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[JournalMultiPeriods.git] / article.tex

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@@ -541,11 +541,7 @@ active nodes.
 %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.
 
 
 %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.
 
 

-\indent Instead of working with the coverage area, we consider for each sensor a set of points called primary points~\cite{idrees2014coverage}. 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 sensing range $R_s$,  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:\\
-Assuming that the point center of a wireless sensor node is located at $(p_x,p_y)$, we can define up to 25 primary points $X_1$ to $X_{25}$.\\
+\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 (point  center: ($p_x,p_y$))  of  a wireless sensor node  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 coordinates of the primary points are the following :\\

 %$(p_x,p_y)$ = point center of wireless sensor node\\  
 $X_1=(p_x,p_y)$ \\ 
 $X_2=( p_x + R_s * (1), p_y + R_s * (0) )$\\           
 %$(p_x,p_y)$ = point center of wireless sensor node\\  
 $X_1=(p_x,p_y)$ \\ 
 $X_2=( p_x + R_s * (1), p_y + R_s * (0) )$\\           


@@ -564,8 +560,8 @@ $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_{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_{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_{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})) $\\


@@ -612,9 +608,17 @@ $X_{25}=( p_x + R_s * (\frac{1}{2}), p_y + R_s * (\frac{-\sqrt{3}}{2})) $.
 
 \subsection{Background idea}
 %%RC : we need to clarify the difference between round and period. Currently it seems to be the same (for me at least).
 
 \subsection{Background idea}
 %%RC : we need to clarify the difference between round and period. Currently it seems to be the same (for me at least).

-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.
+%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.
+
+\textcolor{green}{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,
 
 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,


@@ -801,7 +805,7 @@ Subject to
 \end{equation}
 
 \begin{equation}
 \end{equation}
 
 \begin{equation}

-  \sum_{t=1}^{T}  X_{t,j}   \leq  \floor*{RE_{j}/E_{R}} \hspace{6 mm} \forall j \in J, t = 1,\dots,T
+  \sum_{t=1}^{T}  X_{t,j}   \leq  \floor*{RE_{j}/E_{R}} \hspace{10 mm}\forall j \in J\hspace{6 mm} 

   \label{eq144} 
 \end{equation}
 
   \label{eq144} 
 \end{equation}
 


@@ -849,6 +853,8 @@ 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
 large compared to $W_{\theta}$.
 %% 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
 large compared to $W_{\theta}$.

+
+\textcolor{green}{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.
 
 
 %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.
 
 


@@ -1129,12 +1135,37 @@ $W_{U}$ & $|P|^2$ \\
 \label{table3}
 % is used to refer this table in the text
 \end{table}
 \label{table3}
 % is used to refer this table in the text
 \end{table}

-  
-\textcolor{red}{Our first protocol based GLPK optimization solver 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). }
-%The second protocol based GA is declined into  four versions: GA-MuDiLCO-1,  GA-MuDiLCO-3, GA-MuDiLCO-5,
-%and  GA-MuDiLCO-7 for the same reason of the first protocol. After extensive experiments, we chose the dedicated values for the parameters $P_c$, $P_m$, and $S_{pop}$ because they gave the best results}. 
+
+\textcolor{green}{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}
+
+
+
+

  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
  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


@@ -1321,6 +1352,7 @@ indicate the energy consumed by the whole network in round $t$.
 
 \end{enumerate}
 
 
 \end{enumerate}
 

+\section{Results and analysis}

 \subsection{Performance Analysis for Different Number of Primary Points}
 \label{ch4:sec:04:06}
 
 \subsection{Performance Analysis for Different Number of Primary Points}
 \label{ch4:sec:04:06}
 


@@ -1364,12 +1396,12 @@ As highlighted by Figures~\ref{Figures/ch4/R2/LT}(a) and \ref{Figures/ch4/R2/LT}
   \label{Figures/ch4/R2/LT}
 \end{figure}
 
   \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 Model-5 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. 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. 

 
 %\end{enumerate}
 
 
 
 %\end{enumerate}
 
 

-\subsection{Results and analysis}
+%\subsection{Results and analysis}

 
 \subsubsection{Coverage ratio} 
 
 
 \subsubsection{Coverage ratio} 
 


@@ -1469,8 +1501,9 @@ network sizes, for $Lifetime_{95}$ and $Lifetime_{50}$.
 
 The  results  show  that  MuDiLCO  is  the  most  competitive  from  the  energy
 consumption point of view.  The  other approaches have a high energy consumption
 
 The  results  show  that  MuDiLCO  is  the  most  competitive  from  the  energy
 consumption point of view.  The  other approaches have a high energy consumption

-due  to activating a  larger number  of redundant  nodes as  well as  the energy consumed during  the different  status of the  sensor node. Among  the different versions of our protocol, the MuDiLCO-7  one consumes more energy than the other
-versions. This is  easy to understand since the bigger the  number of rounds and the number of  sensors involved in the integer program are,  the larger the time computation to solve the optimization problem is. To improve the performances of MuDiLCO-7, we  should increase the  number of subregions  in order to  have less sensors to consider in the integer program.
+due  to activating a  larger number  of redundant  nodes as  well as  the energy consumed during  the different  status of the  sensor node.
+% Among  the different versions of our protocol, the MuDiLCO-7  one consumes more energy than the other
+%versions. This is  easy to understand since the bigger the  number of rounds and the number of  sensors involved in the integer program are,  the larger the time computation to solve the optimization problem is. To improve the performances of MuDiLCO-7, we  should increase the  number of subregions  in order to  have less sensors to consider in the integer program.

 %\textcolor{red}{As shown in Figure~\ref{fig7}, GA-MuDiLCO consumes less energy than both DESK and GAF, but a little bit higher than MuDiLCO  because it provides a near optimal solution by activating a larger number of nodes during the sensing phase.  GA-MuDiLCO consumes less energy in comparison with MuDiLCO-7 version, especially for the dense networks. However, MuDiLCO protocol and GA-MuDiLCO protocol are the most competitive from the energy
 %consumption point of view. The other approaches have a high energy consumption
 %due to activating a larger number of redundant nodes.}
 %\textcolor{red}{As shown in Figure~\ref{fig7}, GA-MuDiLCO consumes less energy than both DESK and GAF, but a little bit higher than MuDiLCO  because it provides a near optimal solution by activating a larger number of nodes during the sensing phase.  GA-MuDiLCO consumes less energy in comparison with MuDiLCO-7 version, especially for the dense networks. However, MuDiLCO protocol and GA-MuDiLCO protocol are the most competitive from the energy
 %consumption point of view. The other approaches have a high energy consumption
 %due to activating a larger number of redundant nodes.}