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/wakeup duty cycles) is achieved in each subregion by a
+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
+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 protols.
+longer lifetime coverage for WSNs in comparison with some other protocols.
\end{abstract}
% Note that keywords are not normally used for peerreview papers.
\IEEEpeerreviewmaketitle
-\section{\uppercase{Introduction}}
+\section{Introduction}
\label{sec:introduction}
\noindent The continuous progress in Micro Electro-Mechanical Systems (MEMS) and
networks of tiny sensors, called Wireless Sensor Networks
(WSN)~\cite{akyildiz2002wireless,puccinelli2005wireless}, to fulfill monitoring
tasks. A WSN consists of small low-powered sensors working together by
-communicating with one another through multihop radio communications. Each node
+communicating with one another through multi-hop radio communications. Each node
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
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 measurements which are
-collected; a radio communication unit for data transmission and receiving.
+processing unit which will process and store the collected measurements; a radio
+communication unit for data transmission and receiving.
The energy needed by an active sensor node to perform sensing, processing, and
communication is supplied by a power supply which is a battery. This battery has
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? So, this last years many
energy-efficient mechanisms have been suggested to retain energy and extend the
lifetime of the WSNs~\cite{rault2014energy}.
%\uppercase{\textbf{Our contributions.}}
-% MICHEL - TO CONTINUED FROM HERE
This paper makes the following contributions.
\begin{enumerate}
-\item We devise a framework to schedules nodes to be activated
- alternatively, such that the network lifetime may be prolonged ans
- certain coverage requirement can still be met. This framework is
- based on the division of the area of interest into several smaller
- subregions; on the division of timeline into periods of equal
- length. One leader is elected for each subregion in an independent,
- distributed, and simultaneous way by the cooperation among the
- sensor nodes within each subregion, and this is similar to cluster
- architecture
-\item We propose 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. And a weighted sum of these
- deviations is minimized.
-\item We conducted extensive simulation experiments using the discrete
- event simulator OMNeT++, to demonstrate the efficiency of our
- protocol, compared to two approaches found in the literature, DESK
- \cite{ChinhVu} and GAF \cite{xu2001geography}, and compared to our
- previous work using another optimization model for sensor scheduling
- \cite{Idrees2}.
+\item We devise 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
+ 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
+ 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
+ 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:
+ 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;
% 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 remainder of the paper is organized as follows. The next section
-reviews the related work in the field. Section~\ref{sec:The LiCO
- Protocol Description} is devoted to the LiCO protocol
-Description. Section~\ref{cp} gives the coverage model formulation
-which is used to schedule the activation of sensors.
-Section~\ref{sec:Simulation Results and Analysis} presents simulations
-results. Finally, we give concluding remarks and some suggestions for
-future works in Section~\ref{sec:Conclusion and Future Works}.
+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
+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
+Section~\ref{sec:Conclusion and Future Works}.
% that show that our protocol outperforms others protocols.
-\section{\uppercase{Related Literature}}
+\section{Related Literature}
\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 LiCO protocol from the works presented in
the literature.
-The most discussed coverage problems in literature can be classified into three
-types \cite{li2013survey}: area coverage \cite{Misra} where every point inside
-an area is to be monitored, target coverage \cite{yang2014novel} where the main
-objective is to cover only a finite number of discrete points called targets,
-and barrier coverage \cite{HeShibo}\cite{kim2013maximum} to prevent intruders
-from entering into the region of interest. In \cite{Deng2012} authors transform
-the area coverage problem to the target coverage problem 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, the whole area is sufficiently covered and they provide an algorithm in $O(nd~log~d)$ time to compute the perimeter-coverage of each sensor ($d$ the maximum number of sensors that are neighboring to a sensor, $n$ the total number of sensors in the network). {\it In LiCO 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.}
-
-The major approach to extend network lifetime while preserving coverage is to
-divide/organize the sensors into a suitable number of set covers (disjoint or
-non-disjoint), where each set completely covers a region of interest, and to
-activate these set covers successively. The network activity can be planned in
-advance and scheduled for the entire network lifetime or organized in periods,
+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
+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
+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
+ 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.}
+
+The major approach to extend network lifetime while preserving coverage is to
+divide/organize the sensors into a suitable number of set covers (disjoint or
+non-disjoint), where each set completely covers a region of interest, and to
+activate these set covers successively. The network activity can be planned in
+advance and scheduled for the entire network lifetime or organized in periods,
and the set of active sensor nodes is decided at the beginning of each period
\cite{ling2009energy}. Active node selection is determined based on the problem
-requirements (e.g. area monitoring, connectivity, power efficiency). For
-instance, Jaggi et al. \cite{jaggi2006} address the problem of maximizing
-network lifetime by dividing sensors into the maximum number of disjoint subsets
-such that each subset can ensure both coverage and connectivity. A greedy
+requirements (e.g. area monitoring, connectivity, or power efficiency). For
+instance, Jaggi {\em et al.}~\cite{jaggi2006} address the problem of maximizing
+the lifetime by dividing sensors into the maximum number of disjoint subsets
+such that each subset can ensure both coverage and connectivity. A greedy
algorithm is applied once to solve this problem and the computed sets are
activated in succession to achieve the desired network lifetime. Vu
-\cite{chin2007}, Padmatvathy 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, 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 approaches, including centralized, or distributed algorithms, have been
-proposed to extend the network lifetime. In distributed
-algorithms~\cite{yangnovel,ChinhVu,qu2013distributed}, information is
-disseminated throughout the network and sensors decide cooperatively by
-communicating with their neighbors which of them will remain in sleep mode for a
-certain period of time. The centralized
-algorithms~\cite{cardei2005improving,zorbas2010solving,pujari2011high} always
-provide nearly or close to optimal solution since the algorithm has global view
-of the whole network. But such a method has the disadvantage of requiring high
-communication costs, since the node (located at the base station) making the
-decision needs information from all the sensor nodes in the area and the amount
-of information can be huge. {\it In order to be suitable for large-scale
- network, 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.}
-
-A large variety of coverage scheduling algorithms has been developed. Many of
-the existing algorithms, 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 say targets that are covered by the smallest number of sensors
+\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,
+ 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
+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
+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.
+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
+say targets that are covered by the smallest number of sensors
\cite{berman04,zorbas2010solving}. Other approaches are based on mathematical
programming formulations~\cite{cardei2005energy,5714480,pujari2011high,Yang2014}
-and dedicated techniques (solving with a branch-and-bound algorithms available
-in optimization solver). The problem is formulated as an optimization problem
+and dedicated techniques (solving with a branch-and-bound algorithm available in
+optimization solver). The problem is formulated as an optimization problem
(maximization of the lifetime or number of cover sets) under target coverage and
-energy constraints. Column generation techniques, well-known and widely
+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
- protocol, each leader, in each subregion, solves an integer program with
-the double objective consisting in minimizing the overcoverage and the
- undercoverage of the perimeter of each sensor.
-
-}
-
+used~\cite{castano2013column,rossi2012exact,deschinkel2012column}. {\it In LiCO
+ 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.}
%\noindent Recently, the coverage problem has been received a high attention, which concentrates on how the physical space could be well monitored after the deployment. Coverage is one of the Quality of Service (QoS) parameters in WSNs, which is highly concerned with power depletion~\cite{zhu2012survey}. Most of the works about the coverage protocols have been suggested in the literature focused on three types of the coverage in WSNs~\cite{mulligan2010coverage}: the first, area coverage means that each point in the area of interest within the sensing range of at least one sensor node; the second, target coverage in which a fixed set of targets need to be monitored; the third, barrier coverage refers to detect the intruders crossing a boundary of WSN. The work in this paper emphasized on the area coverage, so, some area coverage protocols have been reviewed in this section, and the shortcomings of reviewed approaches are being summarized.
%\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.
-
\section{ The LiCO Protocol Description}
\label{sec:The LiCO Protocol Description}
-\noindent In this section, we describe our Lifetime Coverage Optimization Protocol which is called LiCO in more detail.
+
+\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.
+
% It is based on two efficient-energy mechanisms: the first, is partitioning the sensing field into smaller subregions, and one leader is elected for each subregion; the second, a sensor activity scheduling based new optimization model so as to produce the optimal cover set of active sensors for the sensing stage during the period. Obviously, these two mechanisms can be contribute in extend the network lifetime coverage efficiently.
%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}
-\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 interested area. We assume that all the sensor nodes are homogeneous in terms of communication, sensing, and processing capabilities and heterogeneous in term of energy supply. The location information is available to the sensor node either through hardware such as embedded GPS or through 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 aerial vehicle (UAV) flying regularly over the sensor field to collect measurements from sensor nodes. A mobile sink node collects the measurements and transmits them to the base station. We consider a boolean disk coverage model which is the most widely used sensor coverage model in the literature. Each sensor has a constant sensing range $R_s$. All space points within a disk centered at the sensor with the radius of the sensing range is said to be covered by this sensor. We also assume that the communication range $R_c \geq 2R_s$. In fact, Zhang and Zhou~\cite{Zhang05} proved that if the transmission range fulfills the previous hypothesis, a complete coverage of a convex area implies connectivity among the working nodes in the active mode.
-
-\indent LiCO protocol uses the perimeter-coverage model which states in ~\cite{huang2005coverage} as following: The sensor is said to be perimeter covered if all the points on its perimeter are covered by at least one sensor other than itself. Huang and Tseng in \cite{huang2005coverage} proves that a network area is $k$-covered if and only if each sensor in the network is $k$-perimeter-covered.
+\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
+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
+aerial vehicle (UAV) flying regularly over the sensor field to collect
+measurements from sensor nodes. A mobile sink node collects the measurements and
+transmits them to the base station. We consider a Boolean disk coverage model,
+which is the most widely used sensor coverage model in the literature, and all
+sensor nodes have a constant sensing range $R_s$. Thus, all the space points
+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
+complete coverage of a convex area implies connectivity among active nodes.
+
+\indent LiCO protocol uses the same perimeter-coverage model than 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.
%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{pcmfig} illuminates the perimeter coverage of the sensor node $0$. On this figure, sensor $0$ has $9$ neighbors. We report for each sensor $i$ having an intersection with sensor $0$, the two intersection points, $iL$ for left point and $iR$ for right point. These intersections points subdivide the perimeter of the sensor $0$ (the perimeter of the disk covered by the sensor) into portions called segments.
+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
-\includegraphics[width=75mm]{pcm.jpg}
-\caption{Perimeter coverage of sensor node 0}
-\label{pcmfig}
+ \centering
+ \begin{tabular}{@{}cr@{}}
+ \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 $v$.}
+ \label{pcm2sensors}
\end{figure}
-Figure~\ref{twosensors} demonstrates the way of locating the left and right points of a segment for a sensor node $u$ covered by a sensor node $v$. This figure assumes that the neighbor sensor node $v$ is located on the west of a sensor $u$. It is assumed that the two sensor nodes $v$ and $u$ are located in the positions $(v_x,v_y)$ and $(u_x,u_y)$, respectively. The distance between $v$ and $u$ is computed by $Dist(u,v) = \sqrt{\vert u_x - v_x \vert^2 + \vert u_y - v_y \vert^2}$. The angle $\alpha$ is computed through the formula $\alpha = arccos \left(\dfrac{Dist(u,v)}{2R_s} \right)$. So, the arch of sensor $u$ falling in the angle $[\pi - \alpha,\pi + \alpha]$, is said to be perimeter-covered by sensor node $v$.
-
-The left and right points of each segment are placed on the line segment $[0,2\pi]$. Figure~\ref{pcmfig} illustrates the segments for the 9 neighbors of sensor $0$. 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{pcmfig}.
+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
+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 $v$.
+
+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 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.
+
+%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!]
-\centering
-\includegraphics[width=75mm]{twosensors.jpg}
-\caption{Locating the segment of $u$$\rq$s perimeter covered by $v$.}
-\label{twosensors}
-\end{figure}
-
-\begin{figure}[ht!]
+\begin{figure*}[ht!]
\centering
-\includegraphics[width=75mm]{expcm.pdf}
-\caption{ Coverage levels for sensor node $0$.}
+\includegraphics[width=137.5mm]{expcm.pdf}
+\caption{Maximum coverage levels for perimeter of sensor node $0$.}
\label{expcm}
-\end{figure}
-
-
-
-
-
-
-
-
+\end{figure*}
%For example, consider the sensor node $0$ in figure~\ref{pcmfig}, which has 9 neighbors. Figure~\ref{expcm} shows the perimeter coverage level for all left and right points of a segment that covered by a neighboring sensor nodes. Based on the figure~\ref{expcm}, the set of sensors for each left and right point of the segments illustrated in figure~\ref{ex2pcm} for the sensor node 0.
\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@{}}The angle \\ $\alpha$ \end{tabular} & \begin{tabular}[c]{@{}c@{}}Segment \\ Left (L) or\\ Right (R)\end{tabular} & \begin{tabular}[c]{@{}c@{}}Sensor \\ Node Id\end{tabular} & \begin{tabular}[c]{@{}c@{}}Interval \\ Coverage\\ Level\end{tabular} & \multicolumn{5}{c|}{\begin{tabular}[c]{@{}c@{}}The Set of Sensors\\ Involved in Interval \\ Coverage\end{tabular}} \\ \hline
-0.0291 & L & 1 & 4 & 0 & 1 & 3 & 4 & \\ \hline
-0.104 & L & 2 & 5 & 0 & 1 & 3 & 4 & 2 \\ \hline
-0.3168 & R & 3 & 4 & 0 & 1 & 4 & 2 & \\ \hline
-0.6752 & R & 4 & 3 & 0 & 1 & 2 & & \\ \hline
-1.8127 & R & 1 & 2 & 0 & 2 & & & \\ \hline
-1.9228 & L & 5 & 3 & 0 & 2 & 5 & & \\ \hline
-2.3959 & L & 6 & 4 & 0 & 2 & 5 & 6 & \\ \hline
-2.4258 & R & 2 & 3 & 0 & 5 & 6 & & \\ \hline
-2.7868 & L & 7 & 4 & 0 & 5 & 6 & 7 & \\ \hline
-2.8358 & L & 8 & 5 & 0 & 5 & 6 & 7 & 8 \\ \hline
-2.9184 & R & 5 & 4 & 0 & 6 & 7 & 8 & \\ \hline
-3.3301 & R & 7 & 3 & 0 & 6 & 8 & & \\ \hline
-3.9464 & L & 9 & 4 & 0 & 6 & 8 & 9 & \\ \hline
-4.767 & R & 6 & 3 & 0 & 8 & 9 & & \\ \hline
-4.8425 & L & 3 & 4 & 0 & 3 & 8 & 9 & \\ \hline
-4.9072 & R & 8 & 3 & 0 & 3 & 9 & & \\ \hline
-5.3804 & L & 4 & 4 & 0 & 3 & 4 & 9 & \\ \hline
-5.9157 & R & 9 & 3 & 0 & 3 & 4 & & \\ \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
+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}
%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.
-In LiCO protocol, scheduling of sensor nodes'activities is formulated with an integer program based on coverage intervals and is detailed in section~\ref{cp}.
-
-In the case of sensor node, which has a part of its sensing range outside the border of the WSN sensing field as in figure~\ref{ex4pcm}, the coverage level for this segment is set to $\infty$, and the corresponding interval will not be taken into account by the optimization algorithm.
-\begin{figure}[ht!]
+In LiCO protocol, scheduling of 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 $\infty$
+and the corresponding interval will not be taken into account by the
+optimization algorithm.
+
+\begin{figure}[t!]
\centering
-\includegraphics[width=75mm]{ex4pcm.jpg}
-\caption{Part of sensing range outside the the border of WSN sensing field.}
+\includegraphics[width=62.5mm]{ex4pcm.jpg}
+\caption{Sensing range outside the WSN's area of interest.}
\label{ex4pcm}
\end{figure}
%Figure~\ref{ex5pcm} gives an example to compute the perimeter coverage levels for the left and right points of the segments for a sensor node $0$, which has a part of its sensing range exceeding the border of the sensing field of WSN, and it has a six neighbors. In figure~\ref{ex5pcm}, the sensor node $0$ has two segments outside the border of the network sensing field, so the left and right points of the two segments called $-1L$, $-1R$, $-2L$, and $-2R$.
%\label{ex5pcm}
%\end{figure}
-
\subsection{The Main Idea}
-\noindent The area of interest can be divided into smaller areas called subregions and
-then our protocol will be implemented in each subregion simultaneously. LiCO protocol works into periods fashion as shown in figure~\ref{fig2}.
-\begin{figure}[ht!]
+
+\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
+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
+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=85mm]{Model.pdf}
+\includegraphics[width=80mm]{Model.pdf}
\caption{LiCO protocol}
\label{fig2}
\end{figure}
-Each period is divided into 4 stages: Information (INFO) Exchange, Leader Election, Optimization Decision, and Sensing. For each period there is exactly one set cover responsible for the sensing task. LiCO is more powerful against an unexpected node failure because it works in periods. On the one hand, if the node failure is discovered before taking the decision of the optimization algorithm, the sensor node would not involved to current stage, and, on the other hand, if the sensor failure takes place 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 for some sensor nodes, even when they do not join the network to monitor the area.
-
-We define two types of packets to be used by LiCO protocol.
+We define two types of packets to be used by LiCO protocol:
%\begin{enumerate}[(a)]
\begin{itemize}
-\item INFO packet: sent by each sensor node to all the nodes inside a same subregion for information exchange.
-\item ActiveSleep packet: sent by the leader to all the nodes in its subregion to inform them to be Active or Sleep during the sensing phase.
+\item INFO packet: sent by each sensor node to all the nodes inside a same
+ subregion for information exchange.
+\item ActiveSleep packet: sent by the leader to all the nodes in its subregion
+ to transmit to them their respective status (stay Active or go Sleep) during
+ sensing phase.
\end{itemize}
%\end{enumerate}
-There are five status for each sensor node in the network :
+Five status are possible for a sensor node in the network:
%\begin{enumerate}[(a)]
\begin{itemize}
-\item LISTENING: Sensor is waiting for a decision (to be active or not)
-\item COMPUTATION: Sensor applies the optimization process as leader
-\item ACTIVE: Sensor is active
-\item SLEEP: Sensor is turned off
-\item COMMUNICATION: Sensor is transmitting or receiving packet
+\item LISTENING: waits for a decision (to be active or not);
+\item COMPUTATION: executes the optimization algorithm as leader to
+ determine the activities scheduling;
+\item ACTIVE: node is sensing;
+\item SLEEP: node is turned off;
+\item COMMUNICATION: transmits or recevives packets.
\end{itemize}
%\end{enumerate}
%Below, we describe each phase in more details.
\subsection{LiCO Protocol Algorithm}
-The pseudo-code for LiCO Protocol is illustrated as follows:
+\noindent The pseudocode implementing the protocol on a node is given below.
+More precisely, Algorithm~\ref{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.
\begin{algorithm}[h!]
% \KwIn{all the parameters related to information exchange}
\If{ $RE_k \geq E_{th}$ }{
\emph{$s_k.status$ = COMMUNICATION}\;
- \emph{Send $INFO()$ packet to other nodes in the subregion}\;
- \emph{Wait $INFO()$ packet from other nodes in the subregion}\;
+ \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 $}{
% \emph{ Determine the segment points using perimeter coverage model}\;
}
- \If{$ (s_k.ID $ is the same Previous Leader) AND (K.CurrentSize = K.PreviousSize)}{
+ \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 $a^j_{ik}$ and prepare data to Algorithm}\;
+ \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}\;
}
\end{algorithm}
-\noindent Algorithm 1 gives a brief description of the protocol applied by each sensor node (denoted by $s_k$ for a sensor node indexed by $k$). In this algorithm, the K.CurrentSize and K.PreviousSize refer to the current size and the previous size of sensor nodes still alive in the subregion respectively.
-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 based Embedded GPS or 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 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 energy, and then in case of equality, larger index. Thereafter the leader collects information to formulate and solve the integer program which allows to construct the set of active sensors in the sensing stage.
-
+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 collect position coordinates,
+remaining energy, sensor node ID, and the number of its 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 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.
%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.
% 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.
-
\section{Lifetime Coverage problem formulation}
\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:
+\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{CI}. For a coverage interval $i$,
+let $a^j_{ik}$ denote 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 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. 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 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 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) 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 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 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 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 not 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{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 set a layout of
+ $N~=~51~\times~26~=~1326$~grid points.
+ % MICHEL TO BE CONTINUED FROM HERE
\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
\section{\uppercase{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.
+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.
