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. \textcolor{blue}{The flowchart of MuDiLCO protocol executed in each sensor node is presented in Figure \ref{flow5}.}
+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
%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.
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\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}
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.
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