-
-\subsection{Background Idea}
-\label{ch4:sec:02:02}
-The area of interest can be divided using the divide-and-conquer strategy into
-smaller areas, called subregions, and then our MuDiLCO protocol will be
-implemented in each subregion in a distributed way.
-
-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. Each sensing phase may be itself divided into $T$ rounds
-and for each round a set of sensors (a cover set) is responsible for the sensing
-task. 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/ch4/GeneralModel.jpg} % 70mm Modelgeneral.pdf
-\caption{The MuDiLCO protocol scheme executed on each node}
-\label{fig2}
-\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 to this phase, and, 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.
-
-
-These phases can be described in more details as follow:
-\subsection{Information Exchange Phase}
-\label{ch4:sec:02:02:01}
-The information exchange among the wireless sensor nodes is similar to that one which is described in chapter 3, sections \ref{ch3:sec:02:03:01}.
-
-\subsection{Leader Election phase}
-\label{ch4:sec:02:02:02}
-The leader election in each subregion is similar to that one which is described in chapter 3, sections\ref{ch3:sec:02:03:02}, but the difference in that the elected leader in each subregion is for each period.
-
-\subsection{Decision phase}
-\label{ch4:sec:02:02:03}
-Each WSNL 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 WSNL will send an Active-Sleep 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. The integer program is based on the model
-proposed by \cite{ref156} with some modifications, where the objective is
-to find a maximum number of 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.
-
-
-For a primary point $p$, let $\alpha_{j,p}$ denote the indicator function of
-whether the point $p$ is covered, that is:
-\begin{equation}
-\alpha_{j,p} = \left \{
-\begin{array}{l l}
- 1 & \mbox{if the primary point $p$ is covered} \\
- & \mbox{by sensor node $j$}, \\
- 0 & \mbox{otherwise.}\\
-\end{array} \right.
-%\label{eq12}
-\end{equation}
-The number of active sensors that cover the primary point $p$ during
-round $t$ is equal to $\sum_{j \in J} \alpha_{j,p} * X_{t,j}$ where:
-\begin{equation}
-X_{t,j} = \left \{
-\begin{array}{l l}
- 1& \mbox{if sensor $j$ is active during round $t$,} \\
- 0 & \mbox{otherwise.}\\
-\end{array} \right.
-%\label{eq11}
-\end{equation}
-We define the Overcoverage variable $\Theta_{t,p}$ as:
-\begin{equation}
- \Theta_{t,p} = \left \{
-\begin{array}{l l}
- 0 & \mbox{if the primary point $p$}\\
- & \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}
-\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
-Undercoverage variable $U_{t,p}$ of the primary point $p$ during round $t$ is
-defined by:
-\begin{equation}
-U_{t,p} = \left \{
-\begin{array}{l l}
- 1 &\mbox{if the primary point $p$ is not covered during round $t$,} \\
- 0 & \mbox{otherwise.}\\
-\end{array} \right.
-\label{eq14}
-\end{equation}
