each round of the sensing phase. Each sensing phase is itself divided into $T$ rounds and for each round a set of sensors (a cover set) is responsible for the sensing task.
%Each sensor node in the subregion will receive an ActiveSleep packet from leader, informing it to stay awake or to go to sleep for each round of the sensing phase.
Algorithm~\ref{alg:MuDiLCO}, which will be executed by each node at the beginning of a period, explains how the ActiveSleep packet is obtained. In this way, a multiround optimization process is performed during each
-period after Information~Exchange and Leader~Election phases, in order to produce $T$ cover sets that will take the mission of sensing for $T$ rounds.
+period after Information~Exchange and Leader~Election phases, in order to produce $T$ cover sets that will take the mission of sensing for $T$ rounds. The flowchart of MuDiLCO protocol executed in each sensor node is presented in Figure \ref{flow5}.
+
+\begin{figure}[ht!]
+\centering
+\includegraphics[scale=0.50]{Figures/ch5/Algo2.png} % 70mm
+\caption{The flowchart of MuDiLCO protocol.}
+\label{flow5}
+\end{figure}
%This protocol minimizes the impact of unexpected node failure (not due to batteries running out of energy), because it works in periods. On the one hand, if a node failure is detected before making the decision, the node will not participate during this phase. On the other hand, if the node failure occurs after the decision, the sensing task of the network will be temporarily affected: only during the period of sensing until a new period starts.
%The energy consumption and some other constraints can easily be taken into account since the sensors can update and then exchange their information (including their residual energy) at the beginning of each period. However, the pre-sensing phases (Information Exchange, Leader Election, and Decision) are energy consuming for some nodes, even when they do not join the network to monitor the area.
-
-
-
-
-
\section{Primary Points based Multiround Coverage Problem Formulation}
\label{ch5:sec:03}
We extend the mathematical formulation given in section \ref{ch4:sec:03} to take into account multiple rounds.
+\newpage
+
For a primary point $p$, let $\alpha_{j,p}$ denote the indicator function of
whether the point $p$ is covered, that is
\begin{equation}
%\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. \\
+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}[t]
We observe the impact of the network size and of the number of rounds on the
computation time. Figure~\ref{fig77} gives the average execution times in
-seconds (needed to solve optimization problem) for different values of $T$. The original execution time is computed as described in chapter 4, section \ref{ch4:sec:04:02}.
+seconds (needed to solve optimization problem) for different values of $T$. \\\\\\
%The original execution time is computed on a laptop DELL with Intel Core~i3~2370~M (2.4 GHz) processor (2 cores) and the MIPS (Million Instructions Per Second) rate equal to 35330. To be consistent with the use of a sensor node with Atmels AVR ATmega103L microcontroller (6 MHz) and a MIPS rate equal to 6 to run the optimization resolution, this time is multiplied by 2944.2 $\left( \frac{35330}{2} \times \frac{1}{6} \right)$ and reported on Figure~\ref{fig77} for different network sizes.
\label{fig77}
\end{figure}
-As expected, the execution time increases with the number of rounds $T$ taken into account to schedule the sensing phase. The times obtained for $T=1,3$ or $5$ seem bearable, but for $T=7$ they become quickly unsuitable for a sensor node, especially when the sensor network size increases. Again, we can notice that if we want to schedule the nodes activities for a large number of rounds,
+The original execution time is computed as described in chapter 4, section \ref{ch4:sec:04:02}. As expected, the execution time increases with the number of rounds $T$ taken into account to schedule the sensing phase. The times obtained for $T=1,3$ or $5$ seem bearable, but for $T=7$ they become quickly unsuitable for a sensor node, especially when the sensor network size increases. Again, we can notice that if we want to schedule the nodes activities for a large number of rounds,
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. %\\ \\ \\ \\ \\ \\ \\
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
+\\\\\\\\This point was already noticed in \ref{subsec:EC} devoted to the
energy consumption, since network lifetime and energy consumption are directly linked.
\end{enumerate}
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
+
