\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.
+%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.
+%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
