\indent Instead of working with the coverage area, we consider for each sensor a set of points called primary points. We also 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 (point center: ($p_x,p_y$)) of a wireless sensor node and it's $R_s$, 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.
We can calculate the positions of the selected primary
points in the circle disk of the sensing range of a wireless sensor
-node (see figure~\ref{fig1}) as follows:\\
+node (see Figure~\ref{fig1}) as follows:\\
$(p_x,p_y)$ = point center of wireless sensor node\\
$X_1=(p_x,p_y)$ \\
\end{figure}
It can be seen that DiLCO protocol (with 4, 8, 16 and 32 subregions) gives nearly similar coverage ratios during the first thirty rounds.
DiLCO-2 protocol gives near similar coverage ratio with other ones for first 10 rounds and then decreased until the died of the network in the round $18^{th}$. In case of only 2 subregions, the energy consumption is high and the network is rapidly disconnected.
-As shown in the figure ~\ref{Figures/ch4/R1/CR}, as the number of subregions increases, the coverage preservation for the area of interest increases for a larger number of rounds. Coverage ratio decreases when the number of rounds increases due to dead nodes. Although some nodes are dead, thanks to DiLCO-8, DiLCO-16, and DiLCO-32 protocols, other nodes are preserved to ensure the coverage. Moreover, when we have a dense sensor network, it leads to maintain the coverage for a larger number of rounds. DiLCO-8, DiLCO-16, and DiLCO-32 protocols are slightly more efficient than other protocols, because they subdivide the area of interest into 8, 16 and 32~subregions; if one of the subregions becomes disconnected, the coverage may be still ensured in the remaining subregions.
+As shown in the Figure ~\ref{Figures/ch4/R1/CR}, as the number of subregions increases, the coverage preservation for the area of interest increases for a larger number of rounds. Coverage ratio decreases when the number of rounds increases due to dead nodes. Although some nodes are dead, thanks to DiLCO-8, DiLCO-16, and DiLCO-32 protocols, other nodes are preserved to ensure the coverage. Moreover, when we have a dense sensor network, it leads to maintain the coverage for a larger number of rounds. DiLCO-8, DiLCO-16, and DiLCO-32 protocols are slightly more efficient than other protocols, because they subdivide the area of interest into 8, 16 and 32~subregions; if one of the subregions becomes disconnected, the coverage may be still ensured in the remaining subregions.
\item {{\bf Active Sensors Ratio}}
%\subsubsection{Active Sensors Ratio}
\label{Figures/ch4/R1/ASR}
\end{figure}
-The results presented in figure~\ref{Figures/ch4/R1/ASR} show the increase of the number of subregions lead to the increase of the number of active nodes. The DiLCO-16 and DiLCO-32 protocols have a larger number of active nodes, but it preserve the coverage for a larger number of rounds. The advantage of the DiLCO-16 and DiLCO-32 protocols are that even if a network is disconnected in one subregion, the other ones usually continues the optimization process, and this extends the lifetime of the network.
+The results presented in Figure~\ref{Figures/ch4/R1/ASR} show the increase of the number of subregions lead to the increase of the number of active nodes. The DiLCO-16 and DiLCO-32 protocols have a larger number of active nodes, but it preserve the coverage for a larger number of rounds. The advantage of the DiLCO-16 and DiLCO-32 protocols are that even if a network is disconnected in one subregion, the other ones usually continues the optimization process, and this extends the lifetime of the network.
\item {{\bf The percentage of stopped simulation runs}}
%\subsubsection{The percentage of stopped simulation runs}
\label{Figures/ch4/R1/T}
\end{figure}
-We can see from figure~\ref{Figures/ch4/R1/T}, that the DiLCO-32 has very low execution times in comparison with other DiLCO versions because it is distributed on larger number of small subregions. Conversely, DiLCO-2 requires to solve an optimization problem considering half the nodes in each subregion presents high execution times.
+We can see from Figure~\ref{Figures/ch4/R1/T}, that the DiLCO-32 has very low execution times in comparison with other DiLCO versions because it is distributed on larger number of small subregions. Conversely, DiLCO-2 requires to solve an optimization problem considering half the nodes in each subregion presents high execution times.
We think that in distributed fashion the solving of the optimization problem in a subregion can be tackled by sensor nodes. Overall, to be able to deal with very large networks, a distributed method is clearly required.
\item {{\bf The Network Lifetime}}
%\subsubsection{The Network Lifetime}
-In figure~\ref{Figures/ch4/R1/LT95} and \ref{Figures/ch4/R1/LT50}, network lifetime, $Lifetime95$ and $Lifetime50$ respectively, are illustrated for different network sizes.
+In Figure~\ref{Figures/ch4/R1/LT95} and \ref{Figures/ch4/R1/LT50}, network lifetime, $Lifetime95$ and $Lifetime50$ respectively, are illustrated for different network sizes.
\begin{figure}[h!]
\centering
\end{figure}
It is shown that all models provide a very near coverage ratios during the network lifetime, with a very small superiority for the models with higher number of primary points. Moreover, when the number of rounds increases, coverage ratio produced by Model-13, Model-17, and Model-21 decreases in comparison with Model-5 and Model-9 due to a larger time computation for the decision process for larger number of primary points.
-As shown in figure ~\ref{Figures/ch4/R2/CR}, Coverage ratio decreases when the number of rounds increases due to dead nodes. Model-9 is slightly more efficient than other models, because it is balanced between the number of rounds and the better coverage ratio in comparison with other Models.
+As shown in Figure ~\ref{Figures/ch4/R2/CR}, Coverage ratio decreases when the number of rounds increases due to dead nodes. Model-9 is slightly more efficient than other models, because it is balanced between the number of rounds and the better coverage ratio in comparison with other Models.
\item {{\bf Active Sensors Ratio}}
%\subsubsection{Active Sensors Ratio}
\label{Figures/ch4/R2/ASR}
\end{figure}
-The results presented in figure~\ref{Figures/ch4/R2/ASR} show the superiority of the proposed Model-5, in comparison with the other models. The model with fewer number of primary points uses fewer active nodes than the other models. According to the results presented in figure~\ref{Figures/ch4/R2/CR}, we observe that although the Model-5 continue to a larger number of rounds, but it has less coverage ratio compared with other models. The advantage of the Model-9 approach is to use fewer number of active nodes for each round compared with Model-13, Model-17, and Model-21. This led to continuing for a larger number of rounds with extending the network lifetime. Model-9 has a better coverage ratio compared to Model-5 and acceptable number of rounds.
+The results presented in Figure~\ref{Figures/ch4/R2/ASR} show the superiority of the proposed Model-5, in comparison with the other models. The model with fewer number of primary points uses fewer active nodes than the other models. According to the results presented in Figure~\ref{Figures/ch4/R2/CR}, we observe that although the Model-5 continue to a larger number of rounds, but it has less coverage ratio compared with other models. The advantage of the Model-9 approach is to use fewer number of active nodes for each round compared with Model-13, Model-17, and Model-21. This led to continuing for a larger number of rounds with extending the network lifetime. Model-9 has a better coverage ratio compared to Model-5 and acceptable number of rounds.
\item {{\bf The percentage of stopped simulation runs}}
\label{Figures/ch4/R3/ASR}
\end{figure}
-The results presented in figure~\ref{Figures/ch4/R3/ASR} show the superiority of the proposed DiLCO-16 protocol and DiLCO-32 protocol, in comparison with the other approaches. DESK and GAF have 37.5 \% and 44.5 \% active nodes and DiLCO-16 protocol and DiLCO-32 protocol compete perfectly with only 17.4 \%, 24.8 \% and 26.8 \% active nodes for the first 14 rounds. Then as the number of rounds increases DiLCO-16 protocol and DiLCO-32 protocol have larger number of active nodes in comparison with DESK and GAF, especially from round $35^{th}$ because they give a better coverage ratio than other approaches. We see that DESK and GAF have less number of active nodes beginning at the rounds $35^{th}$ and $32^{th}$ because there are many nodes are died due to the high energy consumption by the redundant nodes during the sensing phase. \\
+The results presented in Figure~\ref{Figures/ch4/R3/ASR} show the superiority of the proposed DiLCO-16 protocol and DiLCO-32 protocol, in comparison with the other approaches. DESK and GAF have 37.5 \% and 44.5 \% active nodes and DiLCO-16 protocol and DiLCO-32 protocol compete perfectly with only 17.4 \%, 24.8 \% and 26.8 \% active nodes for the first 14 rounds. Then as the number of rounds increases DiLCO-16 protocol and DiLCO-32 protocol have larger number of active nodes in comparison with DESK and GAF, especially from round $35^{th}$ because they give a better coverage ratio than other approaches. We see that DESK and GAF have less number of active nodes beginning at the rounds $35^{th}$ and $32^{th}$ because there are many nodes are died due to the high energy consumption by the redundant nodes during the sensing phase. \\
\item {{\bf The percentage of stopped simulation runs}}
