& \mbox{is not covered during round $t$,}\\
\left( \sum_{j \in J} \alpha_{jp} * X_{tj} \right)- 1 & \mbox{otherwise.}\\
\end{array} \right.
-\label{eq13}
+\label{eq133}
\end{equation}
More precisely, $\Theta_{t,p}$ represents the number of active sensor nodes
minus one that cover the primary point $p$ during round $t$. The
1 &\mbox{if the primary point $p$ is not covered during round $t$,} \\
0 & \mbox{otherwise.}\\
\end{array} \right.
-\label{eq14}
+\label{eq1114}
\end{equation}
Our coverage optimization problem can then be formulated as follows
that the deployed nodes can cover the sensing field with the given sensing
range.
-Our 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). In the following, we will make comparisons with three other methods. DESK \cite{DESK}, GAF~\cite{GAF}, and DiLCO~\cite{Idrees2}, where MuDiLCO-1 is similar to DiLCO.
+Our 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). In the following, we will make comparisons with three other methods. DESK \cite{DESK}, GAF~\cite{GAF}, and DiLCO~\cite{Idrees2}, where MuDiLCO-1 is the same of DiLCO.
%Some preliminary experiments were performed in chapter 4 to study the choice of the number of subregions which subdivides the sensing field, considering different network sizes. They show that as the number of subregions increases, so does the network lifetime. Moreover, it makes the MuDiLCO protocol more robust against random network disconnection due to node failures. However, too many subdivisions reduce the advantage of the optimization. In fact, there is a balance between the benefit from the optimization and the execution time needed to solve it. Therefore,
We set the number of subregions to 16 rather than 32 as explained in section \ref{ch4:sec:04:05}.
We use the modeling language and the optimization solver which are mentioned in section \ref{ch4:sec:04:02}.
\end{frame}
\subsection{Results Analysis and Comparison }
-\label{ch5:sec:04:02}
+\label{ch5:sec:04:03}
\begin{enumerate}[i)]
%\subsection{Stopped simulation runs}
%\label{ch5:sec:03:02:03}
-Figure~\ref{fig6} reports the cumulative percentage of stopped simulations runs
-per round for 150 deployed nodes. This figure gives the breakpoint for each method. DESK stops first, after approximately 45~rounds, because it consumes the
-more energy by turning on a large number of redundant nodes during the sensing
-phase. GAF stops secondly for the same reason than DESK. MuDiLCO overcomes
-DESK and GAF because the optimization process distributed on several subregions
-leads to coverage preservation and so extends the network lifetime. Let us
-emphasize that the simulation continues as long as a network in a subregion is
-still connected.
+Figure~\ref{fig6} reports the cumulative percentage of stopped simulations runs per round for 150 deployed nodes. This figure gives the breakpoint for each method. \\ \\ \\ \\
+DESK stops first, after approximately 45~rounds, because it consumes the more energy by turning on a large number of redundant nodes during the sensing phase. GAF stops secondly for the same reason than DESK. MuDiLCO overcomes DESK and GAF because the optimization process distributed on several subregions leads to coverage preservation and so extends the network lifetime. Let us
+emphasize that the simulation continues as long as a network in a subregion is still connected. \\
-\begin{figure}[h!]
+\begin{figure}[t]
\centering
\includegraphics[scale=0.8]{Figures/ch5/R1/SR.pdf}
\caption{Cumulative percentage of stopped simulation runs for 150 deployed nodes }
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, the larger the time computation to solve the optimization problem. To improve the performances of MuDiLCO-7, we should increase the number of subregions in order to have fewer sensors to consider in the integer program.
-
+\\ \\ \\
\item {{\bf Execution time}}
we need to choose a relevant number of subregions in order to avoid a complicated and cumbersome optimization. On the one hand, a large value for $T$ permits to reduce the energy overhead due to the three pre-sensing phases, on the other hand a leader node may waste a considerable amount of energy to solve the optimization problem. \\
-
\item {{\bf Network lifetime}}
%\subsection{Network lifetime}
%\label{ch5:sec:03:02:06}
-
The next two figures, Figures~\ref{fig8}(a) and \ref{fig8}(b), illustrate the network lifetime for different network sizes, respectively for $Lifetime_{95}$ and $Lifetime_{50}$. Both figures show that the network lifetime increases together with the number of sensor nodes, whatever the protocol, thanks to the node density which results in more and more redundant nodes that can be deactivated and thus save energy. Compared to the other approaches, our MuDiLCO
-protocol maximizes the lifetime of the network. In particular, the gain in lifetime for a coverage over 95\% is greater than 38\% when switching from GAF to MuDiLCO-3. The slight decrease that can be observed for MuDiLCO-7 in case of $Lifetime_{95}$ with large wireless sensor networks results from the difficulty of the optimization problem to be solved by the integer program.
-This point was already noticed in \ref{subsec:EC} devoted to the
-energy consumption, since network lifetime and energy consumption are directly linked.
-
+protocol maximizes the lifetime of the network. In particular, the gain in lifetime for a coverage over 95\% is greater than 38\% when switching from GAF to MuDiLCO-3. \\ \\ \\
\begin{figure}[h!]
\centering
\end{figure}
-
-\end{enumerate}
-
+\end{enumerate}
+The slight decrease that can be observed for MuDiLCO-7 in case of $Lifetime_{95}$ with large wireless sensor networks results from the difficulty of the optimization problem to be solved by the integer program.
+This point was already noticed in \ref{subsec:EC} devoted to the
+energy consumption, since network lifetime and energy consumption are directly linked.
\section{Conclusion}
\label{ch5:sec:05}
-We have addressed the problem of the coverage and of the lifetime optimization in wireless sensor networks. This is a key issue as sensor nodes have limited resources in terms of memory, energy, and computational power. To cope with this problem, the field of sensing is divided into smaller subregions using the concept of divide-and-conquer method, and then we propose a protocol which optimizes coverage and lifetime performances in each subregion. Our protocol,
-called MuDiLCO (Multiround Distributed Lifetime Coverage Optimization) combines two efficient techniques: network leader election and sensor activity scheduling. The activity scheduling in each subregion works in periods, where each period consists of four phases: (i) Information Exchange, (ii) Leader Election, (iii) Decision Phase to plan the activity of the sensors over $T$ rounds, (iv) Sensing Phase itself divided into T rounds.
-
-Simulations results show the relevance of the proposed protocol in terms of lifetime, coverage ratio, active sensors ratio, energy consumption, execution time. Indeed, when dealing with large wireless sensor networks, a distributed approach, like the one we propose, allows to reduce the difficulty of a single global optimization problem by partitioning it into many smaller problems, one per subregion, that can be solved more easily. Nevertheless, results also show that it is not possible to plan the activity of sensors over too many rounds because the resulting optimization problem leads to too high-resolution times and thus to an excessive energy consumption.
-
+We have addressed the problem of the coverage and of the lifetime optimization in wireless sensor networks. This is a key issue as sensor nodes have limited resources in terms of memory, energy, and computational power. To cope with this problem, the field of sensing is divided into smaller subregions using the concept of divide-and-conquer method, and then we propose a protocol which optimizes coverage and lifetime performances in each subregion. Our protocol,
+called MuDiLCO (Multiround Distributed Lifetime Coverage Optimization) combines two efficient techniques: network leader election and sensor activity scheduling. The activity scheduling in each subregion works in periods, where each period consists of four phases: (i) Information exchange, (ii) Leader election, (iii) Decision phase to plan the activity of the sensors over $T$ rounds, (iv) Sensing phase itself divided into T rounds.
+Simulations results show the relevance of the proposed protocol in terms of lifetime, coverage ratio, active sensors ratio, energy consumption, execution time. Indeed, when dealing with large wireless sensor networks, a distributed approach, like the one we propose, allows to reduce the difficulty of a single global optimization problem by partitioning it into many smaller problems, one per subregion, that can be solved more easily. Nevertheless, results also show that it is not possible to plan the activity of sensors over too many rounds because the resulting optimization problem leads to too high-resolution times and thus to an excessive energy consumption. Compared with DiLCO, it is clear that MuDiLCO improves the network lifetime especially for the dense network, but it is less robust than DiLCO under sensor nodes failures. Therefore, choosing the number of rounds $T$ depends on the type of application the WSN is deployed for.
\ No newline at end of file
