-\noindent In this section, we describe in details our Lifetime Coverage
-Optimization protocol. First we present the assumptions we made and the models
-we considered (in particular the perimeter coverage one), second we describe the
-background idea of our protocol, and third we give the outline of the algorithm
-executed by each node.
-
-
-
-\subsection{Assumptions and Models}
-\label{ch5:sec:02:01}
-PeCO protocol uses the same assumptions and network model that presented in chapter 3, section \ref{ch3:sec:02:01}.
-
-The PeCO protocol uses the same perimeter-coverage model as Huang and
-Tseng in~\cite{ref133}. It can be expressed as follows: a sensor is
-said to be perimeter covered if all the points on its perimeter are covered by
-at least one sensor other than itself. They proved that a network area is
-$k$-covered if and only if each sensor in the network is $k$-perimeter-covered (perimeter covered by at least $k$ sensors).
-
-Figure~\ref{pcm2sensors}(a) shows the coverage of sensor node~$0$. On this
-figure, we can see that sensor~$0$ has nine neighbors and we have reported on
-its perimeter (the perimeter of the disk covered by the sensor) for each
-neighbor the two points resulting from intersection of the two sensing
-areas. These points are denoted for neighbor~$i$ by $iL$ and $iR$, respectively
-for left and right from neighbor point of view. The resulting couples of
-intersection points subdivide the perimeter of sensor~$0$ into portions called
-arcs.
-
-\begin{figure}[ht!]
- \centering
- \begin{tabular}{@{}cr@{}}
- \includegraphics[width=95mm]{Figures/ch5/pcm.jpg} & \raisebox{3.25cm}{(a)} \\
- \includegraphics[width=95mm]{Figures/ch5/twosensors.jpg} & \raisebox{2.75cm}{(b)}
- \end{tabular}
- \caption{(a) Perimeter coverage of sensor node 0 and (b) finding the arc of
- $u$'s perimeter covered by $v$.}
- \label{pcm2sensors}
-\end{figure}
-
-Figure~\ref{pcm2sensors}(b) describes the geometric information used to find the
-locations of the left and right points of an arc on the perimeter of a sensor
-node~$u$ covered by a sensor node~$v$. Node~$v$ is supposed to be located on the
-west side of sensor~$u$, with the following respective coordinates in the
-sensing area~: $(v_x,v_y)$ and $(u_x,u_y)$. From the previous coordinates we can
-compute the euclidean distance between nodes~$u$ and $v$: $Dist(u,v)=\sqrt{\vert
- u_x - v_x \vert^2 + \vert u_y-v_y \vert^2}$, while the angle~$\alpha$ is
-obtained through the formula: $$\alpha = \arccos \left(\dfrac{Dist(u,v)}{2R_s}
-\right).$$ The arc on the perimeter of~$u$ defined by the angular interval $[\pi
- - \alpha,\pi + \alpha]$ is said to be perimeter-covered by sensor~$v$.
-
-Every couple of intersection points is placed on the angular interval $[0,2\pi]$
-in a counterclockwise manner, leading to a partitioning of the interval.
-Figure~\ref{pcm2sensors}(a) illustrates the arcs for the nine neighbors of
-sensor $0$ and Figure~\ref{expcm} gives the position of the corresponding arcs
-in the interval $[0,2\pi]$. More precisely, we can see that the points are
-ordered according to the measures of the angles defined by their respective
-positions. The intersection points are then visited one after another, starting
-from the first intersection point after point~zero, and the maximum level of
-coverage is determined for each interval defined by two successive points. The
-maximum level of coverage is equal to the number of overlapping arcs. For
-example,
-between~$5L$ and~$6L$ the maximum level of coverage is equal to $3$
-(the value is highlighted in yellow at the bottom of Figure~\ref{expcm}), which
-means that at most 2~neighbors can cover the perimeter in addition to node $0$.
-Table~\ref{my-label} summarizes for each coverage interval the maximum level of
-coverage and the sensor nodes covering the perimeter. The example discussed
-above is thus given by the sixth line of the table.
-
-
-\begin{figure*}[t!]
-\centering
-\includegraphics[width=150.5mm]{Figures/ch5/expcm2.jpg}
-\caption{Maximum coverage levels for perimeter of sensor node $0$.}
-\label{expcm}
-\end{figure*}
-
-
- \begin{table}[h!]
- \caption{Coverage intervals and contributing sensors for sensor node 0.}
- \centering
-\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
-\hline
-\begin{tabular}[c]{@{}c@{}}Left \\ point \\ angle~$\alpha$ \end{tabular} & \begin{tabular}[c]{@{}c@{}}Interval \\ left \\ point\end{tabular} & \begin{tabular}[c]{@{}c@{}}Interval \\ right \\ point\end{tabular} & \begin{tabular}[c]{@{}c@{}}Maximum \\ coverage\\ level\end{tabular} & \multicolumn{5}{c|}{\begin{tabular}[c]{@{}c@{}}Set of sensors\\ involved \\ in coverage interval\end{tabular}} \\ \hline
-0.0291 & 1L & 2L & 4 & 0 & 1 & 3 & 4 & \\ \hline
-0.104 & 2L & 3R & 5 & 0 & 1 & 3 & 4 & 2 \\ \hline
-0.3168 & 3R & 4R & 4 & 0 & 1 & 4 & 2 & \\ \hline
-0.6752 & 4R & 1R & 3 & 0 & 1 & 2 & & \\ \hline
-1.8127 & 1R & 5L & 2 & 0 & 2 & & & \\ \hline
-1.9228 & 5L & 6L & 3 & 0 & 2 & 5 & & \\ \hline
-2.3959 & 6L & 2R & 4 & 0 & 2 & 5 & 6 & \\ \hline
-2.4258 & 2R & 7L & 3 & 0 & 5 & 6 & & \\ \hline
-2.7868 & 7L & 8L & 4 & 0 & 5 & 6 & 7 & \\ \hline
-2.8358 & 8L & 5R & 5 & 0 & 5 & 6 & 7 & 8 \\ \hline
-2.9184 & 5R & 7R & 4 & 0 & 6 & 7 & 8 & \\ \hline
-3.3301 & 7R & 9R & 3 & 0 & 6 & 8 & & \\ \hline
-3.9464 & 9R & 6R & 4 & 0 & 6 & 8 & 9 & \\ \hline
-4.767 & 6R & 3L & 3 & 0 & 8 & 9 & & \\ \hline
-4.8425 & 3L & 8R & 4 & 0 & 3 & 8 & 9 & \\ \hline
-4.9072 & 8R & 4L & 3 & 0 & 3 & 9 & & \\ \hline
-5.3804 & 4L & 9R & 4 & 0 & 3 & 4 & 9 & \\ \hline
-5.9157 & 9R & 1L & 3 & 0 & 3 & 4 & & \\ \hline
-\end{tabular}
-
-\label{my-label}
-\end{table}
-
-
-In the PeCO protocol, the scheduling of the sensor nodes' activities is formulated with an
-integer program based on coverage intervals. The formulation of the coverage
-optimization problem is detailed in~section~\ref{ch5:sec:03}. Note that when a sensor
-node has a part of its sensing range outside the WSN sensing field, as in
-Figure~\ref{ex4pcm}, the maximum coverage level for this arc is set to $\infty$
-and the corresponding interval will not be taken into account by the
-optimization algorithm.
-
-
-\begin{figure}[h!]
-\centering
-\includegraphics[width=95.5mm]{Figures/ch5/ex4pcm.jpg}
-\caption{Sensing range outside the WSN's area of interest.}
-\label{ex4pcm}
-\end{figure}
-
-
-
-
-
-\subsection{The Main Idea}
