision and it may be unsuitable or impossible to replace or
recharge it in most applications. Therefore it is necessary to deploy WSN with
-high density in order to increase the reliability and to exploit node redundancy
+high density in order to increase reliability and to exploit node redundancy
thanks to energy-efficient activity scheduling approaches. Indeed, the overlap
of sensing areas can be exploited to schedule alternatively some sensors in a
low power sleep mode and thus save energy. Overall, the main question that must
be answered is: how to extend the lifetime coverage of a WSN as long as possible
-while ensuring a high level of coverage? So, this last years many
+while ensuring a high level of coverage? These past few years many
energy-efficient mechanisms have been suggested to retain energy and extend the
lifetime of the WSNs~\cite{rault2014energy}.
This paper makes the following contributions.
\begin{enumerate}
-\item We devise a framework to schedule nodes to be activated alternatively such
+\item We have devised a framework to schedule nodes to be activated alternatively such
that the network lifetime is prolonged while ensuring that a certain level of
- coverage is preserved. A key idea in our framework is to exploit spatial an
- temporal subdivision. On the one hand the area of interest if divided into
- several smaller subregions and on the other hand the time line is divided into
+ coverage is preserved. A key idea in our framework is to exploit spatial and
+ temporal subdivision. On the one hand, the area of interest is divided into
+ several smaller subregions and, on the other hand, the time line is divided into
periods of equal length. In each subregion the sensor nodes will cooperatively
choose a leader which will schedule nodes' activities, and this grouping of
sensors is similar to typical cluster architecture.
-\item We propose a new mathematical optimization model. Instead of trying to
+\item We have proposed a new mathematical optimization model. Instead of trying to
cover a set of specified points/targets as in most of the methods proposed in
the literature, we formulate an integer program based on perimeter coverage of
each sensor. The model involves integer variables to capture the deviations
- between the actual level of coverage and the required level. So that an
+ between the actual level of coverage and the required level. Hence, an
optimal scheduling will be obtained by minimizing a weighted sum of these
deviations.
-\item We conducted extensive simulation experiments, using the discrete event
- simulator OMNeT++, to demonstrate the efficiency of our protocol. We compared
- our LiCO protocol to two approaches found in the literature:
+\item We have conducted extensive simulation experiments, using the discrete event
+ simulator OMNeT++, to demonstrate the efficiency of our protocol. We have compared
+ our PeCO protocol to two approaches found in the literature:
DESK~\cite{ChinhVu} and GAF~\cite{xu2001geography}, and also to our previous
work published in~\cite{Idrees2} which is based on another optimization model
for sensor scheduling.
\end{enumerate}
-%Two combined integrated energy-efficient techniques have been used by LiCO protocol in order to maximize the lifetime coverage in WSN: the first, by dividing the area of interest into several smaller subregions based on divide-and-conquer method and then one leader elected for each subregion in an independent, distributed, and simultaneous way by the cooperation among the sensor nodes within each subregion, and this similar to cluster architecture;
+%Two combined integrated energy-efficient techniques have been used by PeCO protocol in order to maximize the lifetime coverage in WSN: the first, by dividing the area of interest into several smaller subregions based on divide-and-conquer method and then one leader elected for each subregion in an independent, distributed, and simultaneous way by the cooperation among the sensor nodes within each subregion, and this similar to cluster architecture;
% the second, activity scheduling based new optimization model has been used to provide the optimal cover set that will take the mission of sensing during current period. This optimization algorithm is based on a perimeter-coverage model so as to optimize the shared perimeter among the sensors in each subregion, and this represents as a energu-efficient control topology mechanism in WSN.
The rest of the paper is organized as follows. In the next section we review
-some related work in the field. Section~\ref{sec:The LiCO Protocol Description}
-is devoted to the LiCO protocol description and Section~\ref{cp} focuses on the
+some related work in the field. Section~\ref{sec:The PeCO Protocol Description}
+is devoted to the PeCO protocol description and Section~\ref{cp} focuses on the
coverage model formulation which is used to schedule the activation of sensor
nodes. Section~\ref{sec:Simulation Results and Analysis} presents simulations
results and discusses the comparison with other approaches. Finally, concluding
-remarks are drawn and some suggestions given for future works in
+remarks are drawn and some suggestions are given for future works in
Section~\ref{sec:Conclusion and Future Works}.
% that show that our protocol outperforms others protocols.
\label{sec:Literature Review}
\noindent In this section, we summarize some related works regarding the
-coverage problem and distinguish our LiCO protocol from the works presented in
+coverage problem and distinguish our PeCO protocol from the works presented in
the literature.
The most discussed coverage problems in literature can be classified in three
categories~\cite{li2013survey} according to their respective monitoring
objective. Hence, area coverage \cite{Misra} means that every point inside a
-fixed area must be monitored, while target coverage~\cite{yang2014novel} refer
+fixed area must be monitored, while target coverage~\cite{yang2014novel} refers
to the objective of coverage for a finite number of discrete points called
targets, and barrier coverage~\cite{HeShibo}\cite{kim2013maximum} focuses on
preventing intruders from entering into the region of interest. In
-\cite{Deng2012} authors transform the area coverage problem to the target
+\cite{Deng2012} authors transform the area coverage problem into the target
coverage one taking into account the intersection points among disks of sensors
nodes or between disk of sensor nodes and boundaries. In
\cite{Huang:2003:CPW:941350.941367} authors prove that if the perimeters of
sensors are sufficiently covered it will be the case for the whole area. They
provide an algorithm in $O(nd~log~d)$ time to compute the perimeter-coverage of
each sensor, where $d$ denotes the maximum number of sensors that are
-neighboring to a sensor and $n$ is the total number of sensors in the
-network. {\it In LiCO protocol, instead of determining the level of coverage of
+neighbors to a sensor and $n$ is the total number of sensors in the
+network. {\it In PeCO protocol, instead of determining the level of coverage of
a set of discrete points, our optimization model is based on checking the
perimeter-coverage of each sensor to activate a minimal number of sensors.}
activated in succession to achieve the desired network lifetime. Vu
\cite{chin2007}, Padmatvathy {\em et al.}~\cite{pc10}, propose algorithms
working in a periodic fashion where a cover set is computed at the beginning of
-each period. {\it Motivated by these works, LiCO protocol works in periods,
+each period. {\it Motivated by these works, PeCO protocol works in periods,
where each period contains a preliminary phase for information exchange and
decisions, followed by a sensing phase where one cover set is in charge of the
sensing task.}
-Various centralized and distributed approaches, or even a mixing of these two
+Various centralized and distributed approaches, or even a mixing of these two
concepts, have been proposed to extend the network lifetime. In distributed
-algorithms~\cite{yangnovel,ChinhVu,qu2013distributed} each sensors decides of
-its own activity scheduling after an information exchange with its neighbors.
-The main interest of a such approach is to avoid long range communications and
-thus to reduce the energy dedicated to the communications. Unfortunately, since
-each node has only information on its immediate neighbors (usually the one-hop
-ones) it may take a bad decision leading to a global suboptimal solution.
-Conversely, centralized
-algorithms~\cite{cardei2005improving,zorbas2010solving,pujari2011high} always
+algorithms~\cite{yangnovel,ChinhVu,qu2013distributed} each sensor decides of its
+own activity scheduling after an information exchange with its neighbors. The
+main interest of such an approach is to avoid long range communications and thus
+to reduce the energy dedicated to the communications. Unfortunately, since each
+node has only information on its immediate neighbors (usually the one-hop ones)
+it may make a bad decision leading to a global suboptimal solution. Conversely,
+centralized
+algorithms~\cite{cardei2005improving,zorbas2010solving,pujari2011high} always
provide nearly or close to optimal solution since the algorithm has a global
view of the whole network. The disadvantage of a centralized method is obviously
-its high cost in communications needed to transmit to a single node, the base
-station which will globally schedule nodes' activities, data from all the other
-sensor nodes in the area. The price in communications can be very huge since
-long range communications will be needed. In fact the larger the WNS, the higher
-the communication and thus energy cost. {\it In order to be suitable for
- large-scale networks, in the LiCO protocol the area of interest is divided
- into several smaller subregions, and in each one, a node called the leader is
- in charge for selecting the active sensors for the current period. Thus our
- protocol is scalable and a globally distributed method, whereas it is
- centralized in each subregion.}
-
-Various coverage scheduling algorithms have been developed this last years.
+its high cost in communications needed to transmit to a single node, the base
+station which will globally schedule nodes' activities, data from all the other
+sensor nodes in the area. The price in communications can be huge since
+long range communications will be needed. In fact the larger the WNS is, the
+higher the communication and thus the energy cost are. {\it In order to be
+ suitable for large-scale networks, in the PeCO protocol, the area of interest
+ is divided into several smaller subregions, and in each one, a node called the
+ leader is in charge of selecting the active sensors for the current
+ period. Thus our protocol is scalable and is a globally distributed method,
+ whereas it is centralized in each subregion.}
+
+Various coverage scheduling algorithms have been developed these past few years.
Many of them, dealing with the maximization of the number of cover sets, are
heuristics. These heuristics involve the construction of a cover set by
including in priority the sensor nodes which cover critical targets, that is to
energy constraints. Column generation techniques, well-known and widely
practiced techniques for solving linear programs with too many variables, have
also been
-used~\cite{castano2013column,rossi2012exact,deschinkel2012column}. {\it In LiCO
+used~\cite{castano2013column,rossi2012exact,deschinkel2012column}. {\it In the PeCO
protocol, each leader, in charge of a subregion, solves an integer program
which has a twofold objective: minimize the overcoverage and the undercoverage
of the perimeter of each sensor.}
%\uppercase{\textbf{shortcomings}}. In spite of many energy-efficient protocols for maintaining the coverage and improving the network lifetime in WSNs were proposed, non of them ensure the coverage for the sensing field with optimal minimum number of active sensor nodes, and for a long time as possible. For example, in a full centralized algorithms, an optimal solutions can be given by using optimization approaches, but in the same time, a high energy is consumed for the execution time of the algorithm and the communications among the sensors in the sensing field, so, the full centralized approaches are not good candidate to use it especially in large WSNs. Whilst, a full distributed algorithms can not give optimal solutions because this algorithms use only local information of the neighboring sensors, but in the same time, the energy consumption during the communications and executing the algorithm is highly lower. Whatever the case, this would result in a shorter lifetime coverage in WSNs.
-%\uppercase{\textbf{Our Protocol}}. In this paper, a Lifetime Coverage Optimization Protocol, called (LiCO) in WSNs is suggested. The sensing field is divided into smaller subregions by means of divide-and-conquer method, and a LiCO protocol is distributed in each sensor in the subregion. The network lifetime in each subregion is divided into periods, each period includes 4 stages: Information Exchange, Leader election, decision based activity scheduling optimization, and sensing. The leaders are elected in an independent, asynchronous, and distributed way in all the subregions of the WSN. After that, energy-efficient activity scheduling mechanism based new optimization model is performed by each leader in the subregions. This optimization model is based on the perimeter coverage model in order to producing the optimal cover set of active sensors, which are taken the responsibility of sensing during the current period. LiCO protocol merges between two energy efficient mechanisms, which are used the main advantages of the centralized and distributed approaches and avoids the most of their disadvantages.
+%\uppercase{\textbf{Our Protocol}}. In this paper, a Lifetime Coverage Optimization Protocol, called (PeCO) in WSNs is suggested. The sensing field is divided into smaller subregions by means of divide-and-conquer method, and a PeCO protocol is distributed in each sensor in the subregion. The network lifetime in each subregion is divided into periods, each period includes 4 stages: Information Exchange, Leader election, decision based activity scheduling optimization, and sensing. The leaders are elected in an independent, asynchronous, and distributed way in all the subregions of the WSN. After that, energy-efficient activity scheduling mechanism based new optimization model is performed by each leader in the subregions. This optimization model is based on the perimeter coverage model in order to producing the optimal cover set of active sensors, which are taken the responsibility of sensing during the current period. PeCO protocol merges between two energy efficient mechanisms, which are used the main advantages of the centralized and distributed approaches and avoids the most of their disadvantages.
-\section{ The LiCO Protocol Description}
-\label{sec:The LiCO Protocol Description}
+\section{ The PeCO Protocol Description}
+\label{sec:The PeCO Protocol Description}
-\noindent In this section, we describe in details our Lifetime Coverage
+\noindent In this section, we describe in details our Perimeter-based 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
%Before proceeding in the presentation of the main ideas of the protocol, we will briefly describe the perimeter coverage model and give some necessary assumptions and definitions.
\subsection{Assumptions and Models}
+\label{CI}
\noindent A WSN consisting of $J$ stationary sensor nodes randomly and uniformly
distributed in a bounded sensor field is considered. The wireless sensors are
deployed in high density to ensure initially a high coverage ratio of the area
of interest. We assume that all the sensor nodes are homogeneous in terms of
communication, sensing, and processing capabilities and heterogeneous from
-energy provision point of view. The location information is available to a
+the energy provision point of view. The location information is available to a
sensor node either through hardware such as embedded GPS or location discovery
algorithms. We assume that each sensor node can directly transmit its
measurements to a mobile sink node. For example, a sink can be an unmanned
within a disk centered at a sensor with a radius equal to the sensing range are
said to be covered by this sensor. We also assume that the communication range
$R_c$ satisfies $R_c \geq 2 \cdot R_s$. In fact, Zhang and Zhou~\cite{Zhang05}
-proved that if the transmission range fulfills the previous hypothesis, a
+proved that if the transmission range fulfills the previous hypothesis, the
complete coverage of a convex area implies connectivity among active nodes.
-\indent LiCO protocol uses the same perimeter-coverage model than Huang and
+The PeCO protocol uses the same perimeter-coverage model as Huang and
Tseng in~\cite{huang2005coverage}. 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.
+$k$-covered if and only if each sensor in the network is $k$-perimeter-covered (perimeter covered by at least $k$ sensors).
%According to this model, we named the intersections among the sensor nodes in the sensing field as intersection points. Instead of working with the coverage area, we consider for each sensor a set of intersection points which are determined by using perimeter-coverage model.
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
+neighbor the two points resulting from the 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
+for left and right from a neighboing 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=75mm]{pcm.jpg} & \raisebox{3.25cm}{(a)}
- \\ \includegraphics[width=75mm]{twosensors.jpg} & \raisebox{2.75cm}{(b)}
+ \includegraphics[width=75mm]{pcm.jpg} & \raisebox{3.25cm}{(a)} \\
+ \includegraphics[width=75mm]{twosensors.jpg} & \raisebox{2.75cm}{(b)}
\end{tabular}
- \caption{Perimeter coverage of sensor node 0 (a) and finding the arc of $u$'s
- perimeter covered by $v$.}
+ \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~$s$ is supposed to be located on the
+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 $\alpha = arccos \left(\dfrac{Dist(u,v)}{2R_s}
-\right)$. So, the arc on the perimeter of node~$u$ defined by the angular
-interval $[\pi - \alpha,\pi + \alpha]$ is said to be perimeter-covered by sensor
-node $v$.
+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
+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 first intersection point after point~zero, and the maximum level of
+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 $0$.
+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.
%The points reported on the line segment $[0,2\pi]$ separates it in intervals as shown in figure~\ref{expcm}. For example, for each neighboring sensor of sensor 0, place the points $\alpha^ 1_L$, $\alpha^ 1_R$, $\alpha^ 2_L$, $\alpha^ 2_R$, $\alpha^ 3_L$, $\alpha^ 3_R$, $\alpha^ 4_L$, $\alpha^ 4_R$, $\alpha^ 5_L$, $\alpha^ 5_R$, $\alpha^ 6_L$, $\alpha^ 6_R$, $\alpha^ 7_L$, $\alpha^ 7_R$, $\alpha^ 8_L$, $\alpha^ 8_R$, $\alpha^ 9_L$, and $\alpha^ 9_R$; on the line segment $[0,2\pi]$, and then sort all these points in an ascending order into a list. Traverse the line segment $[0,2\pi]$ by visiting each point in the sorted list from left to right and determine the coverage level of each interval of the sensor 0 (see figure \ref{expcm}). For each interval, we sum up the number of parts of segments, and we deduce a level of coverage for each interval. For instance, the interval delimited by the points $5L$ and $6L$ contains three parts of segments. That means that this part of the perimeter of the sensor $0$ may be covered by three sensors, sensor $0$ itself and sensors $2$ and $5$. The level of coverage of this interval may reach $3$ if all previously mentioned sensors are active. Let say that sensors $0$, $2$ and $5$ are involved in the coverage of this interval. Table~\ref{my-label} summarizes the level of coverage for each interval and the sensors involved in for sensor node 0 in figure~\ref{pcm2sensors}(a).
% to determine the level of the perimeter coverage for each left and right point of a segment.
-\begin{figure*}[ht!]
+\begin{figure*}[t!]
\centering
-\includegraphics[width=137.5mm]{expcm.pdf}
+\includegraphics[width=127.5mm]{expcm2.jpg}
\caption{Maximum coverage levels for perimeter of sensor node $0$.}
\label{expcm}
\end{figure*}
\fi
- \begin{table}[h]
+ \begin{table}[h!]
\caption{Coverage intervals and contributing sensors for sensor node 0.}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
-\begin{tabular}[c]{@{}c@{}}Left \\ point \\ angle~$\alpha$ \end{tabular} & \begin{tabular}[c]{@{}c@{}}Interval \\ left \\ point\end{tabular} & \begin{tabular}[c]{@{}c@{}}Interval \\ right \\ point\end{tabular} & \begin{tabular}[c]{@{}c@{}}Maximum \\ coverage\\ level\end{tabular} & \multicolumn{5}{c|}{\begin{tabular}[c]{@{}c@{}}Set of sensors\\ involved \\ in interval coverage\end{tabular}} \\ \hline
+\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
\end{table}
-%The optimization algorithm that used by LiCO protocol based on the perimeter coverage levels of the left and right points of the segments and worked to minimize the number of sensor nodes for each left or right point of the segments within each sensor node. The algorithm minimize the perimeter coverage level of the left and right points of the segments, while, it assures that every perimeter coverage level of the left and right points of the segments greater than or equal to 1.
+%The optimization algorithm that used by PeCO protocol based on the perimeter coverage levels of the left and right points of the segments and worked to minimize the number of sensor nodes for each left or right point of the segments within each sensor node. The algorithm minimize the perimeter coverage level of the left and right points of the segments, while, it assures that every perimeter coverage level of the left and right points of the segments greater than or equal to 1.
-In LiCO protocol, scheduling of sensor nodes' activities is formulated with an
+In 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{cp}. 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 $\infty$
+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}[t!]
+\begin{figure}[h!]
\centering
\includegraphics[width=62.5mm]{ex4pcm.jpg}
\caption{Sensing range outside the WSN's area of interest.}
our protocol will be executed in a distributed way in each subregion
simultaneously to schedule nodes' activities for one sensing period.
-As shown in figure~\label{fig2}, node activity scheduling is produced by our
+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
-LiCO, are more robust against an unexpected node failure. On the one hand, if
-node failure is discovered before taking the decision, the corresponding sensor
-node will not be considered by the optimization algorithm, and, on the other
+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
\begin{figure}[t!]
\centering
\includegraphics[width=80mm]{Model.pdf}
-\caption{LiCO protocol}
+\caption{PeCO protocol.}
\label{fig2}
\end{figure}
-We define two types of packets to be used by LiCO protocol:
+We define two types of packets to be used by PeCO protocol:
%\begin{enumerate}[(a)]
\begin{itemize}
\item INFO packet: sent by each sensor node to all the nodes inside a same
determine the activities scheduling;
\item ACTIVE: node is sensing;
\item SLEEP: node is turned off;
-\item COMMUNICATION: transmits or recevives packets.
+\item COMMUNICATION: transmits or receives packets.
\end{itemize}
%\end{enumerate}
%Below, we describe each phase in more details.
-\subsection{LiCO Protocol Algorithm}
+\subsection{PeCO Protocol Algorithm}
-The pseudocode implementing the protocol on a node is given below. More
-precisely, Algorithm~\label{alg:LiCO} gives a brief description of the protocol
-applied by a sensor node $s_k$ where $k$ is the node index in the WSN.
+\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}
}
\emph{$s_k.status$ = COMMUNICATION}\;
- \emph{Send $ActiveSleep()$ to each node $l$ in subregion} \;
+ \emph{Send $ActiveSleep()$ to each node $l$ in subregion}\;
\emph{Update $RE_k $}\;
}
\Else{
}
}
\Else { Exclude $s_k$ from entering in the current sensing stage}
-
-
-\caption{LiCO($s_k$)}
-\label{alg:LiCO}
-
+\caption{PeCO($s_k$)}
+\label{alg:PeCO}
\end{algorithm}
-\noindent In this algorithm, K.CurrentSize and K.PreviousSize refer to the
-current size and the previous size of the subnetwork in the subregion
-respectively. That means the number of sensor nodes which are still
-alive. Initially, the sensor node checks its remaining energy $RE_k$, which must
-be greater than a threshold $E_{th}$ in order to participate in the current
-period. Each sensor node determines its position and its subregion using an
-embedded GPS or a location discovery algorithm. After that, all the sensors
+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 its one-hop live neighbors during the information exchange. The sensors
+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 number of neighbors, larger remaining
+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
+collects information to formulate and solve the integer program which allows to
construct the set of active sensors in the sensing stage.
-%After the cooperation among the sensor nodes in the same subregion, the leader will be elected in distributed way, where each sensor node and based on it's information decide who is the leader. The selection criteria for the leader in order of priority are: larger number of neighbors, larger remaining energy, and then in case of equality, larger index. Thereafter, if the sensor node is leader, it will execute the perimeter-coverage model for each sensor in the subregion in order to determine the segment points which would be used in the next stage by the optimization algorithm of the LiCO protocol. Every sensor node is selected as a leader, it is executed the perimeter coverage model only one time during it's life in the network.
+%After the cooperation among the sensor nodes in the same subregion, the leader will be elected in distributed way, where each sensor node and based on it's information decide who is the leader. The selection criteria for the leader in order of priority are: larger number of neighbors, larger remaining energy, and then in case of equality, larger index. Thereafter, if the sensor node is leader, it will execute the perimeter-coverage model for each sensor in the subregion in order to determine the segment points which would be used in the next stage by the optimization algorithm of the PeCO protocol. Every sensor node is selected as a leader, it is executed the perimeter coverage model only one time during it's life in the network.
% The leader has the responsibility of applying the integer program algorithm (see section~\ref{cp}), which provides a set of sensors planned to be active in the sensing stage. As leader, it will send an Active-Sleep packet to each sensor in the same subregion to inform it if it has to be active or not. On the contrary, if the sensor is not the leader, it will wait for the Active-Sleep packet to know its state for the sensing stage.
-% MICHEL TO BE CONTINUED
+\section{Perimeter-based Coverage Problem Formulation}
+\label{cp}
-\section{Lifetime Coverage problem formulation}
+\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.
-\label{cp}
-In this section, the coverage model is mathematically formulated.
-For convenience, the notations are described first.
-%Then the lifetime problem of sensor network is formulated.
-\noindent $S :$ the set of all sensors in the network.\\
-\noindent $A :$ the set of alive sensors within $S$.\\
-%\noindent $I :$ the set of segment points.\\
-\noindent $I_j :$ the set of coverage intervals (CI) for sensor $j$.\\
-\noindent $I_j$ refers to the set of intervals which have been defined for each sensor $j$ in section~\ref{sec:The LiCO Protocol Description}.
-\noindent For a coverage interval $i$, let $a^j_{ik}$ denote the indicator function of whether the sensor $k$ is involved in the coverage interval $i$ of sensor $j$, that is:
+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{CI}. 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 the sensor $k$ is involved in the } \\
+ 1 & \mbox{if sensor $k$ is involved in the } \\
& \mbox{coverage interval $i$ of sensor $j$}, \\
- 0 & \mbox{Otherwise.}\\
+ 0 & \mbox{otherwise.}\\
\end{array} \right.
%\label{eq12}
\notag
\end{equation}
-Note that $a^k_{ik}=1$ by definition of the interval.\\
+Note that $a^k_{ik}=1$ by definition of the interval.
%, where the objective is to find the maximum number of non-disjoint sets of sensor nodes such that each set cover can assure the coverage for the whole region so as to extend the network lifetime in WSN. Our model uses the PCL~\cite{huang2005coverage} in order to optimize the lifetime coverage in each subregion.
-%We defined some parameters, which are related to our optimization model. In our model, we consider binary variables $X_{k}$, which determine the activation of sensor $k$ in the sensing round $k$. .
-\noindent We consider binary variables $X_{k}$ ($X_k=1$ if the sensor $k$ is active or 0 otherwise), which determine the activation of sensor $k$ in the sensing phase. We define the integer variable $M^j_i$ which measures the undercoverage for the coverage interval $i$ for sensor $j$. In the same way, we define the integer variable $V^j_i$, which measures the overcoverage for the coverage interval $i$ for sensor $j$. If we decide to sustain a level of coverage equal to $l$ all along the perimeter of the sensor $j$, we have to ensure that at least $l$ sensors involved in each coverage interval $i$ ($i \in I_j$) 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 $\sum_{k \in K} a^j_{ik} X_k$. To extend the network lifetime, the objective is to active 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 satisfy the level of coverage for all covergae intervals. We uses 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 of active sensors. 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 can not be completely satisfied, to reach a coverage level as close as possible that the desired one.
-
-
+%We defined some parameters, which are related to our optimization model. In our model, we consider binary variables $X_{k}$, which determine the activation of sensor $k$ in the sensing round $k$. .
+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.
%A system of linear constraints is imposed to attempt to keep the coverage level in each coverage interval to within specified PCL. Since it is physically impossible to satisfy all constraints simultaneously, each constraint uses a variable to either record when the coverage level is achieved, or to record the deviation from the desired coverage level. These additional variables are embedded into an objective function to be minimized.
%\noindent $V^j_i (overcoverage): $ integer value $\in \mathbb{N}$ for segment point $i$ of sensor $j$.
-
-
-
-
-\noindent Our coverage optimization problem can be mathematically formulated as follows:
+Our coverage optimization problem can then be mathematically expressed as follows:
%Objective:
-\begin{equation} \label{eq:ip2r}
+\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 )&\\
X_{k} \in \{0,1\}, \forall k \in A
\end{array}
\right.
+\notag
\end{equation}
-
-
-\noindent $\alpha^j_i$ 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 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{\uppercase{PERFORMANCE EVALUATION AND ANALYSIS}}
+$\alpha^j_i$ 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{sec:Simulation Results and Analysis}
%\noindent \subsection{Simulation Framework}
\subsection{Simulation Settings}
%\label{sub1}
-In this section, we focus on the performance of LiCO protocol, which is distributed in each sensor node in the sixteen subregions of WSN. We use the same energy consumption model which is used in~\cite{Idrees2}. Table~\ref{table3} gives the chosen parameters setting.
+
+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 initializing.}
+\caption{Relevant parameters for network initialization.}
% title of Table
\centering
% used for centering table
\hline
% inserts single horizontal line
-Sensing Field & $(50 \times 25)~m^2 $ \\
+Sensing field & $(50 \times 25)~m^2 $ \\
-Nodes Number & 100, 150, 200, 250 and 300~nodes \\
+WSN size & 100, 150, 200, 250, and 300~nodes \\
%\hline
-Initial Energy & 500-700~joules \\
+Initial energy & in range 500-700~Joules \\
%\hline
-Sensing Period & 60 Minutes \\
-$E_{th}$ & 36 Joules\\
+Sensing period & duration of 60 minutes \\
+$E_{th}$ & 36~Joules\\
$R_s$ & 5~m \\
%\hline
$\alpha^j_i$ & 0.6 \\
\label{table3}
% is used to refer this table in the text
\end{table}
-Simulations with five different node densities going from 100 to 250~nodes were
-performed considering each time 25~randomly generated networks, to obtain
-experimental results which are relevant. All simulations are repeated 25 times and the results are averaged. 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 it's 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) by 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 rounds.
-
-The values of $\alpha^j_i$ and $\beta^j_i$ have been chosen in a way that ensuring a good network coverage and for a longer time during the lifetime of the WSN. We have given a higher priority for 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 given a little bit lower value for $\beta^j_i$ so as to minimize the number of active sensor nodes that contribute in covering the interval i in sensor j.
-
-In the simulations, we introduce the following performance metrics to evaluate
-the efficiency of our approach:
+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 introduce the following performance metrics to evaluate the efficiency of our
+approach.
%\begin{enumerate}[i)]
\begin{itemize}
-\item {{\bf Network Lifetime}:} we define the network lifetime as the time until
- the coverage ratio drops below a predefined threshold. We denote by
- $Lifetime_{95}$ (respectively $Lifetime_{50}$) the amount of time during which
- the network can satisfy an area coverage greater than $95\%$ (respectively
- $50\%$). We assume that the sensor network can fulfill its task until all its
- nodes have been drained of their energy or it becomes disconnected. Network
- connectivity is crucial because an active sensor node without connectivity
- towards a base station cannot transmit any information regarding an observed
- event in the area that it monitors.
-
-
-\item {{\bf Coverage Ratio (CR)}:} it measures how well the WSN is able to
- observe the area of interest. In our case, we discretized the sensor field
- as a regular grid, which yields the following equation to compute the
- coverage ratio:
-\begin{equation*}
-\scriptsize
-\mbox{CR}(\%) = \frac{\mbox{$n$}}{\mbox{$N$}} \times 100.
-\end{equation*}
-where $n$ is the number of covered grid points by active sensors of every
-subregions during the current sensing phase and $N$ is total number of grid
-points in the sensing field. In our simulations, we have a layout of $N = 51
-\times 26 = 1326$ grid points.
-
-
-\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{$A_r$}}{\mbox{$S$}} \times 100 .
-\end{equation*}
-Where: $A_r^t$ is the number of active sensors in the subregion $r$ in the current sensing stage, $S$ is the total number of sensors in the network, and $R$ is the total number of the subregions in the network.
-
-
-
-\item {{\bf Energy Consumption}:} energy consumption (EC) can be seen as the
- total energy consumed by the sensors during $Lifetime_{95}$ or
- $Lifetime_{50}$, divided by the number of periods. Formally, the computation
- of EC can be expressed as follows:
+\item {\bf Network Lifetime}: the lifetime is defined as the time elapsed until
+ the coverage ratio falls below a fixed threshold. $Lifetime_{95}$ and
+ $Lifetime_{50}$ denote, respectively, the amount of time during which is
+ guaranteed a level of coverage greater than $95\%$ and $50\%$. The WSN can
+ fulfill the expected monitoring task until all its nodes have depleted their
+ energy or if the network is no more connected. This last condition is crucial
+ because without network connectivity a sensor may not be able to send to a
+ base station an event it has sensed.
+\item {\bf Coverage Ratio (CR)} : it measures how well the WSN is able to
+ observe the area of interest. In our case, we discretized the sensor field as
+ a regular grid, which yields the following equation:
\begin{equation*}
\scriptsize
- \mbox{EC} = \frac{\sum\limits_{m=1}^{M} \left( E^{\mbox{com}}_m+E^{\mbox{list}}_m+E^{\mbox{comp}}_m
- + E^{a}_m+E^{s}_m \right)}{M},
+ \mbox{CR}(\%) = \frac{\mbox{$n$}}{\mbox{$N$}} \times 100
\end{equation*}
-
-where $M$ corresponds to the number of periods. The total energy consumed by
-the sensors (EC) comes through taking into consideration four main energy factors. The first one, denoted $E^{\scriptsize \mbox{com}}_m$, represent the
-energy consumption spent by all the nodes for wireless communications during
-period $m$. $E^{\scriptsize \mbox{list}}_m$, the next factor, corresponds to
-the energy consumed by the sensors in LISTENING status before receiving the
-decision to go active or sleep in period $m$. $E^{\scriptsize \mbox{comp}}_m$
-refers to the energy needed by all the leader nodes to solve the integer program
-during a period. Finally, $E^a_{m}$ and $E^s_{m}$ indicate the energy consumed
-by the whole network in the sensing phase (active and sleeping nodes).
-
-
+ where $n$ is the number of covered grid points by active sensors of every
+ subregions during the current sensing phase and $N$ is total number of grid
+ points in the sensing field. In our simulations we have set a layout of
+ $N~=~51~\times~26~=~1326$~grid points.
+\item {\bf Active Sensors Ratio (ASR)}: a major objective of our protocol is to
+ activate as few nodes as possible, in order to minimize the communication
+ overhead and maximize the WSN lifetime. The active sensors ratio is defined as
+ follows:
+ \begin{equation*}
+ \scriptsize
+ \mbox{ASR}(\%) = \frac{\sum\limits_{r=1}^R \mbox{$|A_r^p|$}}{\mbox{$|S|$}} \times 100
+ \end{equation*}
+ where $|A_r^p|$ is the number of active sensors in the subregion $r$ in the
+ current sensing period~$p$, $|S|$ is the number of sensors in the network, and
+ $R$ is the number of subregions.
+\item {\bf Energy Consumption (EC)}: energy consumption can be seen as the total
+ energy consumed by the sensors during $Lifetime_{95}$ or $Lifetime_{50}$,
+ divided by the number of periods. The value of EC is computed according to
+ this formula:
+ \begin{equation*}
+ \scriptsize
+ \mbox{EC} = \frac{\sum\limits_{p=1}^{P} \left( E^{\mbox{com}}_p+E^{\mbox{list}}_p+E^{\mbox{comp}}_p
+ + E^{a}_p+E^{s}_p \right)}{P},
+ \end{equation*}
+ where $P$ corresponds to the number of periods. The total energy consumed by
+ the sensors comes through taking into consideration four main energy
+ factors. The first one, denoted $E^{\scriptsize \mbox{com}}_p$, represents the
+ energy consumption spent by all the nodes for wireless communications during
+ period $p$. $E^{\scriptsize \mbox{list}}_p$, 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 $p$. $E^{\scriptsize \mbox{comp}}_p$
+ refers to the energy needed by all the leader nodes to solve the integer
+ program during a period. Finally, $E^a_{p}$ and $E^s_{p}$ indicate the energy
+ consumed by the WSN during the sensing phase (active and sleeping nodes).
\end{itemize}
%\end{enumerate}
\subsection{Simulation Results}
-In this section, we present the simulation results of LiCO protocol and the other protocols using a discrete event simulator OMNeT++ \cite{varga} to run different series of simulations. We implemented all protocols precisely 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, the original execution time on the laptop is multiplied by 2944.2 $\left(\frac{35330}{2} \times \frac{1}{6} \right)$ so as to use it by the energy consumption model especially, after the computation and listening. Employing the modeling language for Mathematical Programming (AMPL)~\cite{AMPL}, the associated integer program instance is generated 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.
-
-We compared LiCO protocol to three other approaches: the first, called DESK and proposed by ~\cite{ChinhVu} is a fully distributed coverage algorithm; the second, called GAF ~\cite{xu2001geography}, consists in dividing the region
-into fixed squares. During the decision phase, in each square, one sensor is
-chosen to remain active during the sensing phase; the third, DiLCO protocol~\cite{Idrees2} is an improved version on the work presented in ~\cite{idrees2014coverage}. Note that the LiCO protocol is based on the same framework as that of DiLCO. For these two protocols, the division of the region of interest in 16 subregions was chosen since it produces the best results. The difference between the two protocols relies on the use of the integer programming to provide the set of sensors that have to be activated in each sensing phase. Whereas DiLCO protocol tries to satisfy the coverage of a set of primary points, LiCO protocol tries to reach a desired level of coverage $l$ for each sensor's perimeter. In the experimentations, we chose a level of coverage equal to 1 ($l=1$).
-\subsubsection{\textbf{Coverage Ratio}}
-Figure~\ref{fig333} shows the average coverage ratio for 200 deployed nodes obtained with the four methods.
-
+In order to assess and analyze the performance of our protocol we have
+implemented PeCO protocol in OMNeT++~\cite{varga} simulator. Besides PeCO, two
+other protocols, described in the next paragraph, will be evaluated for
+comparison purposes. The simulations were run on a DELL laptop 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 to three other approaches. The first
+one, called DESK, is a fully distributed coverage algorithm proposed by
+\cite{ChinhVu}. The second one, called GAF~\cite{xu2001geography}, 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{idrees2014coverage}. 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 made because
+it corresponds to the configuration producing the best 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 the 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{\bf Coverage Ratio}
+
+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] {R/CR.eps}
-\caption{The coverage ratio for 200 deployed nodes}
+\caption{Coverage ratio for 200 deployed nodes.}
\label{fig333}
\end{figure}
-DESK, GAF, and DiLCO provide a little better coverage ratio with 99.99\%, 99.91\%, and 99.02\% against 98.76\% produced by LiCO for the first periods. This is due to the fact that DiLCO protocol put in sleep mode redundant sensors using optimization (which lightly decreases the coverage ratio) while there are more active nodes in the case of others methods. But when the number of periods exceeds 70 periods, it clearly appears that LiCO 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).
+%When the number of periods increases, coverage ratio produced by DESK and GAF protocols decreases. This is due to dead nodes. However, DiLCO protocol maintains almost a good coverage from the round 31 to the round 63 and it is close to PeCO protocol. The coverage ratio of PeCO protocol is better than other approaches from the period 64.
-%When the number of periods increases, coverage ratio produced by DESK and GAF protocols decreases. This is due to dead nodes. However, DiLCO protocol maintains almost a good coverage from the round 31 to the round 63 and it is close to LiCO protocol. The coverage ratio of LiCO protocol is better than other approaches from the period 64.
+%because the optimization algorithm used by PeCO has been optimized the lifetime coverage based on the perimeter coverage model, so it provided acceptable coverage for a larger number of periods and prolonging the network lifetime based on the perimeter of the sensor nodes in each subregion of WSN. Although some nodes are dead, sensor activity scheduling based optimization of PeCO selected another nodes to ensure the coverage of the area of interest. i.e. DiLCO-16 showed a good coverage in the beginning then PeCO, when the number of periods increases, the coverage ratio decreases due to died sensor nodes. Meanwhile, thanks to sensor activity scheduling based new optimization model, which is used by PeCO protocol to ensure a longer lifetime coverage in comparison with other approaches.
-%because the optimization algorithm used by LiCO has been optimized the lifetime coverage based on the perimeter coverage model, so it provided acceptable coverage for a larger number of periods and prolonging the network lifetime based on the perimeter of the sensor nodes in each subregion of WSN. Although some nodes are dead, sensor activity scheduling based optimization of LiCO selected another nodes to ensure the coverage of the area of interest. i.e. DiLCO-16 showed a good coverage in the beginning then LiCO, when the number of periods increases, the coverage ratio decreases due to died sensor nodes. Meanwhile, thanks to sensor activity scheduling based new optimization model, which is used by LiCO protocol to ensure a longer lifetime coverage in comparison with other approaches.
+\subsubsection{\bf Active Sensors Ratio}
-\subsubsection{\textbf{Active Sensors Ratio}}
-Having active nodes as few as possible in each period is essential in order to minimize the energy consumption and so maximize the network lifetime. Figure~\ref{fig444} shows the average active nodes ratio for 200 deployed nodes.
+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]{R/ASR.eps}
-\caption{The active sensors ratio for 200 deployed nodes }
+\caption{Active sensors ratio for 200 deployed nodes.}
\label{fig444}
\end{figure}
-We observe that DESK and GAF have 30.36 \% and 34.96 \% active nodes for the first fourteen rounds and DiLCO and LiCO protocols compete perfectly with only 17.92 \% and 20.16 \% active nodes during the same time interval. As the number of periods increases, LiCO protocol has a lower number of active nodes in comparison with the three other approaches, while keeping of greater coverage ratio as shown in figure \ref{fig333}.
+\subsubsection{\bf Energy Consumption}
+
+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.
-\subsubsection{\textbf{The Energy Consumption}}
-We study the effect of the energy consumed by the WSN during the communication, computation, listening, active, and sleep modes for different network densities and compare it for the four approaches. Figures~\ref{fig3EC95} and \ref{fig3EC50} illustrate the energy consumption for different network sizes and for $Lifetime95$ and $Lifetime50$.
-
-\begin{figure}[h!]
-\centering
-\includegraphics[scale=0.5]{R/EC95.eps}
-\caption{The Energy Consumption per period with $Lifetime_{95}$}
-\label{fig3EC95}
-\end{figure}
-
\begin{figure}[h!]
-\centering
-\includegraphics[scale=0.5]{R/EC50.eps}
-\caption{The Energy Consumption per period with $Lifetime_{50}$}
-\label{fig3EC50}
+ \centering
+ \begin{tabular}{@{}cr@{}}
+ \includegraphics[scale=0.475]{R/EC95.eps} & \raisebox{2.75cm}{(a)} \\
+ \includegraphics[scale=0.475]{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}
-The results show that our LiCO protocol is the most competitive from the energy consumption point of view. As shown in figures~\ref{fig3EC95} and \ref{fig3EC50}, LiCO 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 this optimization program allows to reduce significantly the number of active sensors and so the energy consumption while keeping a good coverage level.
-%The optimization algorithm, which used by LiCO protocol, was improved the lifetime coverage efficiently based on the perimeter coverage model.
+%The optimization algorithm, which used by PeCO protocol, was improved the lifetime coverage efficiently based on the perimeter coverage model.
%The other approaches have a high energy consumption due to activating a larger number of sensors. In fact, a distributed method on the subregions greatly reduces the number of communications and the time of listening so thanks to the partitioning of the initial network into several independent subnetworks.
%\subsubsection{Execution Time}
-\subsubsection{\textbf{The Network Lifetime}}
-We observe the superiority of LiCO and DiLCO protocols against other two approaches in prolonging the network lifetime. In figures~\ref{fig3LT95} and \ref{fig3LT50}, network lifetime, $Lifetime95$ and $Lifetime50$ respectively, are illustrated for different network sizes.
+\subsubsection{\bf Network Lifetime}
+
+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
-\includegraphics[scale=0.5]{R/LT95.eps}
-\caption{The Network Lifetime for $Lifetime_{95}$}
-\label{fig3LT95}
-\end{figure}
-
-
-\begin{figure}[h!]
-\centering
-\includegraphics[scale=0.5]{R/LT50.eps}
-\caption{The Network Lifetime for $Lifetime_{50}$}
-\label{fig3LT50}
+ \centering
+ \begin{tabular}{@{}cr@{}}
+ \includegraphics[scale=0.475]{R/LT95.eps} & \raisebox{2.75cm}{(a)} \\
+ \includegraphics[scale=0.475]{R/LT50.eps} & \raisebox{2.75cm}{(b)}
+ \end{tabular}
+ \caption{Network Lifetime for (a)~$Lifetime_{95}$ \\
+ and (b)~$Lifetime_{50}$.}
+ \label{fig3LT}
\end{figure}
-As highlighted by figures~\ref{fig3LT95} and \ref{fig3LT50}, the network lifetime obviously increases when the size of the network increases, and it is clearly larger with DiLCO and LiCO protocols compared with the two other methods. For instance, for a network of 300 sensors, the coverage ratio is greater than 50\% about two times longer with LiCO compared to DESK method.
+%By choosing the best suited nodes, for each period, by optimizing the coverage and lifetime of the network to cover the area of interest and by letting the other ones sleep in order to be used later in next rounds, PeCO protocol efficiently prolonged the network lifetime especially for a coverage ratio greater than $50 \%$, whilst it stayed very near to DiLCO-16 protocol for $95 \%$.
-%By choosing the best suited nodes, for each period, by optimizing the coverage and lifetime of the network to cover the area of interest and by letting the other ones sleep in order to be used later in next rounds, LiCO protocol efficiently prolonged the network lifetime especially for a coverage ratio greater than $50 \%$, whilst it stayed very near to DiLCO-16 protocol for $95 \%$.
-Figure~\ref{figLTALL} introduces the comparisons of the lifetime coverage for different coverage ratios for LiCO and DiLCO protocols.
-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. Indeed there are applications that do not require a 100\% coverage of the surveillance region. LiCO might be an interesting method since it achieves a good balance between a high level coverage ratio and network lifetime.
+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]{R/LTa.eps}
-\caption{The Network Lifetime for different coverage ratios}
+\centering \includegraphics[scale=0.5]{R/LTa.eps}
+\caption{Network lifetime for different coverage ratios.}
\label{figLTALL}
\end{figure}
-
-%Comparison shows that LiCO protocol, which are used distributed optimization over the subregions, is the more relevance one for most coverage ratios and WSN sizes because it is robust to network disconnection during the network lifetime as well as it consume less energy in comparison with other approaches. LiCO protocol gave acceptable coverage ratio for a larger number of periods using new optimization algorithm that based on a perimeter coverage model. It also means that distributing the algorithm in each node and subdividing the sensing field into many subregions, which are managed independently and simultaneously, is the most relevant way to maximize the lifetime of a network.
+%Comparison shows that PeCO protocol, which are used distributed optimization over the subregions, is the more relevance one for most coverage ratios and WSN sizes because it is robust to network disconnection during the network lifetime as well as it consume less energy in comparison with other approaches. PeCO protocol gave acceptable coverage ratio for a larger number of periods using new optimization algorithm that based on a perimeter coverage model. It also means that distributing the algorithm in each node and subdividing the sensing field into many subregions, which are managed independently and simultaneously, is the most relevant way to maximize the lifetime of a network.
-\section{\uppercase{Conclusion and Future Works}}
+\section{Conclusion and Future Works}
\label{sec:Conclusion and Future Works}
-In this paper we have studied the problem of lifetime coverage optimization in
-WSNs. We designed a protocol LiCO that schedules node' activities (wakeup and sleep) with the objective of maintaining a good coverage ratio while maximizing the network lifetime. This protocol is applied on each subregion of the area of interest. It works in periods and is based on the resolution of an integer program to select the subset of sensors operating in active mode for each period. Our work is original in so far as it proposes for the first time an integer program scheduling the activation of sensors based on their perimeter coverage level instead of using a set of targets/points to be covered.
-
-
-
-%To cope with this problem, the area of interest is divided into a smaller subregions using divide-and-conquer method, and then a LiCO protocol for optimizing the lifetime coverage in each subregion. LiCO protocol combines two efficient techniques: network
+In this paper 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.
+
+%To cope with this problem, the area of interest is divided into a smaller subregions using divide-and-conquer method, and then a PeCO protocol for optimizing the lifetime coverage in each subregion. PeCO protocol combines two efficient techniques: network
%leader election, which executes the perimeter coverage model (only one time), the optimization algorithm, and sending the schedule produced by the optimization algorithm to other nodes in the subregion ; the second, sensor activity scheduling based optimization in which a new lifetime coverage optimization model is proposed. The main challenges include how to select the most efficient leader in each subregion and the best schedule of sensor nodes that will optimize the network lifetime coverage
%in the subregion.
%The network lifetime coverage in each subregion is divided into
%periods, each period consists of four stages: (i) Information Exchange,
%(ii) Leader Election, (iii) a Decision based new optimization model in order to
%select the nodes remaining active for the last stage, and (iv) Sensing.
-We carried out severals simulations to evaluate the proposed protocol. The simulation results show that LiCO is is more energy-efficient than other approaches, with respect to lifetime, coverage ratio, active sensors ratio, and energy consumption.
-%Indeed, when dealing with large and dense WSNs, a distributed optimization approach on the subregions of WSN like the one we are proposed 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.
-
-We have identified different research directions that arise out of the work presented here.
-We plan to extend our framework such that the schedules are planned for multiple periods in advance.
+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.
+%Indeed, when dealing with large and dense WSNs, a distributed optimization approach on the subregions of WSN like the one we are proposed 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. We have identified different research directions that arise out of the work presented here.
+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 the heterogeneous sensors, which do not have the same energy, processing, sensing and communication capabilities;
+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, our final goal is to implement our protocol using a sensor-testbed to evaluate their performance in real world applications.
+Finally, it would be interesting to implement our protocol using a
+sensor-testbed to evaluate it in real world applications.
-\section*{\uppercase{Acknowledgements}}
-\noindent As a Ph.D. student, Ali Kadhum IDREES would like to gratefully acknowledge the University of Babylon - IRAQ for the financial support and Campus France for the received support.
+\section*{Acknowledgments}
-
+\noindent As a Ph.D. student, Ali Kadhum IDREES would like to gratefully
+acknowledge the University of Babylon - IRAQ for financial support and Campus
+France for the received support. This work is also partially funded by the Labex ACTION program (contract ANR-11-LABX-01-01).
\ifCLASSOPTIONcaptionsoff
