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, Decision, and Sensing.
%The information exchange among wireless sensor nodes is described in chapter 4, section \ref{ch4:sec:02:03:01}. The leader election in each subregion is explained in chapter 4, section \ref{ch4:sec:02:03:02},
-The difference with MuDiLCO in that the elected leader in each subregion is for each period. In the decision phase, each leader will solve an integer program to select which cover sets will be activated in the following sensing phase to cover the subregion to which it belongs. The integer program will produce $T$ cover sets, one for each round. The leader will send an ActiveSleep packet to each sensor in the subregion based on the algorithm's results, indicating if the sensor should be active or not in
-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.
\begin{figure}[ht!]
\centering \includegraphics[width=160mm]{Figures/ch5/GeneralModel.jpg} % 70mm Modelgeneral.pdf
\caption{MuDiLCO protocol.}
\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.
-
-
-
\begin{algorithm}[h!]
% \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)}
\end{algorithm}
+The difference with MuDiLCO in that the elected leader in each subregion is for each period. In the decision phase, each leader will solve an integer program to select which cover sets will be activated in the following sensing phase to cover the subregion to which it belongs. The integer program will produce $T$ cover sets, one for each round. The leader will send an ActiveSleep packet to each sensor in the subregion based on the algorithm's results, indicating if the sensor should be active or not in
+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.
+
+
+%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.
+
+
+
\label{ch5:sec:03}
-According to Algorithm~\ref{alg:MuDiLCO}, the integer program is based on the model
-proposed by \cite{ref156} with some modifications, where the objective of our model is
-to find a maximum number of non-disjoint cover sets.
+%According to Algorithm~\ref{alg:MuDiLCO}, the integer program is based on the model proposed by \cite{ref156} with some modifications, where the objective of our model is to find a maximum number of non-disjoint cover sets.
%To fulfill this goal, the authors proposed an integer program which forces undercoverage and overcoverage of targets to become minimal at the same time. They use binary variables $x_{jl}$ to indicate if sensor $j$ belongs to cover set $l$. In our model,
-We consider binary variables $X_{t,j}$ to determine the possibility of activating
-sensor $j$ during round $t$ of a given sensing phase. We also consider primary
-points as targets. The set of primary points is denoted by $P$ and the set of
-sensors by $J$. Only sensors able to be alive during at least one round are
-involved in the integer program.
+%We consider binary variables $X_{t,j}$ to determine the possibility of activating sensor $j$ during round $t$ of a given sensing phase. We also consider primary points as targets. The set of primary points is denoted by $P$ and the set of sensors by $J$. Only sensors able to be alive during at least one round are involved in the integer program.
+We extend the mathematical formulation given in section \ref{ch4:sec:03} to take into account multiple rounds.
For a primary point $p$, let $\alpha_{j,p}$ denote the indicator function of
whether the point $p$ is covered, that is
covered).
\end{itemize}
-The first group of constraints indicates that some primary point $p$ should be
-covered by at least one sensor and, if it is not always the case, overcoverage
-and undercoverage variables help balancing the restriction equations by taking
-positive values. The constraint given by equation~(\ref{eq144}) guarantees that
-the sensor has enough energy ($RE_j$ corresponds to its remaining energy) to be
-alive during the selected rounds knowing that $E_{th}$ is the amount of energy
-required to be alive during one round.
+%The first group of constraints indicates that some primary point $p$ should be covered by at least one sensor and, if it is not always the case, overcoverage and undercoverage variables help balancing the restriction equations by taking positive values.
+The constraint given by equation~(\ref{eq144}) guarantees that the sensor has enough energy ($RE_j$ corresponds to its remaining energy) to be alive during the selected rounds knowing that $E_{th}$ is the amount of energy required to be alive during one round.
-There are two main objectives. First, we limit the overcoverage of primary
-points in order to activate a minimum number of sensors. Second we prevent the
-absence of monitoring on some parts of the subregion by minimizing the
-undercoverage. The weights $W_\theta$ and $W_U$ must be properly chosen so as
-to guarantee that the maximum number of points are covered during each round.
-%% MS W_theta is smaller than W_u => problem with the following sentence
-In our simulations, priority is given to the coverage by choosing $W_{U}$ very
-large compared to $W_{\theta}$.
+%There are two main objectives. First, we limit the overcoverage of primary points in order to activate a minimum number of sensors. Second we prevent the absence of monitoring on some parts of the subregion by minimizing the undercoverage. The weights $W_\theta$ and $W_U$ must be properly chosen so as to guarantee that the maximum number of points are covered during each round. In our simulations, priority is given to the coverage by choosing $W_{U}$ very large compared to $W_{\theta}$.
%\subsection{Active sensors ratio}
%\label{ch5:sec:03:02:02}
-It is crucial to have as few active nodes as possible in each round, in order to
-minimize the communication overhead and maximize the network
-lifetime. Figure~\ref{fig4} presents the active sensor ratio for 150 deployed
+%It is crucial to have as few active nodes as possible in each round, in order to minimize the communication overhead and maximize the network lifetime.
+Figure~\ref{fig4} presents the active sensor ratio for 150 deployed
nodes all along the network lifetime. It appears that up to round thirteen, DESK
and GAF have respectively 37.6\% and 44.8\% of nodes in active mode, whereas
MuDiLCO clearly outperforms them with only 23.7\% of active nodes. After the
%\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. \\ \\ \\ \\
+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. \\
\label{fig6}
\end{figure}
-
\item {{\bf Energy consumption}} \label{subsec:EC}
%\subsection{Energy consumption}
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}}
\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,
-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. \\
+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. \\ \\ \\
+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}
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.
+\end{enumerate}
+
\section{Conclusion}
\label{ch5:sec:05}
