From: Karine Deschinkel Date: Tue, 5 Jan 2016 13:11:06 +0000 (+0100) Subject: changement de styme, suppression de us, our, we X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/LiCO.git/commitdiff_plain/ef601476e28c74fab20ebf8271fcde3f7ea87541?ds=sidebyside;hp=8672f2758b38ef0ee17391e8ae39be587be825c3 changement de styme, suppression de us, our, we --- diff --git a/PeCO-EO/.directory b/PeCO-EO/.directory index e84f1b1..8478e49 100644 --- a/PeCO-EO/.directory +++ b/PeCO-EO/.directory @@ -1,4 +1,3 @@ [Dolphin] -Timestamp=2015,6,23,15,50,47 +Timestamp=2015,9,29,14,46,5 Version=3 -ViewMode=1 diff --git a/PeCO-EO/ESR.eps b/PeCO-EO/ESR.eps deleted file mode 100644 index ee607d4..0000000 --- a/PeCO-EO/ESR.eps +++ /dev/null @@ -1,1460 +0,0 @@ -%!PS-Adobe-2.0 EPSF-2.0 -%%BoundingBox: 53 53 536 402 -%%HiResBoundingBox: 54 53.5 535 401.5 -%%Creator: gnuplot 4.6 patchlevel 0 -%%CreationDate: Tue Sep 22 23:49:01 2015 -%%EndComments -% EPSF created by ps2eps 1.68 -%%BeginProlog -save 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PDF statistics: 222 PDF objects out of 1000 (max. 8388607) 151 compressed objects within 2 object streams diff --git a/PeCO-EO/articleeo.pdf b/PeCO-EO/articleeo.pdf index 53d187b..3bdbc22 100644 Binary files a/PeCO-EO/articleeo.pdf and b/PeCO-EO/articleeo.pdf differ diff --git a/PeCO-EO/articleeo.tex b/PeCO-EO/articleeo.tex index b6becc3..f9a5f6d 100644 --- a/PeCO-EO/articleeo.tex +++ b/PeCO-EO/articleeo.tex @@ -2,17 +2,12 @@ % v4.0 released April 2013 \documentclass{gENO2e} -%\usepackage[linesnumbered,ruled,vlined,commentsnumbered]{algorithm2e} -%\renewcommand{\algorithmcfname}{ALGORITHM} + \usepackage{indentfirst} \usepackage{color} \usepackage[algo2e,ruled,vlined]{algorithm2e} \begin{document} -%\jvol{00} \jnum{00} \jyear{2013} \jmonth{April} - -%\articletype{GUIDE} - \title{{\itshape Perimeter-based Coverage Optimization \\ to Improve Lifetime in Wireless Sensor Networks}} @@ -29,11 +24,10 @@ 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. We -propose such an approach called Perimeter-based Coverage Optimization protocol -(PeCO). It is a hybrid of centralized and distributed methods: the region of +scheduling which ensures sensing coverage while minimizing the energy cost. An approach called Perimeter-based Coverage Optimization protocol +(PeCO) is proposed. It is a hybrid of centralized and distributed methods: the region of interest is first subdivided into subregions and the protocol is then -distributed among sensor nodes in each subregion. The novelty of our approach +distributed among sensor nodes in each subregion. The novelty of the 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 demonstrate that PeCO can offer longer lifetime @@ -89,15 +83,14 @@ This paper makes the following contributions : architecture. \item A new mathematical optimization model is proposed. Instead of trying to cover a set of specified points/targets as in most of the methods proposed in - the literature, we formulate a mixed-integer program based on the perimeter - coverage of each sensor. The model involves integer variables to capture the + the literature, a mixed-integer program based on the perimeter + coverage of each sensor is formulated. The model involves integer variables to capture the deviations between the actual level of coverage and the required level. Hence, an optimal schedule will be obtained by minimizing a weighted sum of these deviations. \item Extensive simulation experiments are conducted using the discrete event - simulator OMNeT++, to demonstrate the efficiency of our protocol. We have - compared the PeCO protocol to two approaches found in the literature: - DESK~\citep{ChinhVu} and GAF~\citep{xu2001geography}, and also to our previous + simulator OMNeT++, to demonstrate the efficiency of the PeCO protocol. The PeCO protocol has been compared to two approaches found in the literature: + DESK~\citep{ChinhVu} and GAF~\citep{xu2001geography}, and also to the protocol DiLCO published in~\citep{Idrees2}. DiLCO uses the same framework as PeCO but is based on another optimization model for sensor scheduling. \end{enumerate} @@ -134,7 +127,7 @@ algorithm in $O(nd~log~d)$ time to compute the perimeter-coverage of each sensor. $d$ denotes the maximum number of sensors that are 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 + points, the 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 @@ -212,8 +205,8 @@ decomposed into 4 phases: information exchange, leader election, decision, and sensing. The simulations show that DiLCO is able to increase the WSN lifetime and provides improved coverage performance. {\it In the PeCO protocol, a new mathematical optimization model is proposed. Instead of trying to cover a set - of specified points/targets as in the DiLCO protocol, we formulate an integer - program based on the perimeter coverage of each sensor. The model involves + of specified points/targets as in the DiLCO protocol, an integer + program based on the perimeter coverage of each sensor is formulated. The model involves integer variables to capture the deviations between the actual level of coverage and the required level. The idea is that an optimal scheduling will be obtained by minimizing a weighted sum of these deviations.} @@ -221,12 +214,6 @@ and provides improved coverage performance. {\it In the PeCO protocol, a new \section{ The P{\scshape e}CO Protocol Description} \label{sec:The PeCO Protocol Description} -%In this section, the Perimeter-based Coverage -%Optimization protocol is decribed in details. First we present the assumptions we made and the models -%we considered (in particular the perimeter coverage one), second we describe the -%background idea of our protocol, and third we give the outline of the algorithm -%executed by each node. - \subsection{Assumptions and Models} \label{CI} @@ -242,7 +229,7 @@ algorithms. A Boolean disk coverage model, which is the most widely used sensor coverage model in the literature, is considered 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 +this sensor. The communication range $R_c$ is assumed to satisfy : $R_c \geq 2 \cdot R_s$. In fact, \citet{Zhang05} proved that if the transmission range fulfills the previous hypothesis, the complete coverage of a convex area implies connectivity among active nodes. @@ -256,9 +243,9 @@ $k$~sensors) if and only if each sensor in the network is $k$-perimeter-covered (perimeter covered by at least $k$ sensors). Figure~\ref{figure1}(a) shows the coverage of sensor node~$0$. On this figure, -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 the intersection of the two sensing areas. These points are +sensor~$0$ has nine neighbors. For each neighbor the two points +resulting from the intersection of the two sensing areas have been reported on its perimeter (the +perimeter of the disk covered by the sensor~$0$). These points are denoted for neighbor~$i$ by $iL$ and $iR$, respectively for left and right from a neighboring point of view. The resulting couples of intersection points subdivide the perimeter of sensor~$0$ into portions called arcs. @@ -364,7 +351,7 @@ optimization algorithm. \subsection{Main Idea} 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 +subregions using a divide-and-conquer algorithm. In a second step the protocol will be executed in a distributed way in each subregion simultaneously to schedule nodes' activities for one sensing period. Sensor nodes are assumed to be deployed almost uniformly over the region. The regular subdivision is made @@ -397,7 +384,7 @@ of the application. \label{figure4} \end{figure} -We define two types of packets to be used by the PeCO protocol: +Two types of packets used by the PeCO protocol are defined: \begin{itemize} \item INFO packet: sent by each sensor node to all the nodes inside a same subregion for information exchange. @@ -424,10 +411,6 @@ applied by a sensor node $s_k$ where $k$ is the node index in the WSN. \begin{algorithm2e} - % \KwIn{all the parameters related to information exchange} -% \KwOut{$winer-node$ (: the id of the winner sensor node, which is the leader of current round)} -% \BlankLine - %\emph{Initialize the sensor node and determine it's position and subregion} \; \label{alg:PeCO} \caption{PeCO pseudocode} \eIf{$RE_k \geq E_{th}$}{ @@ -462,42 +445,6 @@ applied by a sensor node $s_k$ where $k$ is the node index in the WSN. } \end{algorithm2e} -%\begin{algorithm} -%\noindent{\bf If} $RE_k \geq E_{th}$ {\bf then}\\ -%\hspace*{0.6cm} \emph{$s_k.status$ = COMMUNICATION;}\\ -%\hspace*{0.6cm} \emph{Send $INFO()$ packet to other nodes in subregion;}\\ -%\hspace*{0.6cm} \emph{Wait $INFO()$ packet from other nodes in subregion;}\\ -%\hspace*{0.6cm} \emph{Update K.CurrentSize;}\\ -%\hspace*{0.6cm} \emph{LeaderID = Leader election;}\\ -%\hspace*{0.6cm} {\bf If} $ s_k.ID = LeaderID $ {\bf then}\\ -%\hspace*{1.2cm} \emph{$s_k.status$ = COMPUTATION;}\\ -%\hspace*{1.2cm}{\bf If} \emph{$ s_k.ID $ is Not previously selected as a Leader} {\bf then}\\ -%\hspace*{1.8cm} \emph{ Execute the perimeter coverage model;}\\ -%\hspace*{1.2cm} {\bf end}\\ -%\hspace*{1.2cm}{\bf If} \emph{($s_k.ID $ is the same Previous Leader)~And~(K.CurrentSize = K.PreviousSize)}\\ -%\hspace*{1.8cm} \emph{ Use the same previous cover set for current sensing stage;}\\ -%\hspace*{1.2cm} {\bf end}\\ -%\hspace*{1.2cm} {\bf else}\\ -%\hspace*{1.8cm}\emph{Update $a^j_{ik}$; prepare data for IP~Algorithm;}\\ -%\hspace*{1.8cm} \emph{$\left\{\left(X_{1},\dots,X_{l},\dots,X_{K}\right)\right\}$ = Execute Integer Program Algorithm($K$);}\\ -%\hspace*{1.8cm} \emph{K.PreviousSize = K.CurrentSize;}\\ -%\hspace*{1.2cm} {\bf end}\\ -%\hspace*{1.2cm}\emph{$s_k.status$ = COMMUNICATION;}\\ -%\hspace*{1.2cm}\emph{Send $ActiveSleep()$ to each node $l$ in subregion;}\\ -%\hspace*{1.2cm}\emph{Update $RE_k $;}\\ -%\hspace*{0.6cm} {\bf end}\\ -%\hspace*{0.6cm} {\bf else}\\ -%\hspace*{1.2cm}\emph{$s_k.status$ = LISTENING;}\\ -%\hspace*{1.2cm}\emph{Wait $ActiveSleep()$ packet from the Leader;}\\ -%\hspace*{1.2cm}\emph{Update $RE_k $;}\\ -%\hspace*{0.6cm} {\bf end}\\ -%{\bf end}\\ -%{\bf else}\\ -%\hspace*{0.6cm} \emph{Exclude $s_k$ from entering in the current sensing stage;}\\ -%{\bf end}\\ -%\label{alg:PeCO} -%\end{algorithm} - In this algorithm, $K.CurrentSize$ and $K.PreviousSize$ respectively represent the current number and the previous number of living nodes in the subnetwork of the subregion. At the beginning of the first period $K.PreviousSize$ is @@ -507,7 +454,7 @@ in the current period. Each sensor node determines its position and its subregion using an embedded GPS or a location discovery algorithm. After that, all the sensors collect position coordinates, remaining energy, sensor node ID, and the number of their one-hop live neighbors during the information exchange. -\textcolor{green}{Both INFO packet and ActiveSleep packet contain two parts: header and data payload. The sensor ID is included in the header, where the header size is 8 bits. The data part includes position coordinates (64 bits), remaining energy (32 bits), and the number of one-hop live neighbors (8 bits). Therefore the size of the INFO packet is 112 bits. The ActiveSleep packet is 16 bits size, 8 bits for the header and 8 bits for data part that includes only sensor status (0 or 1).} +Both INFO packet and ActiveSleep packet contain two parts: header and data payload. The sensor ID is included in the header, where the header size is 8 bits. The data part includes position coordinates (64 bits), remaining energy (32 bits), and the number of one-hop live neighbors (8 bits). Therefore the size of the INFO packet is 112 bits. The ActiveSleep packet is 16 bits size, 8 bits for the header and 8 bits for data part that includes only sensor status (0 or 1). The sensors inside a same region cooperate to elect a leader. The selection criteria for the leader are (in order of priority): \begin{enumerate} @@ -569,7 +516,7 @@ coverage level. As the number of alive sensors decreases, it becomes impossible to reach the desired level of coverage for all coverage intervals. Therefore variables $M^j_i$ and $V^j_i$ are introduced 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 +effective number. And these deviations are minimized, 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. @@ -586,18 +533,6 @@ The coverage optimization problem can then be mathematically expressed as follow \end{aligned} \end{equation} -%\begin{equation} -%\left \{ -%\begin{array}{ll} -%\min \sum_{j \in S} \sum_{i \in I_j} (\alpha^j_i ~ M^j_i + \beta^j_i ~ V^j_i ) & \\ -%\textrm{subject to :} &\\ -%\sum_{k \in A} ( a^j_{ik} ~ X_{k}) + M^j_i \geq l \quad \forall i \in I_j, \forall j \in S\\ -%\sum_{k \in A} ( a^j_{ik} ~ X_{k}) - V^j_i \leq l \quad \forall i \in I_j, \forall j \in S\\ -%X_{k} \in \{0,1\}, \forall k \in A \\ -%M^j_i, V^j_i \in \mathbb{R}^{+} -%\end{array} -%\right. -%\end{equation} If a given level of coverage $l$ is required for one sensor, the sensor is said to be undercovered (respectively overcovered) if the level of coverage of one of @@ -634,7 +569,7 @@ be alive during one sensing phase) are considered in the model. \subsection{Simulation Settings} The WSN area of interest is supposed to be divided into 16~regular subregions -and we use the same energy consumption model as in our previous +and the energy consumption model used is described in previous work~\citep{Idrees2}. Table~\ref{table3} gives the chosen parameters settings. \begin{table}[ht] @@ -697,7 +632,7 @@ approach. 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, the sensor field is discretized as + observe the area of interest. Here the sensor field is discretized as a regular grid, which yields the following equation: \begin{equation*} \scriptsize @@ -707,7 +642,7 @@ approach. subregions during the current sensing phase and $N$ is total number of grid points in the sensing field. A layout of $N~=~51~\times~26~=~1326$~grid points is considered in the simulations. -\item {\bf Active Sensors Ratio (ASR)}: a major objective of our protocol is to +\item {\bf Active Sensors Ratio (ASR)}: a major objective of the proposed 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: @@ -719,14 +654,12 @@ approach. sensing period~$p$, $R$ is the number of subregions, and $|J|$ is the number of sensors in the network. -\item {\bf \textcolor{green}{Energy Saving Ratio (ESR)}}: -\textcolor{green}{this metric, which shows the ability of a protocol to save energy, is defined by: +\item {\bf Energy Saving Ratio (ESR)}:this metric, which shows the ability of a protocol to save energy, is defined by: \begin{equation*} \scriptsize \mbox{ESR}(\%) = \frac{\mbox{Number of alive sensors during this round}} {\mbox{Total number of sensors in the network}} \times 100. \end{equation*} - } \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 @@ -751,9 +684,9 @@ approach. \subsection{Simulation Results} -In order to assess and analyze the performance of our protocol we have -implemented the PeCO protocol in OMNeT++~\citep{varga} simulator. The -simulations were run on a DELL laptop with an Intel Core~i3~2370~M (1.8~GHz) + +The PeCO protocol has been implemented in OMNeT++~\citep{varga} simulator in order to assess and analyze its performance. +The simulations were run on a DELL laptop with an Intel Core~i3~2370~M (1.8~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 @@ -785,9 +718,9 @@ 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) \citep{glpk} through a Branch-and-Bound method. In practice, executing GLPK on a sensor node is obviously intractable due to the -huge memory use. Fortunately, to solve the optimization problem we could use +huge memory use. Fortunately, to solve the optimization problem, the use of commercial solvers like CPLEX \citep{iamigo:cplex} which are less memory -consuming and more efficient, or implement a lightweight heuristic. For example, +consuming and more efficient is possible, or a lightweight heuristic may be implemented. For example, for a WSN of 200 sensor nodes, a leader node has to deal with constraints induced by about 12 sensor nodes. In that case, to solve the optimization problem a memory consumption of more than 1~MB can be observed with GLPK, @@ -799,8 +732,8 @@ proposed by \citep{ChinhVu}. The second one, called GAF~\citep{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~\citep{Idrees2}, is an improved version of a research work we presented -in~\citep{idrees2014coverage}. Let us notice that the PeCO and DiLCO protocols +protocol~\citep{Idrees2}, is an improved version of a research work presented +in~\citep{idrees2014coverage}. 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. Of course, this number of @@ -809,8 +742,9 @@ the number of sensors. 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. The DiLCO protocol tries to satisfy the coverage of a set of primary points, whereas the objective of the PeCO protocol -is to reach a desired level of coverage for each sensor perimeter. In our -experimentations, we chose a level of coverage equal to one ($l=1$). +is to reach a desired level of coverage for each sensor perimeter. In the +experimentations, a level of coverage equal to one ($l=1$) is chosen +. \subsubsection{Coverage Ratio} @@ -839,7 +773,7 @@ allows later a substantial increase of the coverage performance. Minimizing the number of active sensor nodes in each period is essential to minimize the energy consumption and thus to maximize the network lifetime. Figure~\ref{figure6} 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 +nodes. DESK and GAF have 30.36~\% and 34.96~\% active nodes for the first fourteen rounds, and the 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, the PeCO protocol has a lower number of active nodes in @@ -853,12 +787,12 @@ keeping a greater coverage ratio as shown in Figure \ref{figure5}. \label{figure6} \end{figure} -\subsubsection{\textcolor{green}{Energy Saving Ratio}} +\subsubsection{Energy Saving Ratio} -\textcolor{green}{The simulation results show that our protocol PeCO saves +The simulation results show that the protocol PeCO saves efficiently energy by turning off some sensors during the sensing phase. As - shown in Figure~\ref{fig5}, GAF provides better energy saving than PeCO for + shown in Figure~\ref{figure7}, GAF provides better energy saving than PeCO for the first fifty rounds. Indeed GAF balances the energy consumption among sensor nodes inside each small fixed grid and thus permits to extend the life of sensors in each grid fairly. However, at the same time it turns on a large @@ -866,22 +800,22 @@ keeping a greater coverage ratio as shown in Figure \ref{figure5}. DESK algorithm shows less energy saving compared with other approaches. In comparison with PeCO, DiLCO protocol usually provides lower energy saving ratios. Moreover, it can be noticed that after round fifty, PeCO protocol - exhibits the slowest decrease among all the considered protocols.} + exhibits the slowest decrease among all the considered protocols. \begin{figure}[h!] %\centering % \begin{multicols}{6} \centering -\includegraphics[scale=0.5]{ESR.eps} %\\~ ~ ~(a) -\caption{Energy Saving Ratio for 200 deployed nodes} -\label{fig5} +\includegraphics[scale=0.5]{figure7.eps} %\\~ ~ ~(a) +\caption{Energy Saving Ratio for 200 deployed nodes.} +\label{figure7} \end{figure} \subsubsection{Energy Consumption} The effect of the energy consumed by the WSN during the communication, computation, listening, active, and sleep status is studied for different -network densities and the four approaches compared. Figures~\ref{figure7}(a) +network densities and the four approaches compared. Figures~\ref{figure8}(a) and (b) illustrate the energy consumption for different network sizes and for $Lifetime_{95}$ and $Lifetime_{50}$. The results show that the PeCO protocol is the most competitive from the energy consumption point of view. As shown by both figures, @@ -890,59 +824,58 @@ 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 also the energy consumption while keeping a good -coverage level. Let us notice that the energy overhead when increasing network +coverage level. The energy overhead when increasing network size is the lowest with PeCO. \begin{figure}[h!] \centering \begin{tabular}{@{}cr@{}} - \includegraphics[scale=0.5]{figure7a.eps} & \raisebox{2.75cm}{(a)} \\ - \includegraphics[scale=0.5]{figure7b.eps} & \raisebox{2.75cm}{(b)} + \includegraphics[scale=0.5]{figure8a.eps} & \raisebox{2.75cm}{(a)} \\ + \includegraphics[scale=0.5]{figure8b.eps} & \raisebox{2.75cm}{(b)} \end{tabular} \caption{Energy consumption per period for (a)~$Lifetime_{95}$ and (b)~$Lifetime_{50}$.} - \label{figure7} + \label{figure8} \end{figure} \subsubsection{Network Lifetime} -We observe the superiority of both the PeCO and DiLCO protocols in comparison with -the two other approaches in prolonging the network lifetime. In -Figures~\ref{figure8}(a) and (b), $Lifetime_{95}$ and $Lifetime_{50}$ are shown for +In comparison with the two other approaches, PeCO and DiLCO protocols are better for prolonging the network lifetime. In +Figures~\ref{figure9}(a) and (b), $Lifetime_{95}$ and $Lifetime_{50}$ are shown for different network sizes. As can be seen in these figures, the lifetime increases with the size of the network, and it is clearly larger for the DiLCO and PeCO protocols. For instance, for a network of 300~sensors and coverage ratio -greater than 50\%, we can see on Figure~\ref{figure8}(b) that the lifetime is +greater than 50\%, it can be observed on Figure~\ref{figure9}(b) that the lifetime is about twice longer with PeCO compared to the DESK protocol. The performance -difference is more obvious in Figure~\ref{figure8}(b) than in -Figure~\ref{figure8}(a) because the gain induced by our protocols increases with +difference is more obvious in Figure~\ref{figure9}(b) than in +Figure~\ref{figure9}(a) because the gain induced by protocols (PeCO and DiLCO) increases with time, and the lifetime with a coverage over 50\% is far longer than with 95\%. \begin{figure}[h!] \centering \begin{tabular}{@{}cr@{}} - \includegraphics[scale=0.5]{figure8a.eps} & \raisebox{2.75cm}{(a)} \\ - \includegraphics[scale=0.5]{figure8b.eps} & \raisebox{2.75cm}{(b)} + \includegraphics[scale=0.5]{figure9a.eps} & \raisebox{2.75cm}{(a)} \\ + \includegraphics[scale=0.5]{figure9b.eps} & \raisebox{2.75cm}{(b)} \end{tabular} \caption{Network Lifetime for (a)~$Lifetime_{95}$ and (b)~$Lifetime_{50}$.} - \label{figure8} + \label{figure9} \end{figure} -Figure~\ref{figure9} compares the lifetime coverage of the DiLCO and PeCO protocols -for different coverage ratios. We denote by Protocol/70, Protocol/80, -Protocol/85, Protocol/90, and Protocol/95 the amount of time during which the +Figure~\ref{figure10} compares the lifetime coverage of the DiLCO and PeCO protocols +for different coverage ratios. Protocol/70, Protocol/80, +Protocol/85, Protocol/90, and Protocol/95 correspond to the amount of time during which the network can satisfy an area coverage greater than $70\%$, $80\%$, $85\%$, $90\%$, and $95\%$ respectively, where the term Protocol refers to DiLCO or -PeCO. \textcolor{green}{Indeed there are applications that do not require a 100\% coverage of the +PeCO. Indeed there are applications that do not require a 100\% coverage of the area to be monitored. For example, forest fire application might require complete coverage in summer seasons while only require 80$\%$ of the area to be covered in rainy seasons~\citep{li2011transforming}. As another example, birds habit study requires only 70$\%$-coverage at nighttime when the birds are sleeping while requires 100$\%$-coverage at daytime when the birds are active~\citep{1279193}. PeCO always outperforms DiLCO for the three lower coverage ratios, moreover the -improvements grow with the network size. DiLCO outperforms PeCO when the coverage ratio is required to be $>90\%$, but PeCO extends the network lifetime significantly when coverage ratio can be relaxed.} +improvements grow with the network size. DiLCO outperforms PeCO when the coverage ratio is required to be $>90\%$, but PeCO extends the network lifetime significantly when coverage ratio can be relaxed. \begin{figure}[h!] -\centering \includegraphics[scale=0.55]{figure9.eps} +\centering \includegraphics[scale=0.55]{figure10.eps} \caption{Network lifetime for different coverage ratios.} -\label{figure9} +\label{figure10} \end{figure} \subsubsection{Impact of $\alpha$ and $\beta$ on PeCO's performance} @@ -954,12 +887,12 @@ hand, the choice of $\beta \gg \alpha$ prevents the overcoverage, and also limits the activation of a large number of sensors, but as $\alpha$ is low, some areas may be poorly covered. This explains the results obtained for $Lifetime_{50}$ with $\beta \gg \alpha$: a large number of periods with low -coverage ratio. On the other hand, when we choose $\alpha \gg \beta$, we favor -the coverage even if some areas may be overcovered, so a high coverage ratio is +coverage ratio. On the other hand, when $\alpha \gg \beta$ is chosen, +the coverage is favored even if some areas may be overcovered, so a high coverage ratio is reached, but a large number of sensors are activated to achieve this goal. Therefore the network lifetime is reduced. The choice $\alpha=0.6$ and $\beta=0.4$ seems to achieve the best compromise between lifetime and coverage -ratio. That explains why we have chosen this setting for the experiments +ratio. That explains why this setting has been chosen for the experiments presented in the previous subsections. @@ -989,14 +922,14 @@ $\alpha$ & $\beta$ & $Lifetime_{50}$ & $Lifetime_{95}$ \\ \hline \section{Conclusion and Future Works} \label{sec:Conclusion and Future Works} -In this paper we have studied the problem of perimeter 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 +In this paper the problem of perimeter coverage optimization in +WSNs has been studied. A new protocol called Perimeter-based Coverage +Optimization is designed. This protocol 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 +subset of sensors operating in active status for each period. This 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. Several simulations have @@ -1004,8 +937,8 @@ been carried out to evaluate the proposed protocol. The simulation results show that PeCO is more energy-efficient than other approaches, with respect to lifetime, coverage ratio, active sensors ratio, and energy consumption. -We plan to extend our framework so that the schedules are planned for multiple -sensing periods. We also want to improve the integer program to take into +This framework will be extented so that the schedules are planned for multiple +sensing periods. The integer program would be improved to take into account heterogeneous sensors from both energy and node characteristics point of views. Finally, it would be interesting to implement the PeCO protocol using a sensor-testbed to evaluate it in real world applications. @@ -1019,6 +952,6 @@ received support. This work is also partially funded by the Labex ACTION program (contract ANR-11-LABX-01-01). \bibliographystyle{gENO} -\bibliography{biblio} %articleeo +\bibliography{biblio} \end{document} diff --git a/PeCO-EO/articleeo.tex~ b/PeCO-EO/articleeo.tex~ index 4faafaa..64383e2 100644 --- a/PeCO-EO/articleeo.tex~ +++ b/PeCO-EO/articleeo.tex~ @@ -2,17 +2,12 @@ % v4.0 released April 2013 \documentclass{gENO2e} -%\usepackage[linesnumbered,ruled,vlined,commentsnumbered]{algorithm2e} -%\renewcommand{\algorithmcfname}{ALGORITHM} + \usepackage{indentfirst} \usepackage{color} \usepackage[algo2e,ruled,vlined]{algorithm2e} \begin{document} -%\jvol{00} \jnum{00} \jyear{2013} \jmonth{April} - -%\articletype{GUIDE} - \title{{\itshape Perimeter-based Coverage Optimization \\ to Improve Lifetime in Wireless Sensor Networks}} @@ -29,11 +24,10 @@ 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. We -propose such an approach called Perimeter-based Coverage Optimization protocol -(PeCO). It is a hybrid of centralized and distributed methods: the region of +scheduling which ensures sensing coverage while minimizing the energy cost. An approach called Perimeter-based Coverage Optimization protocol +(PeCO) is proposed. It is a hybrid of centralized and distributed methods: the region of interest is first subdivided into subregions and the protocol is then -distributed among sensor nodes in each subregion. The novelty of our approach +distributed among sensor nodes in each subregion. The novelty of the 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 demonstrate that PeCO can offer longer lifetime @@ -89,15 +83,14 @@ This paper makes the following contributions : architecture. \item A new mathematical optimization model is proposed. Instead of trying to cover a set of specified points/targets as in most of the methods proposed in - the literature, we formulate a mixed-integer program based on the perimeter - coverage of each sensor. The model involves integer variables to capture the + the literature, a mixed-integer program based on the perimeter + coverage of each sensor is formulated. The model involves integer variables to capture the deviations between the actual level of coverage and the required level. Hence, an optimal schedule will be obtained by minimizing a weighted sum of these deviations. \item Extensive simulation experiments are conducted using the discrete event - simulator OMNeT++, to demonstrate the efficiency of our protocol. We have - compared the PeCO protocol to two approaches found in the literature: - DESK~\citep{ChinhVu} and GAF~\citep{xu2001geography}, and also to our previous + simulator OMNeT++, to demonstrate the efficiency of the PeCO protocol. The PeCO protocol has been compared to two approaches found in the literature: + DESK~\citep{ChinhVu} and GAF~\citep{xu2001geography}, and also to the protocol DiLCO published in~\citep{Idrees2}. DiLCO uses the same framework as PeCO but is based on another optimization model for sensor scheduling. \end{enumerate} @@ -134,7 +127,7 @@ algorithm in $O(nd~log~d)$ time to compute the perimeter-coverage of each sensor. $d$ denotes the maximum number of sensors that are 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 + points, the 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 @@ -212,8 +205,8 @@ decomposed into 4 phases: information exchange, leader election, decision, and sensing. The simulations show that DiLCO is able to increase the WSN lifetime and provides improved coverage performance. {\it In the PeCO protocol, a new mathematical optimization model is proposed. Instead of trying to cover a set - of specified points/targets as in the DiLCO protocol, we formulate an integer - program based on the perimeter coverage of each sensor. The model involves + of specified points/targets as in the DiLCO protocol, an integer + program based on the perimeter coverage of each sensor is formulated. The model involves integer variables to capture the deviations between the actual level of coverage and the required level. The idea is that an optimal scheduling will be obtained by minimizing a weighted sum of these deviations.} @@ -221,12 +214,6 @@ and provides improved coverage performance. {\it In the PeCO protocol, a new \section{ The P{\scshape e}CO Protocol Description} \label{sec:The PeCO Protocol Description} -%In this section, the Perimeter-based Coverage -%Optimization protocol is decribed in details. First we present the assumptions we made and the models -%we considered (in particular the perimeter coverage one), second we describe the -%background idea of our protocol, and third we give the outline of the algorithm -%executed by each node. - \subsection{Assumptions and Models} \label{CI} @@ -242,7 +229,7 @@ algorithms. A Boolean disk coverage model, which is the most widely used sensor coverage model in the literature, is considered 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 +this sensor. The communication range $R_c$ is assumed to satisfy : $R_c \geq 2 \cdot R_s$. In fact, \citet{Zhang05} proved that if the transmission range fulfills the previous hypothesis, the complete coverage of a convex area implies connectivity among active nodes. @@ -256,9 +243,9 @@ $k$~sensors) if and only if each sensor in the network is $k$-perimeter-covered (perimeter covered by at least $k$ sensors). Figure~\ref{figure1}(a) shows the coverage of sensor node~$0$. On this figure, -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 the intersection of the two sensing areas. These points are +sensor~$0$ has nine neighbors. For each neighbor the two points +resulting from the intersection of the two sensing areas have been reported on its perimeter (the +perimeter of the disk covered by the sensor~$0$). These points are denoted for neighbor~$i$ by $iL$ and $iR$, respectively for left and right from a neighboring point of view. The resulting couples of intersection points subdivide the perimeter of sensor~$0$ into portions called arcs. @@ -364,7 +351,7 @@ optimization algorithm. \subsection{Main Idea} 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 +subregions using a divide-and-conquer algorithm. In a second step the protocol will be executed in a distributed way in each subregion simultaneously to schedule nodes' activities for one sensing period. Sensor nodes are assumed to be deployed almost uniformly over the region. The regular subdivision is made @@ -397,7 +384,7 @@ of the application. \label{figure4} \end{figure} -We define two types of packets to be used by the PeCO protocol: +Two types of packets used by the PeCO protocol are defined: \begin{itemize} \item INFO packet: sent by each sensor node to all the nodes inside a same subregion for information exchange. @@ -424,10 +411,6 @@ applied by a sensor node $s_k$ where $k$ is the node index in the WSN. \begin{algorithm2e} - % \KwIn{all the parameters related to information exchange} -% \KwOut{$winer-node$ (: the id of the winner sensor node, which is the leader of current round)} -% \BlankLine - %\emph{Initialize the sensor node and determine it's position and subregion} \; \label{alg:PeCO} \caption{PeCO pseudocode} \eIf{$RE_k \geq E_{th}$}{ @@ -462,42 +445,6 @@ applied by a sensor node $s_k$ where $k$ is the node index in the WSN. } \end{algorithm2e} -%\begin{algorithm} -%\noindent{\bf If} $RE_k \geq E_{th}$ {\bf then}\\ -%\hspace*{0.6cm} \emph{$s_k.status$ = COMMUNICATION;}\\ -%\hspace*{0.6cm} \emph{Send $INFO()$ packet to other nodes in subregion;}\\ -%\hspace*{0.6cm} \emph{Wait $INFO()$ packet from other nodes in subregion;}\\ -%\hspace*{0.6cm} \emph{Update K.CurrentSize;}\\ -%\hspace*{0.6cm} \emph{LeaderID = Leader election;}\\ -%\hspace*{0.6cm} {\bf If} $ s_k.ID = LeaderID $ {\bf then}\\ -%\hspace*{1.2cm} \emph{$s_k.status$ = COMPUTATION;}\\ -%\hspace*{1.2cm}{\bf If} \emph{$ s_k.ID $ is Not previously selected as a Leader} {\bf then}\\ -%\hspace*{1.8cm} \emph{ Execute the perimeter coverage model;}\\ -%\hspace*{1.2cm} {\bf end}\\ -%\hspace*{1.2cm}{\bf If} \emph{($s_k.ID $ is the same Previous Leader)~And~(K.CurrentSize = K.PreviousSize)}\\ -%\hspace*{1.8cm} \emph{ Use the same previous cover set for current sensing stage;}\\ -%\hspace*{1.2cm} {\bf end}\\ -%\hspace*{1.2cm} {\bf else}\\ -%\hspace*{1.8cm}\emph{Update $a^j_{ik}$; prepare data for IP~Algorithm;}\\ -%\hspace*{1.8cm} \emph{$\left\{\left(X_{1},\dots,X_{l},\dots,X_{K}\right)\right\}$ = Execute Integer Program Algorithm($K$);}\\ -%\hspace*{1.8cm} \emph{K.PreviousSize = K.CurrentSize;}\\ -%\hspace*{1.2cm} {\bf end}\\ -%\hspace*{1.2cm}\emph{$s_k.status$ = COMMUNICATION;}\\ -%\hspace*{1.2cm}\emph{Send $ActiveSleep()$ to each node $l$ in subregion;}\\ -%\hspace*{1.2cm}\emph{Update $RE_k $;}\\ -%\hspace*{0.6cm} {\bf end}\\ -%\hspace*{0.6cm} {\bf else}\\ -%\hspace*{1.2cm}\emph{$s_k.status$ = LISTENING;}\\ -%\hspace*{1.2cm}\emph{Wait $ActiveSleep()$ packet from the Leader;}\\ -%\hspace*{1.2cm}\emph{Update $RE_k $;}\\ -%\hspace*{0.6cm} {\bf end}\\ -%{\bf end}\\ -%{\bf else}\\ -%\hspace*{0.6cm} \emph{Exclude $s_k$ from entering in the current sensing stage;}\\ -%{\bf end}\\ -%\label{alg:PeCO} -%\end{algorithm} - In this algorithm, $K.CurrentSize$ and $K.PreviousSize$ respectively represent the current number and the previous number of living nodes in the subnetwork of the subregion. At the beginning of the first period $K.PreviousSize$ is @@ -507,7 +454,7 @@ in the current period. Each sensor node determines its position and its subregion using an embedded GPS or a location discovery algorithm. After that, all the sensors collect position coordinates, remaining energy, sensor node ID, and the number of their one-hop live neighbors during the information exchange. -\textcolor{green}{Both INFO packet and ActiveSleep packet contain two parts: header and data payload. The sensor ID is included in the header, where the header size is 8 bits. The data part includes position coordinates (64 bits), remaining energy (32 bits), and the number of one-hop live neighbors (8 bits). Therefore the size of the INFO packet is 112 bits. The ActiveSleep packet is 16 bits size, 8 bits for the header and 8 bits for data part that includes only sensor status (0 or 1).} +Both INFO packet and ActiveSleep packet contain two parts: header and data payload. The sensor ID is included in the header, where the header size is 8 bits. The data part includes position coordinates (64 bits), remaining energy (32 bits), and the number of one-hop live neighbors (8 bits). Therefore the size of the INFO packet is 112 bits. The ActiveSleep packet is 16 bits size, 8 bits for the header and 8 bits for data part that includes only sensor status (0 or 1). The sensors inside a same region cooperate to elect a leader. The selection criteria for the leader are (in order of priority): \begin{enumerate} @@ -569,7 +516,7 @@ coverage level. As the number of alive sensors decreases, it becomes impossible to reach the desired level of coverage for all coverage intervals. Therefore variables $M^j_i$ and $V^j_i$ are introduced 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 +effective number. And these deviations are minimized, 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. @@ -586,18 +533,6 @@ The coverage optimization problem can then be mathematically expressed as follow \end{aligned} \end{equation} -%\begin{equation} -%\left \{ -%\begin{array}{ll} -%\min \sum_{j \in S} \sum_{i \in I_j} (\alpha^j_i ~ M^j_i + \beta^j_i ~ V^j_i ) & \\ -%\textrm{subject to :} &\\ -%\sum_{k \in A} ( a^j_{ik} ~ X_{k}) + M^j_i \geq l \quad \forall i \in I_j, \forall j \in S\\ -%\sum_{k \in A} ( a^j_{ik} ~ X_{k}) - V^j_i \leq l \quad \forall i \in I_j, \forall j \in S\\ -%X_{k} \in \{0,1\}, \forall k \in A \\ -%M^j_i, V^j_i \in \mathbb{R}^{+} -%\end{array} -%\right. -%\end{equation} If a given level of coverage $l$ is required for one sensor, the sensor is said to be undercovered (respectively overcovered) if the level of coverage of one of @@ -634,7 +569,7 @@ be alive during one sensing phase) are considered in the model. \subsection{Simulation Settings} The WSN area of interest is supposed to be divided into 16~regular subregions -and we use the same energy consumption model as in our previous +and the energy consumption model used is described in previous work~\citep{Idrees2}. Table~\ref{table3} gives the chosen parameters settings. \begin{table}[ht] @@ -697,7 +632,7 @@ approach. 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, the sensor field is discretized as + observe the area of interest. Here the sensor field is discretized as a regular grid, which yields the following equation: \begin{equation*} \scriptsize @@ -707,7 +642,7 @@ approach. subregions during the current sensing phase and $N$ is total number of grid points in the sensing field. A layout of $N~=~51~\times~26~=~1326$~grid points is considered in the simulations. -\item {\bf Active Sensors Ratio (ASR)}: a major objective of our protocol is to +\item {\bf Active Sensors Ratio (ASR)}: a major objective of the proposed 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: @@ -719,14 +654,12 @@ approach. sensing period~$p$, $R$ is the number of subregions, and $|J|$ is the number of sensors in the network. -\item {\bf \textcolor{green}{Energy Saving Ratio (ESR)}}: -\textcolor{green}{this metric, which shows the ability of a protocol to save energy, is defined by: +\item {\bf Energy Saving Ratio (ESR)}:this metric, which shows the ability of a protocol to save energy, is defined by: \begin{equation*} \scriptsize \mbox{ESR}(\%) = \frac{\mbox{Number of alive sensors during this round}} {\mbox{Total number of sensors in the network}} \times 100. \end{equation*} - } \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 @@ -751,9 +684,9 @@ approach. \subsection{Simulation Results} -In order to assess and analyze the performance of our protocol we have -implemented the PeCO protocol in OMNeT++~\citep{varga} simulator. The -simulations were run on a DELL laptop with an Intel Core~i3~2370~M (1.8~GHz) + +The PeCO protocol has been implemented in OMNeT++~\citep{varga} simulator in order to assess and analyze its performance. +The simulations were run on a DELL laptop with an Intel Core~i3~2370~M (1.8~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 @@ -785,9 +718,9 @@ 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) \citep{glpk} through a Branch-and-Bound method. In practice, executing GLPK on a sensor node is obviously intractable due to the -huge memory use. Fortunately, to solve the optimization problem we could use +huge memory use. Fortunately, to solve the optimization problem, the use of commercial solvers like CPLEX \citep{iamigo:cplex} which are less memory -consuming and more efficient, or implement a lightweight heuristic. For example, +consuming and more efficient is possible, or a lightweight heuristic may be implemented. For example, for a WSN of 200 sensor nodes, a leader node has to deal with constraints induced by about 12 sensor nodes. In that case, to solve the optimization problem a memory consumption of more than 1~MB can be observed with GLPK, @@ -799,8 +732,8 @@ proposed by \citep{ChinhVu}. The second one, called GAF~\citep{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~\citep{Idrees2}, is an improved version of a research work we presented -in~\citep{idrees2014coverage}. Let us notice that the PeCO and DiLCO protocols +protocol~\citep{Idrees2}, is an improved version of a research work presented +in~\citep{idrees2014coverage}. 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. Of course, this number of @@ -809,8 +742,9 @@ the number of sensors. 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. The DiLCO protocol tries to satisfy the coverage of a set of primary points, whereas the objective of the PeCO protocol -is to reach a desired level of coverage for each sensor perimeter. In our -experimentations, we chose a level of coverage equal to one ($l=1$). +is to reach a desired level of coverage for each sensor perimeter. In the +experimentations, a level of coverage equal to one ($l=1$) is chosen +. \subsubsection{Coverage Ratio} @@ -839,7 +773,7 @@ allows later a substantial increase of the coverage performance. Minimizing the number of active sensor nodes in each period is essential to minimize the energy consumption and thus to maximize the network lifetime. Figure~\ref{figure6} 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 +nodes. DESK and GAF have 30.36~\% and 34.96~\% active nodes for the first fourteen rounds, and the 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, the PeCO protocol has a lower number of active nodes in @@ -853,12 +787,12 @@ keeping a greater coverage ratio as shown in Figure \ref{figure5}. \label{figure6} \end{figure} -\subsubsection{\textcolor{green}{Energy Saving Ratio}} +\subsubsection{Energy Saving Ratio} -\textcolor{green}{The simulation results show that our protocol PeCO saves +The simulation results show that the protocol PeCO saves efficiently energy by turning off some sensors during the sensing phase. As - shown in Figure~\ref{fig5}, GAF provides better energy saving than PeCO for + shown in Figure~\ref{figure7}, GAF provides better energy saving than PeCO for the first fifty rounds. Indeed GAF balances the energy consumption among sensor nodes inside each small fixed grid and thus permits to extend the life of sensors in each grid fairly. However, at the same time it turns on a large @@ -866,22 +800,22 @@ keeping a greater coverage ratio as shown in Figure \ref{figure5}. DESK algorithm shows less energy saving compared with other approaches. In comparison with PeCO, DiLCO protocol usually provides lower energy saving ratios. Moreover, it can be noticed that after round fifty, PeCO protocol - exhibits the slowest decrease among all the considered protocols.} + exhibits the slowest decrease among all the considered protocols. \begin{figure}[h!] %\centering % \begin{multicols}{6} \centering -\includegraphics[scale=0.5]{ESR.eps} %\\~ ~ ~(a) -\caption{Energy Saving Ratio for 200 deployed nodes} -\label{fig5} +\includegraphics[scale=0.5]{figure7.eps} %\\~ ~ ~(a) +\caption{Energy Saving Ratio for 200 deployed nodes.} +\label{figure7} \end{figure} \subsubsection{Energy Consumption} The effect of the energy consumed by the WSN during the communication, computation, listening, active, and sleep status is studied for different -network densities and the four approaches compared. Figures~\ref{figure7}(a) +network densities and the four approaches compared. Figures~\ref{figure8}(a) and (b) illustrate the energy consumption for different network sizes and for $Lifetime_{95}$ and $Lifetime_{50}$. The results show that the PeCO protocol is the most competitive from the energy consumption point of view. As shown by both figures, @@ -890,59 +824,58 @@ 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 also the energy consumption while keeping a good -coverage level. Let us notice that the energy overhead when increasing network +coverage level. The energy overhead when increasing network size is the lowest with PeCO. \begin{figure}[h!] \centering \begin{tabular}{@{}cr@{}} - \includegraphics[scale=0.5]{figure7a.eps} & \raisebox{2.75cm}{(a)} \\ - \includegraphics[scale=0.5]{figure7b.eps} & \raisebox{2.75cm}{(b)} + \includegraphics[scale=0.5]{figure8a.eps} & \raisebox{2.75cm}{(a)} \\ + \includegraphics[scale=0.5]{figure8b.eps} & \raisebox{2.75cm}{(b)} \end{tabular} \caption{Energy consumption per period for (a)~$Lifetime_{95}$ and (b)~$Lifetime_{50}$.} - \label{figure7} + \label{figure8} \end{figure} \subsubsection{Network Lifetime} -We observe the superiority of both the PeCO and DiLCO protocols in comparison with -the two other approaches in prolonging the network lifetime. In -Figures~\ref{figure8}(a) and (b), $Lifetime_{95}$ and $Lifetime_{50}$ are shown for +In comparison with the two other approaches, PeCO and DiLCO protocols are better for prolonging the network lifetime. In +Figures~\ref{figure9}(a) and (b), $Lifetime_{95}$ and $Lifetime_{50}$ are shown for different network sizes. As can be seen in these figures, the lifetime increases with the size of the network, and it is clearly larger for the DiLCO and PeCO protocols. For instance, for a network of 300~sensors and coverage ratio -greater than 50\%, we can see on Figure~\ref{figure8}(b) that the lifetime is +greater than 50\%, it can be observed on Figure~\ref{figure9}(b) that the lifetime is about twice longer with PeCO compared to the DESK protocol. The performance -difference is more obvious in Figure~\ref{figure8}(b) than in -Figure~\ref{figure8}(a) because the gain induced by our protocols increases with +difference is more obvious in Figure~\ref{figure9}(b) than in +Figure~\ref{figure9}(a) because the gain induced by protocols (PeCO and DiLCO) increases with time, and the lifetime with a coverage over 50\% is far longer than with 95\%. \begin{figure}[h!] \centering \begin{tabular}{@{}cr@{}} - \includegraphics[scale=0.5]{figure8a.eps} & \raisebox{2.75cm}{(a)} \\ - \includegraphics[scale=0.5]{figure8b.eps} & \raisebox{2.75cm}{(b)} + \includegraphics[scale=0.5]{figure9a.eps} & \raisebox{2.75cm}{(a)} \\ + \includegraphics[scale=0.5]{figure9b.eps} & \raisebox{2.75cm}{(b)} \end{tabular} \caption{Network Lifetime for (a)~$Lifetime_{95}$ and (b)~$Lifetime_{50}$.} - \label{figure8} + \label{figure9} \end{figure} -Figure~\ref{figure9} compares the lifetime coverage of the DiLCO and PeCO protocols -for different coverage ratios. We denote by Protocol/70, Protocol/80, -Protocol/85, Protocol/90, and Protocol/95 the amount of time during which the +Figure~\ref{figure10} compares the lifetime coverage of the DiLCO and PeCO protocols +for different coverage ratios. Protocol/70, Protocol/80, +Protocol/85, Protocol/90, and Protocol/95 correspond to the amount of time during which the network can satisfy an area coverage greater than $70\%$, $80\%$, $85\%$, $90\%$, and $95\%$ respectively, where the term Protocol refers to DiLCO or -PeCO. \textcolor{green}{Indeed there are applications that do not require a 100\% coverage of the +PeCO. Indeed there are applications that do not require a 100\% coverage of the area to be monitored. For example, forest fire application might require complete coverage in summer seasons while only require 80$\%$ of the area to be covered in rainy seasons~\citep{li2011transforming}. As another example, birds habit study requires only 70$\%$-coverage at nighttime when the birds are sleeping while requires 100$\%$-coverage at daytime when the birds are active~\citep{1279193}. PeCO always outperforms DiLCO for the three lower coverage ratios, moreover the -improvements grow with the network size. DiLCO outperforms PeCO when the coverage ratio is required to be $>90\%$, but PeCO extends the network lifetime significantly when coverage ratio can be relaxed.} +improvements grow with the network size. DiLCO outperforms PeCO when the coverage ratio is required to be $>90\%$, but PeCO extends the network lifetime significantly when coverage ratio can be relaxed. \begin{figure}[h!] -\centering \includegraphics[scale=0.55]{figure9.eps} +\centering \includegraphics[scale=0.55]{figure10.eps} \caption{Network lifetime for different coverage ratios.} -\label{figure9} +\label{figure10} \end{figure} \subsubsection{Impact of $\alpha$ and $\beta$ on PeCO's performance} @@ -954,15 +887,15 @@ hand, the choice of $\beta \gg \alpha$ prevents the overcoverage, and also limits the activation of a large number of sensors, but as $\alpha$ is low, some areas may be poorly covered. This explains the results obtained for $Lifetime_{50}$ with $\beta \gg \alpha$: a large number of periods with low -coverage ratio. On the other hand, when we choose $\alpha \gg \beta$, we favor -the coverage even if some areas may be overcovered, so a high coverage ratio is +coverage ratio. On the other hand, when $\alpha \gg \beta$ is chosen, +the coverage is favored even if some areas may be overcovered, so a high coverage ratio is reached, but a large number of sensors are activated to achieve this goal. Therefore the network lifetime is reduced. The choice $\alpha=0.6$ and $\beta=0.4$ seems to achieve the best compromise between lifetime and coverage -ratio. That explains why we have chosen this setting for the experiments +ratio. That explains why this setting has been chosen for the experiments presented in the previous subsections. -%As can be seen in Table~\ref{my-labelx}, it is obvious and clear that when $\alpha$ decreased and $\beta$ increased by any step, the network lifetime for $Lifetime_{50}$ increased and the $Lifetime_{95}$ decreased. Therefore, selecting the values of $\alpha$ and $\beta$ depend on the application type used in the sensor nework. In PeCO protocol, $\alpha$ and $\beta$ are chosen based on the largest value of network lifetime for $Lifetime_{95}$. + \begin{table}[h] \centering @@ -989,14 +922,14 @@ $\alpha$ & $\beta$ & $Lifetime_{50}$ & $Lifetime_{95}$ \\ \hline \section{Conclusion and Future Works} \label{sec:Conclusion and Future Works} -In this paper we have studied the problem of perimeter 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 +In this paper the problem of perimeter coverage optimization in +WSNs has been studied. A new protocol called Perimeter-based Coverage +Optimization is designed. This protocol 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 +subset of sensors operating in active status for each period. This 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. Several simulations have @@ -1004,8 +937,8 @@ been carried out to evaluate the proposed protocol. The simulation results show that PeCO is more energy-efficient than other approaches, with respect to lifetime, coverage ratio, active sensors ratio, and energy consumption. -We plan to extend our framework so that the schedules are planned for multiple -sensing periods. We also want to improve the integer program to take into +This framework will be extented so that the schedules are planned for multiple +sensing periods. The integer program would be improved to take into account heterogeneous sensors from both energy and node characteristics point of views. 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-/Pls {stroke [] 0 setdash vpt sub M 0 vpt2 V - currentpoint stroke M - hpt neg vpt neg R hpt2 0 V stroke - } def -/Box {stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M - 0 vpt2 neg V hpt2 0 V 0 vpt2 V - hpt2 neg 0 V closepath stroke - Pnt} def -/Crs {stroke [] 0 setdash exch hpt sub exch vpt add M - hpt2 vpt2 neg V currentpoint stroke M - hpt2 neg 0 R hpt2 vpt2 V stroke} def -/TriU {stroke [] 0 setdash 2 copy vpt 1.12 mul add M - hpt neg vpt -1.62 mul V - hpt 2 mul 0 V - hpt neg vpt 1.62 mul V closepath stroke - Pnt} def -/Star {2 copy Pls Crs} def -/BoxF {stroke [] 0 setdash exch hpt sub exch vpt add M - 0 vpt2 neg V hpt2 0 V 0 vpt2 V - hpt2 neg 0 V closepath fill} def -/TriUF {stroke [] 0 setdash vpt 1.12 mul add M - hpt neg vpt -1.62 mul V - hpt 2 mul 0 V - hpt neg vpt 1.62 mul V closepath fill} def -/TriD {stroke [] 0 setdash 2 copy vpt 1.12 mul sub M - hpt neg vpt 1.62 mul V - hpt 2 mul 0 V - hpt neg vpt -1.62 mul V closepath stroke - Pnt} def -/TriDF {stroke [] 0 setdash vpt 1.12 mul sub M - hpt neg vpt 1.62 mul V - hpt 2 mul 0 V - hpt neg vpt -1.62 mul V closepath fill} def -/DiaF {stroke [] 0 setdash vpt add M - hpt neg vpt neg V hpt vpt neg V - hpt vpt V hpt neg vpt V closepath fill} def -/Pent {stroke [] 0 setdash 2 copy gsave - translate 0 hpt M 4 {72 rotate 0 hpt L} repeat - closepath stroke grestore Pnt} def -/PentF {stroke [] 0 setdash gsave - translate 0 hpt M 4 {72 rotate 0 hpt L} repeat - closepath fill grestore} def -/Circle {stroke [] 0 setdash 2 copy - hpt 0 360 arc stroke Pnt} def -/CircleF {stroke [] 0 setdash hpt 0 360 arc fill} def -/C0 {BL [] 0 setdash 2 copy moveto vpt 90 450 arc} bind def -/C1 {BL [] 0 setdash 2 copy moveto - 2 copy vpt 0 90 arc closepath fill - vpt 0 360 arc closepath} bind def -/C2 {BL [] 0 setdash 2 copy moveto - 2 copy vpt 90 180 arc closepath fill - vpt 0 360 arc closepath} bind def -/C3 {BL [] 0 setdash 2 copy moveto - 2 copy vpt 0 180 arc closepath fill - vpt 0 360 arc closepath} bind def -/C4 {BL [] 0 setdash 2 copy moveto - 2 copy vpt 180 270 arc closepath fill - vpt 0 360 arc closepath} bind def -/C5 {BL [] 0 setdash 2 copy moveto - 2 copy vpt 0 90 arc - 2 copy moveto - 2 copy vpt 180 270 arc closepath fill - vpt 0 360 arc} bind def -/C6 {BL [] 0 setdash 2 copy moveto - 2 copy vpt 90 270 arc closepath fill - vpt 0 360 arc closepath} bind def -/C7 {BL [] 0 setdash 2 copy moveto - 2 copy vpt 0 270 arc closepath fill - vpt 0 360 arc closepath} bind def -/C8 {BL [] 0 setdash 2 copy moveto - 2 copy vpt 270 360 arc closepath fill - vpt 0 360 arc closepath} bind def -/C9 {BL [] 0 setdash 2 copy moveto - 2 copy vpt 270 450 arc closepath fill - vpt 0 360 arc closepath} bind def -/C10 {BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill - 2 copy moveto - 2 copy vpt 90 180 arc closepath fill - vpt 0 360 arc closepath} bind def -/C11 {BL [] 0 setdash 2 copy moveto - 2 copy vpt 0 180 arc closepath fill - 2 copy moveto - 2 copy vpt 270 360 arc closepath fill - vpt 0 360 arc closepath} bind def -/C12 {BL [] 0 setdash 2 copy moveto - 2 copy vpt 180 360 arc closepath fill - vpt 0 360 arc closepath} bind def -/C13 {BL [] 0 setdash 2 copy moveto - 2 copy vpt 0 90 arc closepath fill - 2 copy moveto - 2 copy vpt 180 360 arc closepath fill - vpt 0 360 arc closepath} bind def -/C14 {BL [] 0 setdash 2 copy moveto - 2 copy vpt 90 360 arc closepath fill - vpt 0 360 arc} bind def -/C15 {BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill - vpt 0 360 arc closepath} bind def -/Rec {newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto - neg 0 rlineto closepath} bind def -/Square {dup Rec} bind def -/Bsquare {vpt sub exch vpt sub exch vpt2 Square} bind def -/S0 {BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare} bind def -/S1 {BL [] 0 setdash 2 copy vpt Square fill Bsquare} bind def -/S2 {BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare} bind def -/S3 {BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare} bind def -/S4 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare} bind def -/S5 {BL [] 0 setdash 2 copy 2 copy vpt Square fill - exch vpt sub exch vpt sub vpt Square fill Bsquare} bind def -/S6 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare} bind def -/S7 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill - 2 copy vpt Square fill Bsquare} bind def -/S8 {BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare} bind def -/S9 {BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare} bind def -/S10 {BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill - Bsquare} bind def -/S11 {BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill - Bsquare} bind def -/S12 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare} bind def -/S13 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill - 2 copy vpt Square fill Bsquare} bind def -/S14 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill - 2 copy exch vpt sub exch vpt Square fill Bsquare} bind def -/S15 {BL [] 0 setdash 2 copy Bsquare fill Bsquare} bind def -/D0 {gsave translate 45 rotate 0 0 S0 stroke grestore} bind def -/D1 {gsave translate 45 rotate 0 0 S1 stroke grestore} bind def -/D2 {gsave translate 45 rotate 0 0 S2 stroke grestore} bind def -/D3 {gsave translate 45 rotate 0 0 S3 stroke grestore} bind def -/D4 {gsave translate 45 rotate 0 0 S4 stroke grestore} bind def -/D5 {gsave translate 45 rotate 0 0 S5 stroke grestore} bind def -/D6 {gsave translate 45 rotate 0 0 S6 stroke grestore} bind def -/D7 {gsave translate 45 rotate 0 0 S7 stroke grestore} bind def -/D8 {gsave translate 45 rotate 0 0 S8 stroke grestore} bind def -/D9 {gsave translate 45 rotate 0 0 S9 stroke grestore} bind def -/D10 {gsave translate 45 rotate 0 0 S10 stroke grestore} bind def -/D11 {gsave translate 45 rotate 0 0 S11 stroke grestore} bind def -/D12 {gsave translate 45 rotate 0 0 S12 stroke grestore} bind def -/D13 {gsave translate 45 rotate 0 0 S13 stroke grestore} bind def -/D14 {gsave translate 45 rotate 0 0 S14 stroke grestore} bind def -/D15 {gsave translate 45 rotate 0 0 S15 stroke grestore} bind def -/DiaE {stroke [] 0 setdash vpt add M - hpt neg vpt neg V hpt vpt neg V - hpt vpt V hpt neg vpt V closepath stroke} def -/BoxE {stroke [] 0 setdash exch hpt sub exch vpt add M - 0 vpt2 neg V hpt2 0 V 0 vpt2 V - hpt2 neg 0 V closepath stroke} def -/TriUE {stroke [] 0 setdash vpt 1.12 mul add M - hpt neg vpt -1.62 mul V - hpt 2 mul 0 V - hpt neg vpt 1.62 mul V closepath stroke} def -/TriDE {stroke [] 0 setdash vpt 1.12 mul sub M - hpt neg vpt 1.62 mul V - hpt 2 mul 0 V - hpt neg vpt -1.62 mul V closepath stroke} def -/PentE {stroke [] 0 setdash gsave - translate 0 hpt M 4 {72 rotate 0 hpt L} repeat - closepath stroke grestore} def -/CircE {stroke [] 0 setdash - hpt 0 360 arc stroke} def -/Opaque {gsave closepath 1 setgray fill grestore 0 setgray closepath} def -/DiaW {stroke [] 0 setdash vpt add M - hpt neg vpt neg V hpt vpt neg V - hpt vpt V hpt neg vpt V Opaque stroke} def -/BoxW {stroke [] 0 setdash exch hpt sub exch vpt add M - 0 vpt2 neg V hpt2 0 V 0 vpt2 V - hpt2 neg 0 V Opaque stroke} def -/TriUW {stroke [] 0 setdash vpt 1.12 mul add M - hpt neg vpt -1.62 mul V - hpt 2 mul 0 V - hpt neg vpt 1.62 mul V Opaque stroke} def -/TriDW {stroke [] 0 setdash vpt 1.12 mul sub M - hpt neg vpt 1.62 mul V - hpt 2 mul 0 V - hpt neg vpt -1.62 mul V Opaque stroke} def -/PentW {stroke [] 0 setdash gsave - translate 0 hpt M 4 {72 rotate 0 hpt L} repeat - Opaque stroke grestore} def -/CircW {stroke [] 0 setdash - hpt 0 360 arc Opaque stroke} def -/BoxFill {gsave Rec 1 setgray fill grestore} def -/Density { - /Fillden exch def - currentrgbcolor - /ColB exch def /ColG exch def /ColR exch def - /ColR ColR Fillden mul Fillden sub 1 add def - /ColG ColG Fillden mul Fillden sub 1 add def - /ColB ColB Fillden mul Fillden sub 1 add def - ColR ColG ColB setrgbcolor} def -/BoxColFill {gsave Rec PolyFill} def -/PolyFill {gsave Density fill grestore grestore} def -/h {rlineto rlineto rlineto gsave closepath fill grestore} bind def -% -% PostScript Level 1 Pattern Fill routine for rectangles -% Usage: x y w h s a XX PatternFill -% x,y = lower left corner of box to be filled -% w,h = width and height of box -% a = angle in degrees between lines and x-axis -% XX = 0/1 for no/yes cross-hatch -% -/PatternFill {gsave /PFa [ 9 2 roll ] def - PFa 0 get PFa 2 get 2 div add PFa 1 get PFa 3 get 2 div add translate - PFa 2 get -2 div PFa 3 get -2 div PFa 2 get PFa 3 get Rec - gsave 1 setgray fill grestore clip - currentlinewidth 0.5 mul setlinewidth - /PFs PFa 2 get dup mul PFa 3 get dup mul add sqrt def - 0 0 M PFa 5 get rotate PFs -2 div dup translate - 0 1 PFs PFa 4 get div 1 add floor cvi - {PFa 4 get mul 0 M 0 PFs V} for - 0 PFa 6 get ne { - 0 1 PFs PFa 4 get div 1 add floor cvi - {PFa 4 get mul 0 2 1 roll M PFs 0 V} for - } if - stroke 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{Pat9} {Pat6} ifelse setpattern} bind def -/Pattern7 {PatternBgnd KeepColor Landscape {Pat8} {Pat7} ifelse setpattern} bind def -} def -% -% -%End of PostScript Level 2 code -% -/PatternBgnd { - TransparentPatterns {} {gsave 1 setgray fill grestore} ifelse -} def -% -% Substitute for Level 2 pattern fill codes with -% grayscale if Level 2 support is not selected. -% -/Level1PatternFill { -/Pattern1 {0.250 Density} bind def -/Pattern2 {0.500 Density} bind def -/Pattern3 {0.750 Density} bind def -/Pattern4 {0.125 Density} bind def -/Pattern5 {0.375 Density} bind def -/Pattern6 {0.625 Density} bind def -/Pattern7 {0.875 Density} bind def -} def -% -% Now test for support of Level 2 code -% -Level1 {Level1PatternFill} {Level2PatternFill} ifelse -% -/Symbol-Oblique /Symbol findfont [1 0 .167 1 0 0] makefont -dup length dict begin {1 index /FID eq {pop pop} {def} ifelse} forall -currentdict end definefont pop -% -% Encoding for ISO-8859-1 (also called Latin1) -% -/reencodeISO { -dup dup 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def -/LT2 {PL [2 dl1 3 dl2] LC2 DL} def -/LT3 {PL [1 dl1 1.5 dl2] LC3 DL} def -/LT4 {PL [6 dl1 2 dl2 1 dl1 2 dl2] LC4 DL} def -/LT5 {PL [3 dl1 3 dl2 1 dl1 3 dl2] LC5 DL} def -/LT6 {PL [2 dl1 2 dl2 2 dl1 6 dl2] LC6 DL} def -/LT7 {PL [1 dl1 2 dl2 6 dl1 2 dl2 1 dl1 2 dl2] LC7 DL} def -/LT8 {PL [2 dl1 2 dl2 2 dl1 2 dl2 2 dl1 2 dl2 2 dl1 4 dl2] LC8 DL} def -/Pnt {stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore} def -/Dia {stroke [] 0 setdash 2 copy vpt add M - hpt neg vpt neg V hpt vpt neg V - hpt vpt V hpt neg vpt V closepath stroke - Pnt} def -/Pls {stroke [] 0 setdash vpt sub M 0 vpt2 V - currentpoint stroke M - hpt neg vpt neg R hpt2 0 V stroke - } def -/Box {stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M - 0 vpt2 neg V hpt2 0 V 0 vpt2 V - hpt2 neg 0 V closepath stroke - Pnt} def -/Crs {stroke [] 0 setdash exch hpt sub exch vpt add M - hpt2 vpt2 neg V currentpoint stroke M - hpt2 neg 0 R hpt2 vpt2 V stroke} def -/TriU {stroke [] 0 setdash 2 copy vpt 1.12 mul add M 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closepath} bind def -/C8 {BL [] 0 setdash 2 copy moveto - 2 copy vpt 270 360 arc closepath fill - vpt 0 360 arc closepath} bind def -/C9 {BL [] 0 setdash 2 copy moveto - 2 copy vpt 270 450 arc closepath fill - vpt 0 360 arc closepath} bind def -/C10 {BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill - 2 copy moveto - 2 copy vpt 90 180 arc closepath fill - vpt 0 360 arc closepath} bind def -/C11 {BL [] 0 setdash 2 copy moveto - 2 copy vpt 0 180 arc closepath fill - 2 copy moveto - 2 copy vpt 270 360 arc closepath fill - vpt 0 360 arc closepath} bind def -/C12 {BL [] 0 setdash 2 copy moveto - 2 copy vpt 180 360 arc closepath fill - vpt 0 360 arc closepath} bind def -/C13 {BL [] 0 setdash 2 copy moveto - 2 copy vpt 0 90 arc closepath fill - 2 copy moveto - 2 copy vpt 180 360 arc closepath fill - vpt 0 360 arc closepath} bind def -/C14 {BL [] 0 setdash 2 copy moveto - 2 copy vpt 90 360 arc closepath fill - vpt 0 360 arc} bind def -/C15 {BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill - vpt 0 360 arc closepath} bind def -/Rec {newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto - neg 0 rlineto closepath} bind def -/Square {dup Rec} bind def -/Bsquare {vpt sub exch vpt sub exch vpt2 Square} bind def -/S0 {BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare} bind def -/S1 {BL [] 0 setdash 2 copy vpt Square fill Bsquare} bind def -/S2 {BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare} bind def -/S3 {BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare} bind def -/S4 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare} bind def -/S5 {BL [] 0 setdash 2 copy 2 copy vpt Square fill - exch vpt sub exch vpt sub vpt Square fill Bsquare} bind def -/S6 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare} bind def -/S7 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill - 2 copy vpt Square fill Bsquare} bind def -/S8 {BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare} bind def -/S9 {BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare} bind def -/S10 {BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill - Bsquare} bind def -/S11 {BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill - Bsquare} bind def -/S12 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare} bind def -/S13 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill - 2 copy vpt Square fill Bsquare} bind def -/S14 {BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill - 2 copy exch vpt sub exch vpt Square fill Bsquare} bind def -/S15 {BL [] 0 setdash 2 copy Bsquare fill Bsquare} bind def -/D0 {gsave translate 45 rotate 0 0 S0 stroke grestore} bind def -/D1 {gsave translate 45 rotate 0 0 S1 stroke grestore} bind def -/D2 {gsave translate 45 rotate 0 0 S2 stroke grestore} bind def -/D3 {gsave translate 45 rotate 0 0 S3 stroke grestore} bind def -/D4 {gsave translate 45 rotate 0 0 S4 stroke grestore} bind def -/D5 {gsave translate 45 rotate 0 0 S5 stroke grestore} bind def -/D6 {gsave translate 45 rotate 0 0 S6 stroke grestore} bind def -/D7 {gsave translate 45 rotate 0 0 S7 stroke grestore} bind def -/D8 {gsave translate 45 rotate 0 0 S8 stroke grestore} bind def -/D9 {gsave translate 45 rotate 0 0 S9 stroke grestore} bind def -/D10 {gsave translate 45 rotate 0 0 S10 stroke grestore} bind def -/D11 {gsave translate 45 rotate 0 0 S11 stroke grestore} bind def -/D12 {gsave translate 45 rotate 0 0 S12 stroke grestore} bind def -/D13 {gsave translate 45 rotate 0 0 S13 stroke grestore} bind def -/D14 {gsave translate 45 rotate 0 0 S14 stroke grestore} bind def -/D15 {gsave translate 45 rotate 0 0 S15 stroke grestore} bind def -/DiaE {stroke [] 0 setdash vpt add M - hpt neg vpt neg V hpt vpt neg V - hpt vpt V hpt neg vpt V closepath stroke} def -/BoxE {stroke [] 0 setdash exch hpt sub exch vpt add M - 0 vpt2 neg V hpt2 0 V 0 vpt2 V - hpt2 neg 0 V closepath stroke} def -/TriUE {stroke [] 0 setdash vpt 1.12 mul add M - hpt neg vpt -1.62 mul V - hpt 2 mul 0 V - hpt neg vpt 1.62 mul V closepath stroke} def -/TriDE {stroke [] 0 setdash vpt 1.12 mul sub M - hpt neg vpt 1.62 mul V - hpt 2 mul 0 V - hpt neg vpt -1.62 mul V closepath stroke} def -/PentE {stroke [] 0 setdash gsave - translate 0 hpt M 4 {72 rotate 0 hpt L} repeat - closepath stroke grestore} def -/CircE {stroke [] 0 setdash - hpt 0 360 arc stroke} def -/Opaque {gsave closepath 1 setgray fill grestore 0 setgray closepath} def -/DiaW {stroke [] 0 setdash vpt add M - hpt neg vpt neg V hpt vpt neg V - hpt vpt V hpt neg vpt V Opaque stroke} def -/BoxW {stroke [] 0 setdash exch hpt sub exch vpt add M - 0 vpt2 neg V hpt2 0 V 0 vpt2 V - hpt2 neg 0 V Opaque stroke} def -/TriUW {stroke [] 0 setdash vpt 1.12 mul add M - hpt neg vpt -1.62 mul V - hpt 2 mul 0 V - hpt neg vpt 1.62 mul V Opaque stroke} def -/TriDW {stroke [] 0 setdash vpt 1.12 mul sub M - hpt neg vpt 1.62 mul V - hpt 2 mul 0 V - hpt neg vpt -1.62 mul V Opaque stroke} def -/PentW {stroke [] 0 setdash gsave - translate 0 hpt M 4 {72 rotate 0 hpt L} repeat - Opaque stroke grestore} def -/CircW {stroke [] 0 setdash - hpt 0 360 arc Opaque stroke} def -/BoxFill {gsave Rec 1 setgray fill grestore} def -/Density { - /Fillden exch def - currentrgbcolor - /ColB exch def /ColG exch def /ColR exch def - /ColR ColR Fillden mul Fillden sub 1 add def - /ColG ColG Fillden mul Fillden sub 1 add def - /ColB ColB Fillden mul Fillden sub 1 add def - ColR ColG ColB setrgbcolor} def -/BoxColFill {gsave Rec PolyFill} def -/PolyFill {gsave Density fill grestore grestore} def -/h {rlineto rlineto rlineto gsave closepath fill grestore} bind def -% -% PostScript Level 1 Pattern Fill routine for rectangles -% Usage: x y w h s a XX PatternFill -% x,y = lower left corner of box to be filled -% w,h = width and height of box -% a = angle in degrees between lines and x-axis -% XX = 0/1 for no/yes cross-hatch -% 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