-%Our work will concentrate on the area coverage by design
-%and implementation of a strategy, which efficiently selects the active
-%nodes that must maintain both sensing coverage and network
-%connectivity and at the same time improve the lifetime of the wireless
-%sensor network. But, requiring that all physical points of the
-%considered region are covered may be too strict, especially where the
-%sensor network is not dense. Our approach represents an area covered
-%by a sensor as a set of primary points and tries to maximize the total
-%number of primary points that are covered in each round, while
-%minimizing overcoverage (points covered by multiple active sensors
-%simultaneously).
-
-%In this section, we introduce a Multiperiod Distributed Lifetime Coverage Optimization protocol, which is called MuDiLCO. It is distributed on each subregion in the area of interest. It is based on two efficient techniques: network
-%leader election and sensor activity scheduling for coverage preservation and energy conservation continuously and efficiently to maximize the lifetime in the network.
-%The main features of our MuDiLCO protocol:
-%i)It divides the area of interest into subregions by using divide-and-conquer concept, ii)It requires only the information of the nodes within the subregion, iii) it divides the network lifetime into periods, which consists in round(s), iv)It based on the autonomous distributed decision by the nodes in the subregion to elect the Leader, v)It apply the activity scheduling based optimization on the subregion, vi) it achieves an energy consumption balancing among the nodes in the subregion by selecting different nodes as a leader during the network lifetime, vii) It uses the optimization to select the best representative non-disjoint sets of sensors in the subregion by optimize the coverage and the lifetime over the area of interest, viii)It uses our proposed primary point coverage model, which represent the sensing range of the sensor as a set of points, which are used by the our optimization algorithm, ix) It uses a simple energy model that takes communication, sensing and computation energy consumptions into account to evaluate the performance of our Protocol.
-
-\subsection{Assumptions}
-
-We consider a randomly and uniformly deployed network consisting of static
-wireless sensors. The sensors are deployed in high density to ensure initially
-a high coverage ratio of the interested area. We assume that all nodes are
-homogeneous in terms of communication and processing capabilities, and
-heterogeneous from the point of view of energy provision. Each sensor is
-supposed to get information on its location either through hardware such as
-embedded GPS or through location discovery algorithms.
-
-To model a sensor node's coverage area, we consider the boolean disk coverage
-model which is the most widely used sensor coverage model in the
-literature. Thus, each sensor has a constant sensing range $R_s$ and all space
-points within the disk centered at the sensor with the radius of the sensing
-range is said to be covered by this sensor. We also assume that the
-communication range satisfies $R_c \geq 2R_s$. In fact, Zhang and
-Zhou~\cite{Zhang05} proved that if the transmission range fulfills the previous
-hypothesis, a complete coverage of a convex area implies connectivity among the
-working nodes in the active mode.
-
-Instead of working with a continuous coverage area, we make it discrete by
-considering for each sensor a set of points called primary points. Consequently,
-we assume that the sensing disk defined by a sensor is covered if all of its
-primary points are covered. The choice of number and locations of primary points
-is the subject of another study not presented here.
-
-%By knowing the position (point center: ($p_x,p_y$)) of a wireless
-%sensor node and its $R_s$, we calculate the primary points directly
-%based on the proposed model. We use these primary points (that can be
-%increased or decreased if necessary) as references to ensure that the
-%monitored region of interest is covered by the selected set of
-%sensors, instead of using all the points in the area.
-
-%The MuDiLCO protocol works in periods and executed at each sensor node in the network, each sensor node can still sense data while being in
-%LISTENING mode. Thus, by entering the LISTENING mode at the beginning of each round,
-%sensor nodes still executing sensing task while participating in the leader election and decision phases. More specifically, The MuDiLCO protocol algorithm works as follow:
-%Initially, the sensor node check it's remaining energy in order to participate in the current round. Each sensor node determines it's position and it's subregion based Embedded GPS or Location Discovery Algorithm. After that, All the sensors collect position coordinates, current remaining energy, sensor node id, and the number of its one-hop live neighbors during the information exchange. It stores this information into a list $L$.
-%The sensor node enter in listening mode waiting to receive ActiveSleep packet from the leader after the decision to apply multi-round activity scheduling during the sensing phase. Each sensor node will execute the Algorithm~1 to know who is the leader. After that, if the sensor node is leader, It will execute the integer program algorithm ( see section~\ref{cp}) to optimize the coverage and the lifetime in it's subregion. After the decision, the optimization approach will produce the cover sets of sensor nodes to take the mission of coverage during the sensing phase for $T$ rounds. The leader will send ActiveSleep packet to each sensor node in the subregion to inform him to it's schedule for $T$ rounds during the period of sensing, either Active or sleep until the starting of next period. Based on the decision, the leader as other nodes in subregion, either go to be active or go to be sleep based on it's schedule for $T$ rounds during current sensing phase. the other nodes in the same subregion will stay in listening mode waiting the ActiveSleep packet from the leader. After finishing the time period for sensing, which are includes $T$ rounds, all the sensor nodes in the same subregion will start new period by executing the MuDiLCO protocol and the lifetime in the subregion will continue until all the sensor nodes are died or the network becomes disconnected in the subregion.
+\subsection{Assumptions and primary points}
+\label{pp}
+
+\textcolor{black}{The assumptions and the coverage model are identical to those presented
+ in~\cite{idrees2015distributed}. We consider a scenario in which sensors are deployed in high
+ density to initially ensure a high coverage ratio of the interested area. Each
+ sensor has a predefined sensing range $R_s$, an initial energy supply
+ (eventually different from each other) and is supposed to be equipped with
+ a module to locate its geographical positions. All space points within the
+ disk centered at the sensor with the radius of the sensing range are said to be
+ covered by this sensor.}
+
+\indent Instead of working with the coverage area, we consider for each sensor a
+set of points called primary points~\cite{idrees2014coverage}. We assume that
+the sensing disk defined by a sensor is covered if all the primary points of
+this sensor are covered. By knowing the position of wireless sensor node
+(centered at the the position $\left(p_x,p_y\right)$) and its sensing range
+$R_s$, we define up to 25 primary points $X_1$ to $X_{25}$ as described on
+Figure~\ref{fig1}. The optimal number of primary points is investigated in
+section~\ref{ch4:sec:04:06}.
+
+The coordinates of the primary points are defined as follows:\\
+%$(p_x,p_y)$ = point center of wireless sensor node\\
+$X_1=(p_x,p_y)$ \\
+$X_2=( p_x + R_s * (1), p_y + R_s * (0) )$\\
+$X_3=( p_x + R_s * (-1), p_y + R_s * (0)) $\\
+$X_4=( p_x + R_s * (0), p_y + R_s * (1) )$\\
+$X_5=( p_x + R_s * (0), p_y + R_s * (-1 )) $\\
+$X_6=( p_x + R_s * (\frac{-\sqrt{2}}{2}), p_y + R_s * (\frac{\sqrt{2}}{2})) $\\
+$X_7=( p_x + R_s * (\frac{\sqrt{2}}{2}), p_y + R_s * (\frac{\sqrt{2}}{2})) $\\
+$X_8=( p_x + R_s * (\frac{-\sqrt{2}}{2}), p_y + R_s * (\frac{-\sqrt{2}}{2})) $\\
+$X_9=( p_x + R_s * (\frac{\sqrt{2}}{2}), p_y + R_s * (\frac{-\sqrt{2}}{2})) $\\
+$X_{10}= ( p_x + R_s * (\frac{-\sqrt{2}}{2}), p_y + R_s * (0)) $\\
+$X_{11}=( p_x + R_s * (\frac{\sqrt{2}}{2}), p_y + R_s * (0))$\\
+$X_{12}=( p_x + R_s * (0), p_y + R_s * (\frac{\sqrt{2}}{2})) $\\
+$X_{13}=( p_x + R_s * (0), p_y + R_s * (\frac{-\sqrt{2}}{2})) $\\
+$X_{14}=( p_x + R_s * (\frac{\sqrt{3}}{2}), p_y + R_s * (\frac{1}{2})) $\\
+$X_{15}=( p_x + R_s * (\frac{-\sqrt{3}}{2}), p_y + R_s * (\frac{1}{2})) $\\
+$X_{16}=( p_x + R_s * (\frac{\sqrt{3}}{2}), p_y + R_s * (\frac{- 1}{2})) $\\
+$X_{17}=( p_x + R_s * (\frac{-\sqrt{3}}{2}), p_y + R_s * (\frac{- 1}{2})) $\\
+$X_{18}=( p_x + R_s * (\frac{\sqrt{3}}{2}), p_y + R_s * (0)) $\\
+$X_{19}=( p_x + R_s * (\frac{-\sqrt{3}}{2}), p_y + R_s * (0)) $\\
+$X_{20}=( p_x + R_s * (0), p_y + R_s * (\frac{1}{2})) $\\
+$X_{21}=( p_x + R_s * (0), p_y + R_s * (-\frac{1}{2})) $\\
+$X_{22}=( p_x + R_s * (\frac{1}{2}), p_y + R_s * (\frac{\sqrt{3}}{2})) $\\
+$X_{23}=( p_x + R_s * (\frac{- 1}{2}), p_y + R_s * (\frac{\sqrt{3}}{2})) $\\
+$X_{24}=( p_x + R_s * (\frac{- 1}{2}), p_y + R_s * (\frac{-\sqrt{3}}{2})) $\\
+$X_{25}=( p_x + R_s * (\frac{1}{2}), p_y + R_s * (\frac{-\sqrt{3}}{2})) $.
+
+\begin{figure}[h]
+ \centering
+ \includegraphics[scale=0.375]{fig26.pdf}
+ \label{fig1}
+ \caption{Wireless sensor node represented by up to 25~primary points}
+\end{figure}
