%Instead of working with a continuous coverage area, we make it discrete by considering for each sensor a set of points called primary points. Consequently, we assume that the sensing disk defined by a sensor is covered if all of its primary points are covered. The choice of number and locations of primary points is the subject of another study not presented here.
-\indent Instead of working with the coverage area, we consider for each sensor a set of points called primary points~\cite{idrees2014coverage}. We also assume that the sensing disk defined by a sensor is covered if all the primary points of this sensor are covered. By knowing the position (point center: ($p_x,p_y$)) of a wireless sensor node and it's sensing range $R_s$, we calculate the primary points directly based on the proposed model. We use these primary points (that can be increased or decreased if necessary) as references to ensure that the monitored region of interest is covered by the selected set of sensors, instead of using all the points in the area.
-We can calculate the positions of the selected primary
-points in the circle disk of the sensing range of a wireless sensor
-node (see Figure~\ref{fig1}) as follows:\\
-Assuming that the point center of a wireless sensor node is located at $(p_x,p_y)$, we can define up to 25 primary points $X_1$ to $X_{25}$.\\
+\indent Instead of working with the coverage area, we consider for each sensor a set of points called primary points~\cite{idrees2014coverage}. We assume that the sensing disk defined by a sensor is covered if all the primary points of this sensor are covered. By knowing the position (point center: ($p_x,p_y$)) of a wireless sensor node and it's sensing range $R_s$, we define up to 25 primary points $X_1$ to $X_{25}$ as decribed on Figure~\ref{fig1}. The coordinates of the primary points are the following :\\
%$(p_x,p_y)$ = point center of wireless sensor node\\
$X_1=(p_x,p_y)$ \\
$X_2=( p_x + R_s * (1), p_y + R_s * (0) )$\\
$X_{15}=( p_x + R_s * (\frac{-\sqrt{3}}{2}), p_y + R_s * (\frac{1}{2})) $\\
$X_{16}=( p_x + R_s * (\frac{\sqrt{3}}{2}), p_y + R_s * (\frac{- 1}{2})) $\\
$X_{17}=( p_x + R_s * (\frac{-\sqrt{3}}{2}), p_y + R_s * (\frac{- 1}{2})) $\\
-$X_{18}=( p_x + R_s * (\frac{\sqrt{3}}{2}), p_y + R_s * (0) $\\
-$X_{19}=( p_x + R_s * (\frac{-\sqrt{3}}{2}), p_y + R_s * (0) $\\
+$X_{18}=( p_x + R_s * (\frac{\sqrt{3}}{2}), p_y + R_s * (0)) $\\
+$X_{19}=( p_x + R_s * (\frac{-\sqrt{3}}{2}), p_y + R_s * (0)) $\\
$X_{20}=( p_x + R_s * (0), p_y + R_s * (\frac{1}{2})) $\\
$X_{21}=( p_x + R_s * (0), p_y + R_s * (-\frac{1}{2})) $\\
$X_{22}=( p_x + R_s * (\frac{1}{2}), p_y + R_s * (\frac{\sqrt{3}}{2})) $\\
\subsection{Background idea}
%%RC : we need to clarify the difference between round and period. Currently it seems to be the same (for me at least).
-The area of interest can be divided using the divide-and-conquer strategy into
-smaller areas, called subregions, and then our MuDiLCO protocol will be
-implemented in each subregion in a distributed way.
+%The area of interest can be divided using the divide-and-conquer strategy into
+%smaller areas, called subregions, and then our MuDiLCO protocol will be
+%implemented in each subregion in a distributed way.
+
+\textcolor{green}{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. Sensor nodes are assumed to
+be deployed almost uniformly over the region. The regular subdivision is made
+such that the number of hops between any pairs of sensors inside a subregion is
+less than or equal to 3.}
As can be seen in Figure~\ref{fig2}, our protocol works in periods fashion,
where each is divided into 4 phases: Information~Exchange, Leader~Election,
% is used to refer this table in the text
\end{table}
-\textcolor{red}{Our first protocol based GLPK optimization solver is declined into four versions: MuDiLCO-1, MuDiLCO-3, MuDiLCO-5,
-and MuDiLCO-7, corresponding respectively to $T=1,3,5,7$ ($T$ the number of
-rounds in one sensing period). }
-%The second protocol based GA is declined into four versions: GA-MuDiLCO-1, GA-MuDiLCO-3, GA-MuDiLCO-5,
-%and GA-MuDiLCO-7 for the same reason of the first protocol. After extensive experiments, we chose the dedicated values for the parameters $P_c$, $P_m$, and $S_{pop}$ because they gave the best results}.
+\textcolor{green}{The MuDilLCO protocol is declined into four versions: MuDiLCO-1, MuDiLCO-3, MuDiLCO-5,
+and MuDiLCO-7, corresponding respectively to $T=1,3,5,7$ ($T$ the number of rounds in one sensing period). Since the time resolution may be prohibitif when the size of the problem increases, a time limit treshold has been fixed to solve large instances. In these cases, the solver returns the best solution found, which is not necessary the optimal solution.
+ Table \ref{tl} shows time limit values. These time limit treshold have been set empirically. The basic idea consists in considering the average execution time to solve the integer programs to optimality, then by dividing this average time by three to set the threshold value. After that, this treshold value is increased if necessary such that the solver is able to deliver a feasible solution within the time limit. In fact, selecting the optimal values for the time limits will be investigated in future. In Table \ref{tl}, "NO" indicates that the problem has been solved to optimality without time limit. }.
+
+\begin{table}[ht]
+\caption{Time limit values for MuDiLCO protocol versions }
+\centering
+\begin{tabular}{|c|c|c|c|c|}
+ \hline
+ WSN size & MuDiLCO-1 & MuDiLCO-3 & MuDiLCO-5 & MuDiLCO-7 \\ [0.5ex]
+\hline
+ 50 & NO & NO & NO & NO \\
+ \hline
+100 & NO & NO & NO & NO \\
+\hline
+150 & NO & NO & NO & 0.03 \\
+\hline
+200 & NO & 0.01 & 0.02 & 0.06 \\
+ \hline
+ 250 & NO & 0.02 & 0.03 & 0.08 \\
+ \hline
+\end{tabular}
+
+\label{tl}
+
+\end{table}
+
+
+
+
In the following, we will make comparisons with
two other methods. The first method, called DESK and proposed by \cite{ChinhVu},
is a full distributed coverage algorithm. The second method, called
\end{enumerate}
+\section{Results and analysis}
\subsection{Performance Analysis for Different Number of Primary Points}
\label{ch4:sec:04:06}
Moreover, when the number of periods increases, coverage ratio produced by all models decrease, but Model-5 is the one with the slowest decrease due to a smaller time computation of decision process for a smaller number of primary points.
As shown in Figure ~\ref{Figures/ch4/R2/CR}, coverage ratio decreases when the number of periods increases due to dead nodes. Model-5 is slightly more efficient than other models, because it offers a good coverage ratio for a larger number of periods in comparison with other models.
-%\item {{\bf Active Sensors Ratio}}
-\subsubsection{Active Sensors Ratio}
-
-Figure~\ref{Figures/ch4/R2/ASR} shows the average active nodes ratio for 150 deployed nodes.
-\begin{figure}[h!]
-\centering
-\includegraphics[scale=0.5]{R2/ASR.pdf}
-\caption{Active sensors ratio for 150 deployed nodes }
-\label{Figures/ch4/R2/ASR}
-\end{figure}
-The results presented in Figure~\ref{Figures/ch4/R2/ASR} show the superiority of the proposed Model-5, in comparison with the other models. The model with fewer number of primary points uses fewer active nodes than the other models.
-According to the results presented in Figure~\ref{Figures/ch4/R2/CR}, we observe that Model-5 continues for a larger number of periods with a better coverage ratio compared with other models. The advantage of Model-5 is to use fewer number of active nodes for each period compared with Model-9, Model-13, Model-17, and Model-21. This led to continuing for a larger number of periods and thus extending the network lifetime.
-
-
-%\item {{\bf Stopped simulation runs}}
-\subsubsection{Stopped simulation runs}
-
-Figure~\ref{Figures/ch4/R2/SR} illustrates the percentage of stopped simulation runs per period for 150 deployed nodes.
-
-\begin{figure}[h!]
-\centering
-\includegraphics[scale=0.5]{R2/SR.pdf}
-\caption{Percentage of stopped simulation runs for 150 deployed nodes }
-\label{Figures/ch4/R2/SR}
-\end{figure}
-
-When the number of primary points is increased, the percentage of the stopped simulation runs per period is increased. The reason behind the increase is the increasing number of dead sensors when the primary points increase. Model-5 is better than other models because it conserves more energy by turning on less sensors during the sensing phase and in the same time it preserves a good coverage for a larger number of periods in comparison with other models. Model~5 seems to be more suitable to be used in wireless sensor networks. \\
-
-
-%\item {{\bf Energy Consumption}}
-\subsubsection{Energy Consumption}
-
-In this experiment, we study the effect of increasing the primary points to represent the area of the sensor on the energy consumed by the wireless sensor network for different network densities. Figures~\ref{Figures/ch4/R2/EC}(a) and~\ref{Figures/ch4/R2/EC}(b) illustrate the energy consumption for different network sizes for $Lifetime_{95}$ and $Lifetime_{50}$.
-
-\begin{figure}[h!]
-\centering
- %\begin{multicols}{1}
-\centering
-\includegraphics[scale=0.5]{R2/EC95.pdf}\\~ ~ ~ ~ ~(a) \\
-%\vfill
-\includegraphics[scale=0.5]{R2/EC50.pdf}\\~ ~ ~ ~ ~(b)
-
-%\end{multicols}
-\caption{Energy consumption for (a) $Lifetime_{95}$ and (b) $Lifetime_{50}$}
-\label{Figures/ch4/R2/EC}
-\end{figure}
-
-We see from the results presented in both figures that the energy consumed by the network for each period increases when the number of primary points increases. Indeed, the decision for the optimization process requires more time, which leads to consuming more energy during the listening mode. The results show that Model-5 is the most competitive from the energy consumption point of view and the coverage ratio point of view. The other models have a high energy consumption due to the increase in the primary points. In fact, Model-5 is a good candidate to be used by wireless sensor network because it preserves a good coverage ratio with a suitable energy consumption in comparison with other models.
-
-%\item {{\bf Execution Time}}
-\subsubsection{Execution Time}
-
-In this experiment, we study the impact of the increase in primary points on the execution time of DiLCO protocol. Figure~\ref{Figures/ch4/R2/T} gives the average execution times in seconds for the decision phase (solving of the optimization problem) during one period. The original execution time is computed as described in section \ref{et}.
-
-\begin{figure}[h!]
-\centering
-\includegraphics[scale=0.5]{R2/T.pdf}
-\caption{Execution Time (in seconds)}
-\label{Figures/ch4/R2/T}
-\end{figure}
-
-They are given for the different primary point models and various numbers of sensors. We can see from Figure~\ref{Figures/ch4/R2/T}, that Model-5 has lower execution time in comparison with other models because it uses the smaller number of primary points to represent the area of the sensor. Conversely, the other primary point models have presented higher execution times.
-Moreover, Model-5 has more suitable execution times and coverage ratio that lead to continue for a larger number of period extending the network lifetime. We think that a good primary point model is one that balances between the coverage ratio and the number of periods during the lifetime of the network.
%\item {{\bf Network Lifetime}}
\subsubsection{Network Lifetime}
\label{Figures/ch4/R2/LT}
\end{figure}
-Comparison shows that Model-5, which uses less number of primary points, is the best one because it is less energy consuming during the network lifetime. It is also the better one from the point of view of coverage ratio. Our proposed Model-5 efficiently prolongs the network lifetime with a good coverage ratio in comparison with other models. Therefore, we have chosen Model-5 for all the experiments presented thereafter.
+Comparison shows that Model-5, which uses less number of primary points, is the best one because it is less energy consuming during the network lifetime. It is also the better one from the point of view of coverage ratio. Our proposed Model-5 efficiently prolongs the network lifetime with a good coverage ratio in comparison with other models. Therefore, we have chosen the model with five primary points for all the experiments presented thereafter.
%\end{enumerate}
-\subsection{Results and analysis}
+%\subsection{Results and analysis}
\subsubsection{Coverage ratio}
\begin{figure}[ht!]
\centering
- \includegraphics[scale=0.5] {R/CR.pdf}
+ \includegraphics[scale=0.5] {F/CR.pdf}
\caption{Average coverage ratio for 150 deployed nodes}
\label{fig3}
\end{figure}
\begin{figure}[ht!]
\centering
-\includegraphics[scale=0.5]{R/ASR.pdf}
+\includegraphics[scale=0.5]{F/ASR.pdf}
\caption{Active sensors ratio for 150 deployed nodes}
\label{fig4}
\end{figure}
\begin{figure}[ht!]
\centering
-\includegraphics[scale=0.5]{R/SR.pdf}
+\includegraphics[scale=0.5]{F/SR.pdf}
\caption{Cumulative percentage of stopped simulation runs for 150 deployed nodes }
\label{fig6}
\end{figure}
\begin{figure}[h!]
\centering
\begin{tabular}{cl}
- \parbox{9.5cm}{\includegraphics[scale=0.5]{R/EC95.pdf}} & (a) \\
+ \parbox{9.5cm}{\includegraphics[scale=0.5]{F/EC95.pdf}} & (a) \\
\verb+ + \\
- \parbox{9.5cm}{\includegraphics[scale=0.5]{R/EC50.pdf}} & (b)
+ \parbox{9.5cm}{\includegraphics[scale=0.5]{F/EC50.pdf}} & (b)
\end{tabular}
\caption{Energy consumption for (a) $Lifetime_{95}$ and
(b) $Lifetime_{50}$}
The results show that MuDiLCO is the most competitive from the energy
consumption point of view. The other approaches have a high energy consumption
-due to activating a larger number of redundant nodes as well as the energy consumed during the different status of the sensor node. Among the different versions of our protocol, the MuDiLCO-7 one consumes more energy than the other
-versions. This is easy to understand since the bigger the number of rounds and the number of sensors involved in the integer program are, the larger the time computation to solve the optimization problem is. To improve the performances of MuDiLCO-7, we should increase the number of subregions in order to have less sensors to consider in the integer program.
+due to activating a larger number of redundant nodes as well as the energy consumed during the different status of the sensor node.
+% Among the different versions of our protocol, the MuDiLCO-7 one consumes more energy than the other
+%versions. This is easy to understand since the bigger the number of rounds and the number of sensors involved in the integer program are, the larger the time computation to solve the optimization problem is. To improve the performances of MuDiLCO-7, we should increase the number of subregions in order to have less sensors to consider in the integer program.
%\textcolor{red}{As shown in Figure~\ref{fig7}, GA-MuDiLCO consumes less energy than both DESK and GAF, but a little bit higher than MuDiLCO because it provides a near optimal solution by activating a larger number of nodes during the sensing phase. GA-MuDiLCO consumes less energy in comparison with MuDiLCO-7 version, especially for the dense networks. However, MuDiLCO protocol and GA-MuDiLCO protocol are the most competitive from the energy
%consumption point of view. The other approaches have a high energy consumption
%due to activating a larger number of redundant nodes.}
\begin{figure}[ht!]
\centering
-\includegraphics[scale=0.5]{R/T.pdf}
+\includegraphics[scale=0.5]{F/T.pdf}
\caption{Execution Time (in seconds)}
\label{fig77}
\end{figure}
\begin{figure}[t!]
\centering
\begin{tabular}{cl}
- \parbox{9.5cm}{\includegraphics[scale=0.5]{R/LT95.pdf}} & (a) \\
+ \parbox{9.5cm}{\includegraphics[scale=0.5]{F/LT95.pdf}} & (a) \\
\verb+ + \\
- \parbox{9.5cm}{\includegraphics[scale=0.5]{R/LT50.pdf}} & (b)
+ \parbox{9.5cm}{\includegraphics[scale=0.5]{F/LT50.pdf}} & (b)
\end{tabular}
\caption{Network lifetime for (a) $Lifetime_{95}$ and
(b) $Lifetime_{50}$}
