X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/LiCO.git/blobdiff_plain/f65c2462e1842e661a546d573a5e717d160c5fc7..1c2762bf6f7508f4f418890d2903ad6f008233f0:/LiCO_Journal.tex?ds=sidebyside

diff --git a/LiCO_Journal.tex b/LiCO_Journal.tex
index c8e2eb5..e9d10ac 100644
--- a/LiCO_Journal.tex
+++ b/LiCO_Journal.tex
@@ -293,6 +293,7 @@ executed by each node.
 %Before proceeding in the presentation of the main ideas of the protocol, we will briefly describe the perimeter coverage model and give some necessary assumptions and definitions.
 
 \subsection{Assumptions and Models}
+\label{CI}
 
 \noindent A WSN consisting of $J$ stationary sensor nodes randomly and uniformly
 distributed in  a bounded sensor field  is considered. The wireless  sensors are
@@ -452,7 +453,7 @@ 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.
 
-As shown  in figure~\label{fig2},  node activity scheduling  is produced  by our
+As  shown in  figure~\ref{fig2}, node  activity  scheduling is  produced by  our
 protocol in a periodic manner. Each period is divided into 4 stages: Information
 (INFO)  Exchange,  Leader Election,  Decision  (the  result of  an  optimization
 problem),  and  Sensing.   For  each  period there  is  exactly  one  set  cover
@@ -503,9 +504,9 @@ Five status are possible for a sensor node in the network:
 
 \subsection{LiCO Protocol Algorithm}
 
-The  pseudocode  implementing the  protocol  on  a  node  is given  below.  More
-precisely, Algorithm~\label{alg:LiCO} gives a  brief description of the protocol
-applied by a sensor node $s_k$ where $k$ is the node index in the WSN.
+\noindent The  pseudocode implementing the  protocol on  a node is  given below.
+More  precisely,  Algorithm~\ref{alg:LiCO}  gives  a brief  description  of  the
+protocol applied by a sensor node $s_k$ where $k$ is the node index in the WSN.
 
 \begin{algorithm}[h!]                
  % \KwIn{all the parameters related to information exchange}
@@ -555,55 +556,77 @@ applied by a sensor node $s_k$ where $k$ is the node index in the WSN.
 
 \end{algorithm}
 
-\noindent  In this  algorithm,  K.CurrentSize and  K.PreviousSize  refer to  the
-current  size  and  the  previous  size  of  the  subnetwork  in  the  subregion
-respectively.   That  means   the  number  of  sensor  nodes   which  are  still
-alive. Initially, the sensor node checks its remaining energy $RE_k$, which must
-be greater  than a  threshold $E_{th}$  in order to  participate in  the current
-period. Each  sensor node  determines its  position and  its subregion  using an
-embedded GPS  or a  location discovery  algorithm. After  that, all  the sensors
-collect position coordinates,  remaining energy, sensor node ID,  and the number
-of  its one-hop  live neighbors  during  the information  exchange. The  sensors
-inside a same region cooperate to elect a leader. The selection criteria for the
-leader, in order of priority, are:  larger number of neighbors, larger remaining
-energy, and  then in case  of equality, larger  index.  Once chosen,  the leader
-collects information to formulate and solve  the integer program which allows to
-construct the set of active sensors in the sensing stage.
+In this  algorithm, K.CurrentSize and  K.PreviousSize refer to the  current size
+and the  previous size of  the subnetwork  in the subregion  respectively.  That
+means the  number of sensor nodes  which are still alive.  Initially, the sensor
+node checks its remaining energy $RE_k$,  which must be greater than a threshold
+$E_{th}$  in order  to  participate  in the  current  period.  Each sensor  node
+determines its  position and its subregion  using an embedded GPS  or a location
+discovery algorithm. After  that, all the sensors  collect position coordinates,
+remaining energy, sensor  node ID, and the number of  its one-hop live neighbors
+during the information  exchange. The sensors inside a same  region cooperate to
+elect a  leader. The selection  criteria for the  leader, in order  of priority,
+are: larger  number of neighbors, larger  remaining energy, and then  in case of
+equality,  larger  index.   Once  chosen, the  leader  collects  information  to
+formulate and  solve the integer  program which allows  to construct the  set of
+active sensors in the sensing stage.
 
 %After the cooperation among the sensor nodes in the same subregion, the leader will be elected in distributed way, where each sensor node and based on it's information decide who is the leader. The selection criteria for the leader in order  of priority  are: larger number of neighbors,  larger remaining  energy, and  then in  case of equality, larger index. Thereafter,  if the sensor node is leader, it will execute the perimeter-coverage model for each sensor in the subregion in order to determine the segment points which would be used in the next stage by the optimization algorithm of the LiCO protocol. Every sensor node is selected as a leader, it is executed the perimeter coverage model only one time during it's life in the network.
 
 % The leader has the responsibility of applying the integer program algorithm (see section~\ref{cp}), which provides a set of sensors planned to be active in the sensing stage.  As leader, it will send an Active-Sleep packet to each sensor in the same subregion to inform it if it has to be active or not. On the contrary, if the sensor is not the leader, it will wait for the Active-Sleep packet to know its state for the sensing stage.
 
-% MICHEL TO BE CONTINUED
-
 \section{Lifetime Coverage problem formulation}
-
 \label{cp}
-In this section, the coverage model is mathematically formulated.
-For convenience, the notations are described first. 
-%Then the lifetime problem of sensor network is formulated. 
-\noindent $S :$ the set of all sensors in the network.\\
-\noindent $A :$ the set of alive sensors within $S$.\\
-%\noindent $I :$ the set of segment points.\\
-\noindent $I_j :$ the set of coverage intervals (CI)  for sensor $j$.\\
-\noindent $I_j$ refers to the set of intervals which have been defined for each sensor $j$ in section~\ref{sec:The LiCO Protocol Description}.
-\noindent For a coverage interval  $i$,  let  $a^j_{ik}$ denote the indicator function of whether the sensor $k$ is involved in the coverage interval $i$ of sensor $j$, that is:
+
+\noindent In this  section, the coverage model is  mathematically formulated. We
+start  with a  description of  the notations  that will  be used  throughout the
+section.
+
+First, we have the following sets:
+\begin{itemize}
+\item $S$ represents the set of WSN sensor nodes;
+\item $A \subseteq S $ is the subset of alive sensors;
+\item  $I_j$  designates  the  set  of  coverage  intervals  (CI)  obtained  for
+  sensor~$j$.
+\end{itemize}
+$I_j$ refers to the set of  coverage intervals which have been defined according
+to the  method introduced in  subsection~\ref{CI}. For a coverage  interval $i$,
+let $a^j_{ik}$ denote  the indicator function of whether  sensor~$k$ is involved
+in coverage interval~$i$ of sensor~$j$, that is:
 \begin{equation}
 a^j_{ik} = \left \{ 
 \begin{array}{lll}
-  1 & \mbox{if the sensor $k$ is involved in the } \\
+  1 & \mbox{if sensor $k$ is involved in the } \\
   	&	\mbox{coverage interval $i$ of sensor $j$}, \\
-  0 & \mbox{Otherwise.}\\
+  0 & \mbox{otherwise.}\\
 \end{array} \right.
 %\label{eq12} 
 \notag
 \end{equation}
-Note that $a^k_{ik}=1$ by definition of the interval.\\
+Note that $a^k_{ik}=1$ by definition of the interval.
 %, where the objective is to find the maximum number of non-disjoint sets of sensor nodes such that each set cover can assure the coverage for the whole region so as to extend the network lifetime in WSN. Our model uses the PCL~\cite{huang2005coverage} in order to optimize the lifetime coverage in each subregion.
-%We defined some parameters, which are related to our optimization model. In our model,  we  consider binary variables $X_{k}$, which determine the activation of sensor $k$ in the sensing round $k$. .   
-\noindent We  consider binary variables $X_{k}$ ($X_k=1$ if the sensor $k$ is active or 0 otherwise), which determine the activation of sensor $k$ in the sensing phase. We define the integer variable $M^j_i$ which measures the undercoverage for the coverage interval $i$ for sensor $j$. In the same way, we define the integer variable $V^j_i$, which measures the overcoverage for the coverage interval $i$ for sensor $j$. If we decide to sustain a level of coverage equal to $l$ all along the perimeter of the sensor $j$, we have to ensure that at least $l$ sensors involved in each coverage interval $i$ ($i \in I_j$) of sensor $j$ are active. According to the previous notations, the number of active sensors in the coverage interval $i$ of sensor $j$ is given by $\sum_{k \in K} a^j_{ik} X_k$. To extend the network lifetime, the objective is to active a minimal number of sensors in each period to ensure the desired coverage level. As the number of alive sensors decreases, it becomes impossible to satisfy the level of coverage for all covergae intervals. We uses variables $M^j_i$ and $V^j_i$ as a measure of the deviation between the desired number of active sensors in a coverage interval and the effective number of active sensors. And we try to minimize these deviations, first to force the activation of a minimal number of sensors to ensure the desired coverage level, and if the desired level can not be completely  satisfied, to reach a coverage level as close as possible that the desired one.
-
-
+%We defined some parameters, which are related to our optimization model. In our model,  we  consider binary variables $X_{k}$, which determine the activation of sensor $k$ in the sensing round $k$. .
+Second,  we define  several binary  and integer  variables.  Hence,  each binary
+variable $X_{k}$  determines the activation of  sensor $k$ in the  sensing phase
+($X_k=1$ if  the sensor $k$  is active or 0  otherwise).  $M^j_i$ is  an integer
+variable  which  measures  the  undercoverage  for  the  coverage  interval  $i$
+corresponding to  sensor~$j$. In  the same  way, the  overcoverage for  the same
+coverage interval is given by the variable $V^j_i$.
+
+If we decide to sustain a level of coverage equal to $l$ all along the perimeter
+of sensor  $j$, we have  to ensure  that at least  $l$ sensors involved  in each
+coverage  interval $i  \in I_j$  of  sensor $j$  are active.   According to  the
+previous notations, the number of active sensors in the coverage interval $i$ of
+sensor $j$  is given by  $\sum_{k \in A} a^j_{ik}  X_k$.  To extend  the network
+lifetime,  the objective  is to  activate a  minimal number  of sensors  in each
+period to  ensure the  desired coverage  level. As the  number of  alive sensors
+decreases, it becomes impossible to reach  the desired level of coverage for all
+coverage intervals. Therefore we uses variables $M^j_i$ and $V^j_i$ as a measure
+of the  deviation between  the desired  number of active  sensors in  a coverage
+interval and  the effective  number. And  we try  to minimize  these deviations,
+first to  force the  activation of  a minimal  number of  sensors to  ensure the
+desired coverage level, and if the desired level cannot be completely satisfied,
+to reach a coverage level as close as possible to the desired one.
 
 %A system of linear constraints is imposed to attempt to keep the coverage level in each coverage interval to within specified PCL. Since it is physically impossible to satisfy all constraints simultaneously, each constraint uses a variable to either record when the coverage level is achieved, or to record the deviation from the desired coverage level. These additional variables are embedded into an objective function to be minimized. 
 
@@ -625,13 +648,9 @@ Note that $a^k_{ik}=1$ by definition of the interval.\\
 
 %\noindent $V^j_i (overcoverage): $ integer value $\in  \mathbb{N}$ for segment point $i$ of sensor $j$.
 
-
-
-
-
-\noindent Our coverage optimization problem can be mathematically formulated as follows: 
+Our coverage optimization problem can then be mathematically expressed as follows: 
 %Objective:
-\begin{equation} \label{eq:ip2r}
+\begin{equation} %\label{eq:ip2r}
 \left \{
 \begin{array}{ll}
 \min \sum_{j \in S} \sum_{i \in I_j} (\alpha^j_i ~ M^j_i + \beta^j_i ~ V^j_i )&\\
@@ -645,27 +664,35 @@ Note that $a^k_{ik}=1$ by definition of the interval.\\
 X_{k} \in \{0,1\}, \forall k \in A
 \end{array}
 \right.
+\notag
 \end{equation}
-
-
-\noindent $\alpha^j_i$ and $\beta^j_i$ are nonnegative weights selected according to the
-relative importance of satisfying the associated
-level of coverage. For example, weights associated with coverage intervals of a specified part of a region
-may be given a relatively
-larger magnitude than weights associated
-with another region. This kind of integer program is inspired from the model developed for brachytherapy treatment planning for optimizing dose distribution \cite{0031-9155-44-1-012}. The integer program must be solved by the leader in each subregion at the beginning of each sensing phase, whenever the environment has changed (new leader, death of some sensors). Note that the number of constraints in the model is constant (constraints of coverage expressed for all sensors), whereas the number of variables $X_k$ decreases over periods, since we consider only alive sensors (sensors with enough energy to be alive during one sensing phase) in the model. 
-
-
-\section{\uppercase{PERFORMANCE EVALUATION AND ANALYSIS}}  
+$\alpha^j_i$ and $\beta^j_i$  are nonnegative weights selected  according to the
+relative importance of satisfying the associated level of coverage. For example,
+weights associated with  coverage intervals of a specified part  of a region may
+be  given a  relatively larger  magnitude than  weights associated  with another
+region. This  kind of integer program  is inspired from the  model developed for
+brachytherapy    treatment   planning    for   optimizing    dose   distribution
+\cite{0031-9155-44-1-012}. The integer  program must be solved by  the leader in
+each subregion at the beginning of  each sensing phase, whenever the environment
+has  changed (new  leader,  death of  some  sensors). Note  that  the number  of
+constraints in the model is constant  (constraints of coverage expressed for all
+sensors), whereas the number of variables $X_k$ decreases over periods, since we
+consider only alive  sensors (sensors with enough energy to  be alive during one
+sensing phase) in the model.
+
+\section{Performance Evaluation and Analysis}  
 \label{sec:Simulation Results and Analysis}
 %\noindent \subsection{Simulation Framework}
 
 \subsection{Simulation Settings}
 %\label{sub1}
-In this section, we focus on the performance of LiCO protocol, which is distributed in each sensor node in the sixteen subregions of WSN. We use the same energy consumption model which is used in~\cite{Idrees2}. Table~\ref{table3} gives the chosen parameters setting.
+
+The WSN  area of interest is  supposed to be divided  into 16~regular subregions
+and we use the same energy consumption than in our previous work~\cite{Idrees2}.
+Table~\ref{table3} gives the chosen parameters settings.
 
 \begin{table}[ht]
-\caption{Relevant parameters for network initializing.}
+\caption{Relevant parameters for network initialization.}
 % title of Table
 \centering
 % used for centering table
@@ -676,14 +703,14 @@ Parameter & Value  \\ [0.5ex]
    
 \hline
 % inserts single horizontal line
-Sensing  Field  & $(50 \times 25)~m^2 $   \\
+Sensing field & $(50 \times 25)~m^2 $   \\
 
-Nodes Number &  100, 150, 200, 250 and 300~nodes   \\
+WSN size &  100, 150, 200, 250, and 300~nodes   \\
 %\hline
-Initial Energy  & 500-700~joules  \\  
+Initial energy  & in range 500-700~Joules  \\  
 %\hline
-Sensing Period & 60 Minutes \\
-$E_{th}$ & 36 Joules\\
+Sensing period & duration of 60 minutes \\
+$E_{th}$ & 36~Joules\\
 $R_s$ & 5~m   \\     
 %\hline
 $\alpha^j_i$ & 0.6   \\
@@ -695,51 +722,54 @@ $\beta^j_i$ & 0.4
 \label{table3}
 % is used to refer this table in the text
 \end{table}
-Simulations with five  different node densities going from  100 to 250~nodes were
-performed  considering  each  time  25~randomly generated  networks,  to  obtain
-experimental results  which are relevant. All simulations are repeated 25 times and the results are averaged. The  nodes are deployed on a field of interest of $(50 \times 25)~m^2 $ in such a way that they cover the field with a high coverage ratio.
-
-Each node has an initial energy level, in Joules, which is randomly drawn in the
-interval  $[500-700]$.  If  it's  energy  provision reaches  a  value below  the
-threshold  $E_{th}=36$~Joules, the  minimum energy  needed  for a  node to  stay
-active during one period, it will no more participate in the coverage task. This
-value  corresponds  to the  energy  needed by  the  sensing  phase, obtained  by
-multiplying the energy consumed in active  state (9.72 mW) by the time in seconds
-for one period (3600 seconds), and  adding the energy for the pre-sensing phases.
-According to  the interval of initial energy,  a sensor may be  active during at
-most 20 rounds.
-
-The values of $\alpha^j_i$ and $\beta^j_i$ have been chosen in a way that ensuring a good network coverage and for a longer time during the lifetime of the WSN.  We have given a higher priority for the undercoverage ( by setting the $\alpha^j_i$ with a larger value than $\beta^j_i$) so as to prevent the non-coverage for the interval i of the sensor j. On the other hand, we have given a little bit lower value for $\beta^j_i$ so as to minimize the number of active sensor nodes that contribute in covering the interval i in sensor j.
-
-In the simulations,  we introduce the following performance  metrics to evaluate
-the efficiency of our approach:
+To  obtain  experimental  results  which are  relevant,  simulations  with  five
+different node densities going from  100 to 300~nodes were performed considering
+each time 25~randomly  generated networks. The nodes are deployed  on a field of
+interest of $(50 \times 25)~m^2 $ in such a way that they cover the field with a
+high coverage ratio. Each node has an  initial energy level, in Joules, which is
+randomly drawn in the interval $[500-700]$.   If it's energy provision reaches a
+value below  the threshold $E_{th}=36$~Joules,  the minimum energy needed  for a
+node  to stay  active during  one period,  it will  no more  participate in  the
+coverage task. This value corresponds to the energy needed by the sensing phase,
+obtained by multiplying  the energy consumed in active state  (9.72 mW) with the
+time in  seconds for one  period (3600 seconds), and  adding the energy  for the
+pre-sensing phases.  According  to the interval of initial energy,  a sensor may
+be active during at most 20 periods.
+
+The values  of $\alpha^j_i$ and  $\beta^j_i$ have been  chosen to ensure  a good
+network coverage and a longer WSN lifetime.  We have given a higher priority for
+the  undercoverage  (by  setting  the  $\alpha^j_i$ with  a  larger  value  than
+$\beta^j_i$) so as to prevent the non-coverage  for the interval i of the sensor
+j. On the other hand, we have given  a little bit lower value for $\beta^j_i$ so
+as to  minimize the number of  active sensor nodes which  contribute in covering
+the interval.
+
+We introduce the following performance metrics to evaluate the efficiency of our
+approach.
 
 %\begin{enumerate}[i)]
 \begin{itemize}
-\item {{\bf Network Lifetime}:} we define the network lifetime as the time until
-  the  coverage  ratio  drops  below  a  predefined  threshold.   We  denote  by
-  $Lifetime_{95}$ (respectively $Lifetime_{50}$) the amount of time during which
-  the  network can  satisfy an  area coverage  greater than  $95\%$ (respectively
-  $50\%$). We assume that the sensor  network can fulfill its task until all its
-  nodes have  been drained of their  energy or it  becomes disconnected. Network
-  connectivity  is crucial because  an active  sensor node  without connectivity
-  towards a base  station cannot transmit any information  regarding an observed
-  event in the area that it monitors.
-  
-    
-\item {{\bf Coverage Ratio (CR)}:} it measures how well the WSN is able to 
-  observe the area of interest. In our case, we discretized the sensor field
-  as a regular grid, which yields the following equation to compute the
-  coverage ratio: 
-\begin{equation*}
-\scriptsize
-\mbox{CR}(\%) = \frac{\mbox{$n$}}{\mbox{$N$}} \times 100.
-\end{equation*}
-where  $n$ is  the number  of covered  grid points  by active  sensors  of every
-subregions during  the current  sensing phase  and $N$ is  total number  of grid
-points in  the sensing field. In  our simulations, we have  a layout of  $N = 51
-\times 26 = 1326$ grid points.
+\item {\bf Network Lifetime}: the lifetime  is defined as the time elapsed until
+  the  coverage  ratio  falls  below a  fixed  threshold.   $Lifetime_{95}$  and
+  $Lifetime_{50}$  denote, respectively,  the  amount of  time  during which  is
+  guaranteed a  level of coverage  greater than $95\%$  and $50\%$. The  WSN can
+  fulfill the expected  monitoring task until all its nodes  have depleted their
+  energy or if the network is not more connected. This last condition is crucial
+  because without  network connectivity a  sensor may not be  able to send  to a
+  base station an event it has sensed.
+\item {{\bf  Coverage Ratio  (CR)}:} it  measures how  well the  WSN is  able to
+  observe the area of interest. In our  case, we discretized the sensor field as
+  a regular grid, which yields the following equation:
+  \begin{equation*}
+    \scriptsize
+    \mbox{CR}(\%) = \frac{\mbox{$n$}}{\mbox{$N$}} \times 100.
+  \end{equation*}
+  where $n$  is the  number of covered  grid points by  active sensors  of every
+  subregions during  the current sensing phase  and $N$ is total  number of grid
+  points  in the  sensing field.  In our  simulations we  have set  a layout  of
+  $N~=~51~\times~26~=~1326$~grid points.
 
+  % MICHEL TO BE CONTINUED FROM HERE
 
 \item{{\bf Number of Active Sensors Ratio(ASR)}:} It is important to have as few active nodes as possible in each round,
 in  order to  minimize  the communication  overhead  and maximize  the
