low processing power from the sensor nodes, which usually have limited
processing capabilities. The main drawback of this kind of approach is its
higher cost in communications, since the node that will make the decision needs
-information from all the sensor nodes. \textcolor{blue} {Exact or heuristics
+information from all the sensor nodes. \textcolor{blue} {Exact or heuristic
approaches are designed to provide cover sets.
%(Moreover, centralized approaches usually
%suffer from the scalability problem, making them less competitive as the network
(centered at the the position $\left(p_x,p_y\right)$) and it's sensing range
$R_s$, we define up to 25 primary points $X_1$ to $X_{25}$ as decribed on
Figure~\ref{fig1}. The optimal number of primary points is investigated in
-subsection~\ref{ch4:sec:04:06}.
+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_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 * (0)) $\\
-$X_7=( p_x + R_s * (\frac{\sqrt{2}}{2}), p_y + R_s * (0))$\\
+$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 * (\frac{\sqrt{2}}{2})) $\\
-$X_{11}=( 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})) $\\
%smaller areas, called subregions, and then our MuDiLCO protocol will be
%implemented in each subregion in a distributed way.
-\textcolor{blue}{The WSN area of interest is, in a first step, divided into regular homogeneous
-subregions using a divide-and-conquer algorithm. In a second step our protocol
-will be executed in a distributed way in each subregion simultaneously to
-schedule nodes' activities for one sensing period. Sensor nodes are assumed to
-be deployed almost uniformly over the region. The regular subdivision is made
-such that the number of hops between any pairs of sensors inside a subregion is
-less than or equal to 3.}
-
-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
-\textcolor{blue} {of equal duration} 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=100mm]{Modelgeneral.pdf} % 70mm
+\textcolor{blue}{The WSN area of interest is, at first, divided into
+ regular homogeneous subregions using a divide-and-conquer algorithm. Then, our protocol will be executed in a distributed way in each
+ subregion simultaneously to schedule nodes' activities for one sensing
+ period. Sensor nodes are assumed to be deployed almost uniformly and with high
+ density over the region. The regular subdivision is made so that the number
+ of hops between any pairs of sensors inside a subregion is less than or equal
+ to 3.}
+
+As can be seen in Figure~\ref{fig2}, our protocol works in periods fashion,
+where each period is divided into 4~phases: Information~Exchange,
+Leader~Election, Decision, and Sensing. Each sensing phase may be itself
+divided into $T$ rounds \textcolor{blue} {of equal duration} 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}[t!]
+\centering \includegraphics[width=125mm]{Modelgeneral.pdf} % 70mm
\caption{The MuDiLCO protocol scheme executed on each node}
\label{fig2}
\end{figure}
% set cover responsible for the sensing task.
%For each round a set of sensors (said a cover set) is responsible for the sensing task.
-This protocol minimizes the impact of unexpected node failure (not due to batteries
-running out of energy), because it works in periods.
+This protocol minimizes the impact of unexpected node failure (not due to
+batteries running out of energy), because it works in periods.
%This protocol is reliable against an unexpected node failure, because it works in periods.
%%RC : why? I am not convinced
- 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. \textcolor{blue}{The duration of the rounds are predefined parameters. Round duration should be long enough to hide the system control overhead and short enough to minimize the negative effects in case of node failure.}
+ 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. \textcolor{blue}{The duration of the rounds is a predefined
+ parameter. Round duration should be long enough to hide the system control
+ overhead and short enough to minimize the negative effects in case of node
+ failures.}
%%RC so if there are at least one failure per period, the coverage is bad...
%%MS if we want to be reliable against many node failures we need to have an
\subsection{Leader Election phase}
-This step consists in choosing the Wireless Sensor Node Leader (WSNL), which
+This step consists in choosing the Wireless Sensor Node Leader (WSNL), which
will be responsible for executing the coverage algorithm. Each subregion in the
area of interest will select its own WSNL independently for each period. All
-the sensor nodes cooperate to elect a WSNL. The nodes in the same subregion
-will select the leader based on the received information from all other nodes
-in the same subregion. The selection criteria are, in order of importance:
-larger number of neighbors, larger remaining energy, and then in case of
-equality, larger index. Observations on previous simulations suggest to use the
-number of one-hop neighbors as the primary criterion to reduce energy
-consumption due to the communications.
+the sensor nodes cooperate to elect a WSNL. The nodes in the same subregion
+will select the leader based on the received information from all other nodes in
+the same subregion. The selection criteria are, in order of importance: larger
+number of neighbors, larger remaining energy, and then in case of equality,
+larger index. Observations on previous simulations suggest to use the number of
+one-hop neighbors as the primary criterion to reduce energy consumption due to
+the communications.
%the more priority selection factor is the number of $1-hop$ neighbors, $NBR j$, which can minimize the energy consumption during the communication Significantly.
%The pseudo-code for leader election phase is provided in Algorithm~1.
\subsection{Decision phase}
-Each WSNL will \textcolor{blue}{ 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. $T$ cover sets will be produced, 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. }
+Each WSNL will \textcolor{blue}{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. $T$ cover sets will be produced, 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.}
%Each WSNL will \textcolor{red}{ execute an optimization algorithm (see section \ref{oa})} to select which cover sets will be
%activated in the following sensing phase to cover the subregion to which it
%belongs. The \textcolor{red}{optimization algorithm} 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
%\section{\textcolor{red}{ Optimization Algorithm for Multiround Lifetime Coverage Optimization}}
%\label{oa}
-As shown in Algorithm~\ref{alg:MuDiLCO}, the leader will execute an optimization algorithm based on an integer program. The integer program is based on the model
-proposed by \cite{pedraza2006} 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
+As shown in Algorithm~\ref{alg:MuDiLCO}, the leader will execute an optimization
+algorithm based on an integer program. The integer program is based on the model
+proposed by \cite{pedraza2006} 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
+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.
Our coverage optimization problem can then be formulated as follows:
\begin{equation}
- \min \sum_{t=1}^{T} \sum_{p=1}^{P} \left(W_{\theta}* \Theta_{t,p} + W_{U} * U_{t,p} \right) \label{eq15}
+ \min \sum_{t=1}^{T} \sum_{p=1}^{|P|} \left(W_{\theta}* \Theta_{t,p} + W_{U} * U_{t,p} \right) \label{eq15}
\end{equation}
Subject to
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
+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 $E_{R}$ 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.
+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
+In our simulations, priority is given to the coverage by choosing $W_{U}$ very
large compared to $W_{\theta}$.
-\textcolor{blue}{The size of the problem depends on the number of variables and constraints. The number of variables is linked to the number of alive sensors $A \subset J$, the number of rounds $T$, and the number of primary points $P$. Thus the integer program contains $A*T$ variables of type $X_{t,j}$, $P*T$ overcoverage variables and $P*T$ undercoverage variables. The number of constraints is equal to $P*T$ (for constraints (\ref{eq16})) $+$ $A$ (for constraints (\ref{eq144})).}
-%The Active-Sleep packet includes the schedule vector with the number of rounds that should be applied by the receiving sensor node during the sensing phase.
-
+\textcolor{blue}{The size of the problem depends on the number of variables and
+ constraints. The number of variables is linked to the number of alive sensors
+ $A \subseteq J$, the number of rounds $T$, and the number of primary points
+ $P$. Thus the integer program contains $A*T$ variables of type $X_{t,j}$,
+ $P*T$ overcoverage variables and $P*T$ undercoverage variables. The number of
+ constraints is equal to $P*T$ (for constraints (\ref{eq16})) $+$ $A$ (for
+ constraints (\ref{eq144})).}
+%The Active-Sleep packet includes the schedule vector with the number of rounds that should be applied by the receiving sensor node during the sensing phase
\subsection{Sensing phase}
The sensing phase consists of $T$ rounds. Each sensor node in the subregion will
receive an Active-Sleep packet from WSNL, 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
-Active-Sleep packet is obtained.
+sleep for each round of the sensing phase. Algorithm~\ref{alg:MuDiLCO}, which
+will be executed by each sensor node~$s_j$ at the beginning of a period,
+explains how the Active-Sleep packet is obtained.
% In each round during the sensing phase, there is a cover set of sensor nodes, in which the active sensors will execute their sensing task to preserve maximal coverage and lifetime in the subregion and this will continue until finishing the round $T$ and starting new period.
\If{$ s_j.ID = LeaderID $}{
\emph{$s_j.status$ = COMPUTATION}\;
\emph{$\left\{\left(X_{1,k},\dots,X_{T,k}\right)\right\}_{k \in J}$ =
- Execute \textcolor{red}{Optimization Algorithm}($T,J$)}\;
+ Execute Integer Program Algorithm($T,J$)}\;
\emph{$s_j.status$ = COMMUNICATION}\;
- \emph{Send $ActiveSleep()$ to each node $k$ in subregion a packet \\
+ \emph{Send $ActiveSleep()$ packet to each node $k$ in subregion: a packet \\
with vector of activity scheduling $(X_{1,k},\dots,X_{T,k})$}\;
\emph{Update $RE_j $}\;
}
\fi
+%% EXPERIMENTAL STUDY
+
\section{Experimental study}
\label{exp}
\subsection{Simulation setup}
-We conducted a series of simulations to evaluate the efficiency and the
-relevance of our approach, using the discrete event simulator OMNeT++
-\cite{varga}. The simulation parameters are summarized in
-Table~\ref{table3}. Each experiment for a network is run over 25~different
-random topologies and the results presented hereafter are the average of these
-25 runs.
+We conducted a series of simulations to evaluate the efficiency and the
+relevance of our approach, using the discrete event simulator OMNeT++
+\cite{varga}. The simulation parameters are summarized in Table~\ref{table3}.
+Each experiment for a network is run over 25~different random topologies and the
+results presented hereafter are the average of these 25 runs.
%Based on the results of our proposed work in~\cite{idrees2014coverage}, we found as the region of interest are divided into larger subregions as the network lifetime increased. In this simulation, the network are divided into 16 subregions.
We performed simulations for five different densities varying from 50 to
-250~nodes deployed over a $50 \times 25~m^2 $ sensing field. More
-precisely, the deployment is controlled at a coarse scale in order to ensure
-that the deployed nodes can cover the sensing field with the given sensing
-range.
+250~nodes deployed over a $50 \times 25~m^2 $ sensing field. More precisely,
+the deployment is controlled at a coarse scale in order to ensure that the
+deployed nodes can cover the sensing field with the given sensing range.
%%RC these parameters are realistic?
%% maybe we can increase the field and sensing range. 5mfor Rs it seems very small... what do the other good papers consider ?
% is used to refer this table in the text
\end{table}
-\textcolor{blue}{The MuDilLCO protocol is declined into four versions: MuDiLCO-1, MuDiLCO-3, MuDiLCO-5,
-and MuDiLCO-7, corresponding respectively to $T=1,3,5,7$ ($T$ the number of rounds in one sensing period). Since the time resolution may be prohibitif when the size of the problem increases, a time limit treshold has been fixed to solve large instances. In these cases, the solver returns the best solution found, which is not necessary the optimal solution.
- Table \ref{tl} shows time limit values. These time limit treshold have been set empirically. The basic idea consists in considering the average execution time to solve the integer programs to optimality, then by dividing this average time by three to set the threshold value. After that, this treshold value is increased if necessary such that the solver is able to deliver a feasible solution within the time limit. In fact, selecting the optimal values for the time limits will be investigated in future. In Table \ref{tl}, "NO" indicates that the problem has been solved to optimality without time limit. }.
-
-\begin{table}[ht]
-\caption{Time limit values for MuDiLCO protocol versions }
-\centering
-\begin{tabular}{|c|c|c|c|c|}
- \hline
- WSN size & MuDiLCO-1 & MuDiLCO-3 & MuDiLCO-5 & MuDiLCO-7 \\ [0.5ex]
-\hline
- 50 & NO & NO & NO & NO \\
- \hline
-100 & NO & NO & NO & NO \\
-\hline
-150 & NO & NO & NO & 0.03 \\
-\hline
-200 & NO & NO & NO & 0.06 \\
- \hline
- 250 & NO & NO & NO & 0.08 \\
- \hline
-\end{tabular}
-
-\label{tl}
-
-\end{table}
-
+\textcolor{blue}{Our protocol is declined into four versions: MuDiLCO-1,
+ MuDiLCO-3, MuDiLCO-5, and MuDiLCO-7, corresponding respectively to $T=1,3,5,7$
+ ($T$ the number of rounds in one sensing period). Since the time resolution
+ may be prohibitive when the size of the problem increases, a time limit
+ threshold has been fixed when solving large instances. In these cases, the
+ solver returns the best solution found, which is not necessary the optimal
+ one. In practice, we only set time limit values for the three largest network
+ sizes when $T=7$, using the following respective values (in second): 0.03 for
+ 150~nodes, 0.06 for 200~nodes, and 0.08 for 250~nodes.
+% Table \ref{tl} shows time limit values.
+ These time limit thresholds have been set empirically. The basic idea consists
+ in considering the average execution time to solve the integer programs to
+ optimality, then in dividing this average time by three to set the threshold
+ value. After that, this threshold value is increased if necessary so that
+ the solver is able to deliver a feasible solution within the time limit. In
+ fact, selecting the optimal values for the time limits will be investigated in
+ the future.}
+%In Table \ref{tl}, "NO" indicates that the problem has been solved to optimality without time limit.}
+
+%\begin{table}[ht]
+%\caption{Time limit values for MuDiLCO protocol versions }
+%\centering
+%\begin{tabular}{|c|c|c|c|c|}
+% \hline
+% WSN size & MuDiLCO-1 & MuDiLCO-3 & MuDiLCO-5 & MuDiLCO-7 \\ [0.5ex]
+%\hline
+% 50 & NO & NO & NO & NO \\
+% \hline
+%100 & NO & NO & NO & NO \\
+%\hline
+%150 & NO & NO & NO & 0.03 \\
+%\hline
+%200 & NO & NO & NO & 0.06 \\
+% \hline
+% 250 & NO & NO & NO & 0.08 \\
+% \hline
+%\end{tabular}
+%\label{tl}
+%\end{table}
- In the following, we will make comparisons with
-two other methods. The first method, called DESK and proposed by \cite{ChinhVu},
-is a full distributed coverage algorithm. The second method, called
-GAF~\cite{xu2001geography}, consists in dividing the region into fixed squares.
-During the decision phase, in each square, one sensor is then chosen to remain
-active during the sensing phase time.
+ In the following, we will make comparisons with two other methods. The first
+ method, called DESK and proposed by \cite{ChinhVu}, is a full distributed
+ coverage algorithm. The second method, called GAF~\cite{xu2001geography},
+ consists in dividing the region into fixed squares. During the decision phase,
+ in each square, one sensor is then chosen to remain active during the sensing
+ phase time.
Some preliminary experiments were performed to study the choice of the number of
subregions which subdivides the sensing field, considering different network
% is used to refer this table in the text
\end{table}
-For the sake of simplicity we ignore the energy needed to turn on the radio, to
+For the sake of simplicity we ignore the energy needed to turn on the radio, to
start up the sensor node, to move from one status to another, etc.
%We also do not consider the need of collecting sensing data. PAS COMPRIS
-Thus, when a sensor becomes active (i.e., it has already chosen its status), it can
-turn its radio off to save battery. MuDiLCO uses two types of packets for
-communication. The size of the INFO packet and Active-Sleep packet are 112~bits
-and 24~bits respectively. The value of energy spent to send a 1-bit-content
+Thus, when a sensor becomes active (i.e., it has already chosen its status), it
+can turn its radio off to save battery. MuDiLCO uses two types of packets for
+communication. The size of the INFO packet and Active-Sleep packet are 112~bits
+and 24~bits respectively. The value of energy spent to send a 1-bit-content
message is obtained by using the equation in ~\cite{raghunathan2002energy} to
-calculate the energy cost for transmitting messages and we propose the same
-value for receiving the packets. The energy needed to send or receive a 1-bit
+calculate the energy cost for transmitting messages and we propose the same
+value for receiving the packets. The energy needed to send or receive a 1-bit
packet is equal to 0.2575~mW.
-The initial energy of each node is randomly set in the interval $[500;700]$. A
-sensor node will not participate in the next round if its remaining energy is
+The initial energy of each node is randomly set in the interval $[500;700]$. A
+sensor node will not participate in the next round if its remaining energy is
less than $E_{R}=36~\mbox{Joules}$, the minimum energy needed for the node to
-stay alive during one round. This value has been computed by multiplying the
+stay alive during one round. This value has been computed by multiplying the
energy consumed in active state (9.72 mW) by the time in second for one round
-(3600 seconds). According to the interval of initial energy, a sensor may be
+(3600 seconds). According to the interval of initial energy, a sensor may be
alive during at most 20 rounds.
\subsection{Metrics}
% Old version -> where $M_L$ and $T_L$ are respectively the number of periods and rounds during
%$Lifetime_{95}$ or $Lifetime_{50}$.
% New version
-where $M$ is the number of periods and $T_m$ the number of rounds in a
+where $M$ is the number of periods and $T_m$ the number of rounds in a
period~$m$, both during $Lifetime_{95}$ or $Lifetime_{50}$. The total energy
-consumed by the sensors (EC) comes through taking into consideration four main
+consumed by the sensors (EC) comes through taking into consideration four main
energy factors. The first one , denoted $E^{\scriptsize \mbox{com}}_m$,
-represents the energy consumption spent by all the nodes for wireless
-communications during period $m$. $E^{\scriptsize \mbox{list}}_m$, the next
-factor, corresponds to the energy consumed by the sensors in LISTENING status
-before receiving the decision to go active or sleep in period $m$.
+represents the energy consumption spent by all the nodes for wireless
+communications during period $m$. $E^{\scriptsize \mbox{list}}_m$, the next
+factor, corresponds to the energy consumed by the sensors in LISTENING status
+before receiving the decision to go active or sleep in period $m$.
$E^{\scriptsize \mbox{comp}}_m$ refers to the energy needed by all the leader
nodes to solve the integer program during a period. Finally, $E^a_t$ and $E^s_t$
indicate the energy consumed by the whole network in round $t$.
\subsection{Performance analysis for different number of primary points}
\label{ch4:sec:04:06}
-In this section, we study the performance of MuDiLCO-1 approach for different numbers of primary points. The objective of this comparison is to select the suitable primary point model to be used by a MuDiLCO protocol. In this comparison, MuDiLCO-1 protocol is used with five models, which are called Model-5 (it uses 5 primary points), Model-9, Model-13, Model-17, and Model-21.
-
+In this section, we study the performance of MuDiLCO-1 approach for different
+numbers of primary points. The objective of this comparison is to select the
+suitable number of primary points to be used by a MuDiLCO protocol. In this
+comparison, MuDiLCO-1 protocol is used with five primary point models, each
+model corresponding to a number of primary points, which are called Model-5 (it
+uses 5 primary points), Model-9, Model-13, Model-17, and Model-21.
%\begin{enumerate}[i)]
%\item {{\bf Coverage Ratio}}
\subsubsection{Coverage ratio}
-Figure~\ref{Figures/ch4/R2/CR} shows the average coverage ratio for 150 deployed nodes.
-\parskip 0pt
-\begin{figure}[h!]
+Figure~\ref{Figures/ch4/R2/CR} shows the average coverage ratio for 150 deployed
+nodes. As can be seen, at the beginning the models which use a larger number of
+primary points provide slightly better coverage ratios, but latter they are the
+worst.
+%Moreover, when the number of periods increases, coverage ratio produced by Model-9, Model-13, Model-17, and Model-21 decreases in comparison with Model-5 due to a larger time computation for the decision process for larger number of primary points.
+Moreover, when the number of periods increases, the coverage ratio produced by
+all models decrease due to dead nodes. However, Model-5 is the one with the
+slowest decrease due to lower numbers of active sensors in the earlier periods.
+% smaller time computation of decision process for a smaller number of primary points.
+Overall this model is slightly more efficient than the other ones, because it
+offers a good coverage ratio for a larger number of periods.
+%\parskip 0pt
+\begin{figure}[t!]
\centering
\includegraphics[scale=0.5] {R2/CR.pdf}
\caption{Coverage ratio for 150 deployed nodes}
\label{Figures/ch4/R2/CR}
\end{figure}
-As can be seen in Figure~\ref{Figures/ch4/R2/CR}, at the beginning the models which use a larger number of primary points provide slightly better coverage ratios, but latter they are the worst.
-%Moreover, when the number of periods increases, coverage ratio produced by Model-9, Model-13, Model-17, and Model-21 decreases in comparison with Model-5 due to a larger time computation for the decision process for larger number of primary points.
-Moreover, when the number of periods increases, coverage ratio produced by all models decrease, but Model-5 is the one with the slowest decrease due to a smaller time computation of decision process for a smaller number of primary points.
-As shown in Figure ~\ref{Figures/ch4/R2/CR}, coverage ratio decreases when the number of periods increases due to dead nodes. Model-5 is slightly more efficient than other models, because it offers a good coverage ratio for a larger number of periods in comparison with other models.
%\item {{\bf Network Lifetime}}
\subsubsection{Network lifetime}
-Finally, we study the effect of increasing the primary points on the lifetime of the network.
+Finally, we study the effect of increasing the number of primary points on the lifetime of the network.
%In Figure~\ref{Figures/ch4/R2/LT95} and in Figure~\ref{Figures/ch4/R2/LT50}, network lifetime, $Lifetime95$ and $Lifetime50$ respectively, are illustrated for different network sizes.
-As highlighted by Figures~\ref{Figures/ch4/R2/LT}(a) and \ref{Figures/ch4/R2/LT}(b), the network lifetime obviously increases when the size of the network increases, with Model-5 that leads to the larger lifetime improvement.
+As highlighted by Figures~\ref{Figures/ch4/R2/LT}(a) and
+\ref{Figures/ch4/R2/LT}(b), the network lifetime obviously increases when the
+size of the network increases, with Model-5 which leads to the largest lifetime
+improvement.
\begin{figure}[h!]
\centering
\label{Figures/ch4/R2/LT}
\end{figure}
-Comparison shows that Model-5, which uses less number of primary points, is the best one because it is less energy consuming during the network lifetime. It is also the better one from the point of view of coverage ratio. Our proposed Model-5 efficiently prolongs the network lifetime with a good coverage ratio in comparison with other models. Therefore, we have chosen the model with five primary points for all the experiments presented thereafter.
+Comparison shows that Model-5, which uses less number of primary points, is the
+best one because it is less energy consuming during the network lifetime. It is
+also the better one from the point of view of coverage ratio, as stated
+before. Therefore, we have chosen the model with five primary points for all the
+experiments presented thereafter.
%\end{enumerate}
-\subsection{Results and analysis}
+% MICHEL => TO BE CONTINUED
+
+\subsection{Experimental results and analysis}
\subsubsection{Coverage ratio}
