X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/UIC2013.git/blobdiff_plain/856f104be1ac9636e82c13f72052ec7850f8be2b..f88b58680a85cba2913b1c5a9381077051bee9eb:/bare_conf.tex

diff --git a/bare_conf.tex b/bare_conf.tex
index 59b3d4b..74ed6fc 100755
--- a/bare_conf.tex
+++ b/bare_conf.tex
@@ -12,13 +12,12 @@
 
 \usepackage{float}
 \usepackage{epsfig}
-\usepackage{subfigure}
 \usepackage{calc}
  \usepackage{times,amssymb,amsmath,latexsym}
 \usepackage{graphics}
 \usepackage{graphicx}
 \usepackage{amsmath}
-\usepackage{txfonts}
+%\usepackage{txfonts}
 \usepackage{algorithmic}
 \usepackage[T1]{fontenc}
 \usepackage{tikz}
@@ -36,7 +35,7 @@
 
 \begin{document}
 
-\title{Distributed Coverage Optimization Protocol to Improve the Lifetime in Heterogeneous Energy Wireless Sensor Networks}
+\title{Energy-Efficient Activity Scheduling in Heterogeneous Energy Wireless Sensor Networks}
 
 
 % author names and affiliations
@@ -61,9 +60,9 @@ Email:$\lbrace$ali.idness, karine.deschinkel, michel.salomon,raphael.couturier$\
 
 \begin{abstract}
 %\boldmath
-One of the fundamental challenges in Wireless Sensor Networks (WSNs) is Coverage preservation and extension of network lifetime continuously and effectively during monitoring a certain geographical area.In this paper 
-a distributed coverage optimization protocol to improve the lifetime in in Heterogeneous Energy Wireless Sensor Networks is proposed. The area of interest is divided into subregions using Divide-and-conquer method and an activity scheduling for sensor nodes is planned for each subregion.Our protocol is distributed in each subregion. It divides the network lifetime into activity rounds. In each round a small
-number of active nodes is selected to ensure coverage.Each round includes four phases: INFO Exchange, Leader election, decision and sensing.Simulation results show that the proposed protocol can prolong the network
+One of the fundamental challenges in Wireless Sensor Networks (WSNs) is coverage preservation and extension of network lifetime continuously and effectively during monitoring a certain area of interest. In this paper 
+a coverage optimization protocol to improve the lifetime in Heterogeneous Energy Wireless Sensor Networks is proposed. The area of interest is divided into subregions using Divide-and-conquer method and an activity scheduling for sensor nodes is planned for each subregion. The proposed scheduling works in round. In each round a small
+number of active nodes is selected to ensure coverage. Each round includes four phases: Information Exchange, Leader election, decision and sensing. The decision process is carried out by a leader node with the resolution of an integer program. Simulation results show that the proposed approach can prolong the network
 lifetime and improve network coverage effectively.
 
 
@@ -95,7 +94,7 @@ the lifetime of the network. In such a high density network, if all sensor nodes
 were to be activated at the same time, the lifetime would be reduced. Consequently, 
 future software may need to adapt appropriately to achieve acceptable quality of service for applications.  
 In this paper we concentrate on area coverage problem, with the objective of maximizing the network lifetime by using an adaptive scheduling. Area of interest is divided into subregions and an activity scheduling for sensor nodes is planned for each subregion.
-Our scheduling scheme works in period which includes a discovery phase to exchange information between sensors of the subregion, then a sensor is chosen in suitable manner to carry out a coverage strategy.  This coverage strategy involves the resolution of an integer program which provides the activation of the sensors for the $t$ next round.
+Our scheduling scheme works in round which includes a discovery phase to exchange information between sensors of the subregion, then a sensor is chosen in suitable manner to carry out a coverage strategy.  This coverage strategy involves the resolution of an integer program which provides the activation of the sensors for the next round.
 
 
 The remainder  of the paper is  organized as follows.
@@ -122,16 +121,16 @@ This section is dedicated to the various approaches proposed in the literature f
 
 The most discussed coverage problems in literature can be classified into two types \cite{} : area coverage and targets coverage. An area coverage problem is to find a minimum number of sensors to work such that each physical point in the area is monitored by at least a working sensor. Target coverage problem is to cover only a finite number of discrete points called targets. 
  Our work will concentrate on the area coverage by design and implement a strategy which efficiently select the active nodes that must maintain both sensing coverage and network connectivity and in the same time improve the lifetime of the wireless sensor network. But requiring that all physical points are covered may be too strict, specially where the sensor network is not dense. 
-Our approach represents an area covered by a sensor as a set of principle points and tries to maximize the total number of principles points that are covered in each round, while minimizing overcoverage (points covered by multiple active sensors simultaneously).\\
+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).\\
 {\bf Lifetime}\\
-Various definitions exist for the lifetime of a sensor network. Main definitions proposed in the literature are related to the remaining energy of the nodes \cite{} or to the percentage of coverage \cite{}. The lifetime of the network is mainly defined as the amount of time that the network can satisfy its coverage objective (the amount of time that the network can cover a given percentage of its area or targets of interest) . In our simulation we assume that the network is alive until all sensor nodes are died and we measure the coverage ratio during the process.
+Various definitions exist for the lifetime of a sensor network. Main definitions proposed in the literature are related to the remaining energy of the nodes \cite{} or to the percentage of coverage \cite{}. The lifetime of the network is mainly defined as the amount of time that the network can satisfy its coverage objective (the amount of time that the network can cover a given percentage of its area or targets of interest). In our simulation we assume that the network is alive until all sensor nodes are died and we measure the coverage ratio during the process.
 
 {\bf Activity scheduling}\\
 Activity scheduling is to schedule the activation and deactivation of nodes 'sensor units. The basic objective is to decide which sensors are in which states (active or sleeping mode) and for how long a time such that the application coverage requirement can be guaranteed and network lifetime can be prolonged. Various approaches, including centralized, distributed and localized algorithms, have been proposed for activity scheduling.  In the distributed algorithms, each node in the network autonomously makes decisions on whether to turn on or turn off itself only using local neighbor information. In centralized algorithms, a central controller (node or base station) informs every sensor of the time intervals to be activated.
 
 {\bf Distributed approaches}
 
-Some distributed algorithms have been developed in~\cite{Gallais06,Tian02,Ye03,Zhang05,HeinzelmanCB02}. Distributed algorithms typically operate in roundsf predetermined duration. At the beginning of each round, a sensor exchange information with its neighbors and makes a decision to either turn on or go to sleep for the round. This decision is basically based on simple greedy criteria like the largest uncovered area \cite{Berman05efficientenergy}, maximum uncovered targets \cite{1240799}. 
+Some distributed algorithms have been developed in~\cite{Gallais06,Tian02,Ye03,Zhang05,HeinzelmanCB02}. Distributed algorithms typically operate in rounds for predetermined duration. At the beginning of each round, a sensor exchange information with its neighbors and makes a decision to either turn on or go to sleep for the round. This decision is basically based on simple greedy criteria like the largest uncovered area \cite{Berman05efficientenergy}, maximum uncovered targets \cite{1240799}. 
 In \cite{Tian02}, the sheduling scheme is divided into rounds, where each round has a self-scheduling phase followed by a sensing phase. Each sensor broadcasts  a message to its neighbors containing node ID and node location at the beginning of each round. Sensor determines its status by a rule named off-duty eligible rule which tells him to turn off if its sensing area is covered by its neighbors. A back-off scheme is introduced to let each sensor delay the decision process with a random period of time, in order to avoid that nodes make conflicting decisions simultaneously and that a part of the area is no longer covered.
 \cite{Prasad:2007:DAL:1782174.1782218} propose a model for capturing the dependencies between different cover sets and propose localized heuristic based on this dependency. The algorithm consists of two phases, an initial setup phase during which each sensor calculates and prioritize the covers and a sensing phase  during which each sensor first decides its on/off status and then remains on or off  for the rest of the duration. 
 Authors in \cite{chin2007} propose a novel distributed heuristic named distributed Energy-efficient Scheduling for k-coverage (DESK) so that the energy consumption among all the sensors is balanced, and network lifetime is maximized while the coverage requirements being maintained. This algorithm works in round, requires only 1-sensing-hop-neigbor information, and a sensor decides its status (active/sleep) based on its perimeter coverage computed through the k-Non-Unit-disk coverage algorithm  proposed in \cite{Huang:2003:CPW:941350.941367}.\\
@@ -191,14 +190,14 @@ There are three main questions which should be answered  to build a scheduling s
 phases for information exchange, decision and sensing over time?}
 Our algorithm partitions the time line into a number of periods. Each period contains 4 phases : information Exchange, Leader Election, Decision, and Sensing. Our work further divides sensing phase into a number of rounds of predetermined length.
 \item {\bf What are the rules to decide which node has to turn on or off?}
-Our algorithm tends to limit the overcoverage of points of interest to avoid turning on too much sensors covering the same areas at the same time, and tries to prevent undercoverage. The decision is a good compromise between these two conflicting objectives and is made for the next $T$ rounds of sensing. In our experimentations we will check which value of $T$ is the most appropriate.
+Our algorithm tends to limit the overcoverage of points of interest to avoid turning on too much sensors covering the same areas at the same time, and tries to prevent undercoverage. The decision is a good compromise between these two conflicting objectives. 
 \item {\bf Which node should make such decision ?}
 As mentioned in \cite{pc10}, both centralized and distributed algorithms have their own advantages and disadvantages. Centralized coverage algorithms have the advantage of requiring very low processing power from the sensor nodes which have usually limited processing capabilities. Distributed algorithms are very adaptable to the dynamic and scalable nature of sensors network. Authors in \cite{pc10} concludes that there is a threshold in terms of network size to switch from a localized to a centralized algorithm. Indeed the exchange of messages in large networks may consume  a considerable amount of energy in a localized approach compared to a centralized one. Our work does not consider only one leader to compute and to broadcast the schedule decision to all the sensors. When the size of network increases, the network is divided in many subregions and the decision is made by a leader in each subregion.
 \end{itemize}
 
 
 
- \section{\uppercase{Distributed coverage model}}
+ \section{\uppercase{Activity scheduling}}
 \label{pd}
 We consider a randomly and uniformly deployed network consisting of static wireless sensors. The wireless sensors are deployed in high density to ensure initially a full coverage of the interested area. We assume that all nodes are homogeneous in terms of  communication and processing capabilities and heterogeneous in term of energy.  The location information is available to the sensor node either through hardware such as embedded GPS or through location discovery algorithms.
 The area of interest can be divided using the divide-and-conquer strategy into smaller area called subregions and then our coverage protocol will be implemented in each subregion simultaneously. Our protocol works in rounds fashion as in figure \ref{fig:4}. 
@@ -211,13 +210,11 @@ The area of interest can be divided using the divide-and-conquer strategy into s
 \label{fig:4}
 \end{figure} 
 
-Each round is divided into 4 phases : INFO Exchange, Leader Election, Decision, and Sensing. For each round there is exactly one set cover responsible for sensing task. This protocol is more reliable against the  unexpectedly node failure because it works into rounds,and if the node failure detected before taking the decision, the node will not participate in decision and if the the node failure obtain after the decision   the sensing task of the network will be affected temporarily only during the period of sensing until starting new round, since a new set cover will take charge of the sensing task in the next round. The energy consumption and some other constraints can easily be taken into account since the sensors can update and then exchange the information (including their residual energy) at the beginning of each round. However, the preprocessing phase (INFO Exchange, leader Election, Decision) are energy consuming for some nodes even when they not join the network to monitor the area. We describe each phase in more detail.
+Each round is divided into 4 phases : Information (INFO) Exchange, Leader Election, Decision, and Sensing. For each round there is exactly one set cover responsible for sensing task. This protocol is more reliable against the  unexpectedly node failure because it works into rounds, and if the node failure is detected before taking the decision, the node will not participate in decision and if the node failure occurs after the decision, the sensing task of the network will be affected temporarily only during the period of sensing until starting new round, since a new set cover will take charge of the sensing task in the next round. The energy consumption and some other constraints can easily be taken into account since the sensors can update and then exchange the information (including their residual energy) at the beginning of each round. However, the preprocessing phase (INFO Exchange, leader Election, Decision) are energy consuming for some nodes even when they not join the network to monitor the area. We describe each phase in more detail.
 
 \subsection{\textbf INFO Exchange Phase}
 
-Each sensor node $j$  sends its position, remaining energy $RE_j$, number of local neighbours $NBR_j$  to all wireless sensor nodes in its subregion by using INFO packet and listen to the packets sent from other nodes. After that, each node will have information about all the sensor nodes in the subregion. In our model.
-
-% the remaining energy corresponds to the time that a sensor can live in the active mode.
+Each sensor node $j$  sends its position, remaining energy $RE_j$, number of local neighbours $NBR_j$  to all wireless sensor nodes in its subregion by using INFO packet and listen to the packets sent from other nodes. After that, each node will have information about all the sensor nodes in the subregion. In our model, the remaining energy corresponds to the time that a sensor can live in the active mode.
 
 
 %\subsection{\textbf Working Phase:}
@@ -227,18 +224,18 @@ Each sensor node $j$  sends its position, remaining energy $RE_j$, number of loc
 \subsection{\textbf Leader Election Phase}
 This step includes choosing the Wireless Sensor Node Leader (WSNL) which will be responsible of executing coverage algorithm  to choose the list of active sensor nodes that contribute in covering the subregion.
 % The sensors in the same region are capable to communicate with each others using a routing protocol provided by the simulator OMNET++ in order to provide multi-hop communication protocol.
-The WSNL will be chosen based on the number of local neighbours $NBR_j$  of sensor node $s_j$ and it's remaining energy $RE_j$.
-If we have more than one node has the same $NBR_j$ and $RE_j$, this leads to choose WSNL based on the largest index among them. Each subregion in the area of interest will select its WSNL independently for each round. 
+The WSNL will be chosen based on the number of local neighbours $NBR_j$  of sensor node $j$ and it's remaining energy $RE_j$.
+If we have more than one node with the same $NBR_j$ and $RE_j$, this leads to choose WSNL based on the largest index among them. Each subregion in the area of interest will select its WSNL independently for each round. 
 
 
 \subsection{\textbf Decision Phase}
-The WSNL will execute the GLPK algorithm to select which sensors will be activated in the next rounds to cover the subregion. WSNL will send Active-Sleep packet to each sensor in the subregion based on algorithm's results. 
+The WSNL will solve an integer program (see section \ref{cp}) to select which sensors will be activated in the next round to cover the subregion. WSNL will send Active-Sleep packet to each sensor in the subregion based on algorithm's results. 
 %The main goal in this step after choosing the WSNL is to produce the best representative active nodes set that will take the responsibility of covering the whole region $A^k$ with minimum number of sensor nodes to prolong the lifetime in the wireless sensor network. For our problem, in each round we need to select the minimum set of sensor nodes to improve the lifetime of the network and in the same time taking into account covering the region $A^k$ . We need an optimal solution with tradeoff between our two conflicting objectives.
 %The above region coverage problem can be formulated as a Multi-objective optimization problem and we can use the Binary Particle Swarm Optimization technique to solve it. 
 \\
 
 \subsection{\textbf Sensing Phase}
- 	The algorithm will produce the best representative set of the active nodes that will take the mission of coverage preservation in the subregion during the Sensing phase. Since that we use a homogeneous wireless sensor network, we will assume that the cost of keeping a node awake (or sleep) for sensing task is the same for all wireless sensor nodes in the network.  
+ 	Active sensors in the round will execute their sensing task to preserve maximal coverage  in the region of interest. We will assume that the cost of keeping a node awake (or sleep) for sensing task is the same for all wireless sensor nodes in the network.  
 
 
 
@@ -261,10 +258,10 @@ The WSNL will execute the GLPK algorithm to select which sensors will be activat
 %\end{figure}
 
 %By using the Unit Circle in figure~\ref{fig:cluster1}, 
-%We choose to representEach wireless sensor node will be represented into a selected number of principle points by which we can know if the sensor node is covered or not.
-% Figure ~\ref{fig:cluster2} shows the selected principle points that represents the area of the sensor node and according to the sensing range of the wireless sensor node.
+%We choose to representEach wireless sensor node will be represented into a selected number of primary points by which we can know if the sensor node is covered or not.
+% Figure ~\ref{fig:cluster2} shows the selected primary points that represents the area of the sensor node and according to the sensing range of the wireless sensor node.
 
-\noindent Instead of working with area coverage, we consider for each sensor a set of points called principal points. And we assume the sensing disk defined by a sensor is covered  if all principal points of this sensor are covered.
+\noindent Instead of working with area coverage, we consider for each sensor a set of points called primary points. And we assume the sensing disk defined by a sensor is covered  if all primary points of this sensor are covered.
 
 %\begin{figure}[h!]
 %\centering
@@ -278,7 +275,7 @@ The WSNL will execute the GLPK algorithm to select which sensors will be activat
 
 
 
-\noindent By knowing the position (point center :($p_x,p_y$) of the Wireless sensor node and its $R_s$ , we calculate the principle points directly based on proposed model. We use these principle points (that can be increased or decreased as if it is necessary) as references to ensure that the monitoring area of the region is covered by the selected set of sensors instead of using the all points in the area.
+\noindent By knowing the position (point center :($p_x,p_y$) of the wireless sensor node and its $R_s$ , we calculate the primary points directly based on proposed model. We use these primary points (that can be increased or decreased as if it is necessary) as references to ensure that the monitoring area of the region is covered by the selected set of sensors instead of using the all points in the area.
 
  \begin{figure}[h!]
 %\centering
@@ -291,11 +288,11 @@ The WSNL will execute the GLPK algorithm to select which sensors will be activat
 %\includegraphics[scale=0.10]{fig26.pdf}\\~ ~ ~(e)
 %\includegraphics[scale=0.10]{fig27.pdf}\\~ ~ ~(f)
 %\end{multicols} 
-\caption{Wireless Sensor node represented by 13 principle points }
+\caption{Wireless Sensor node represented by 13 primary points }
 \label{fig3}
 \end{figure}
 
-\noindent We can calculate the positions of the selected principle points in the circle disk of the sensing range of wireless sensor node in figure ~\ref{fig3} as follow:\\
+\noindent We can calculate the positions of the selected primary points in the circle disk of the sensing range of wireless sensor node in figure ~\ref{fig3} as follow:\\
 $p_x,p_y$ = point center of wireless sensor node. \\  
 $X_1=(p_x,p_y)$ \\ 
 $X_2=( p_x + R_s * (1), p_y + R_s * (0) )$\\           
@@ -322,47 +319,46 @@ $X_{13}=( p_x + R_s * (0), p_y + R_s * (\frac{-\sqrt{2}}{2})) $\\
 %To satisfy the coverage requirement, the set of the principal points that will represent all the sensor nodes in the monitored region as $PSET= \lbrace P_1,\ldots ,P_p, \ldots , P_{N_P^k} \rbrace $, where $N_P^k = N_{sp} * N^k $ and according to the proposed model in figure ~\ref{fig:cluster2}. These points can be used by the wireless sensor node leader which will be chosen in each region in A to build a new parameter $\alpha_{jp}$  that represents the coverage possibility for each principal point $P_p$ of each wireless sensor node $s_j$ in $A^k$ as in \eqref{eq12}:\\
 
 
-\noindent Our model is based on the model proposed by \cite{pedraza2006} where the objective is to find a maximum number of disjoint cover sets. To accomplish this goal, authors propose a integer program  which forces undercoverage and overcoverage of targets to become minimal at the same time. They use variables $x_{s,l}$ to indicate if the sensor $s$ belongs to cover set $l$. In our model, we consider binary variables $X_{j,t}$ which determine the activation of sensor $j$ in round $t$. We replace the constraint guarantying that each sensor is a member of only one cover of the entire set of disjoint covers by a constraint specifying that the sum of energy consumed by the activation of sensor during several rounds is less than or equal to the remaining energy of the sensor. We also consider principle points as targets.  \\
-\noindent For a principle point $p$, let $\alpha_{jp}$ denote the indicator function of whether the point $p$ is covered, that is, \\
+\noindent Our model is based on the model proposed by \cite{pedraza2006} where the objective is to find a maximum number of disjoint cover sets. To accomplish this goal, authors propose a integer program  which forces undercoverage and overcoverage of targets to become minimal at the same time. They use binary variables $x_{s,l}$ to indicate if the sensor $s$ belongs to cover set $l$.  In our model, we consider binary variables $X_{j}$ which determine the activation of sensor $j$ in the round.  We also consider primary points as targets. The set of primary points is denoted by P, and the set of sensors by J.  \\
+\noindent For a primary point $p$, let $\alpha_{jp}$ denote the indicator function of whether the point $p$ is covered, that is, \\
 \begin{equation}
 \alpha_{jp} = \left \{ 
 \begin{array}{l l}
-  1 & \mbox{if the principal point $p$ is covered} \\
+  1 & \mbox{if the primary point $p$ is covered} \\
  & \mbox{by active sensor node $j$} \\
   0 & \mbox{Otherwise}\\
 \end{array} \right.
 %\label{eq12} 
 \end{equation}
-The number of sensors that are covering point $p$ during a round $t$ is equal to $\sum_{j \in J} \alpha_{jp} * X_{j,t}$ where :
+The number of sensors that are covering point $p$ is equal to $\sum_{j \in J} \alpha_{jp} * X_{j}$ where :
 \begin{equation}
-X_{j,t} = \left \{ 
+X_{j} = \left \{ 
 \begin{array}{l l}
-  1& \mbox{if sensor $s_j$  is active during round } t\\
-  0 &  \mbox{Otherwise}\\
+  1& \mbox{if sensor $j$  is active} \\
+  0 &  \mbox{otherwise}\\
 \end{array} \right.
 %\label{eq11} 
 \end{equation}
-We define the Overcoverage variable $\Theta_{p,t}$ .\\
+We define the Overcoverage variable $\Theta_{p}$ .\\
 
 \begin{equation}
- \Theta_{p,t} = \left \{ 
+ \Theta_{p} = \left \{ 
 \begin{array}{l l}
-  0            & \mbox{if point p is not }\\
-&\mbox{covered during round } t\\
-  \left( \sum_{j \in J} \alpha_{jp} * X_{j,t} \right)- 1 & \mbox{Otherwise}\\
+  0            & \mbox{if point p is not covered}\\
+  \left( \sum_{j \in J} \alpha_{jp} * X_{j} \right)- 1 & \mbox{otherwise}\\
 \end{array} \right.
 \label{eq13} 
 \end{equation}
 
 
-\noindent$\Theta_{p}$ represents the number of active sensor nodes minus one that cover the principle point $p$.\\
-The Undercoverage variable $U_{p,t}$ of the principle point $p$ is defined as follow :\\
+\noindent$\Theta_{p}$ represents the number of active sensor nodes minus one that cover the primary point $p$.\\
+The Undercoverage variable $U_{p}$ of the primary point $p$ is defined as follow :\\
 
 \begin{equation}
-U_{p,t} = \left \{ 
+U_{p} = \left \{ 
 \begin{array}{l l}
-  1 &\mbox{if point } $p$ \mbox{ is not covered during round } $t$\\
-  0 & \mbox{Otherwise}\\
+  1 &\mbox{if point } $p$ \mbox{ is not covered} \\
+  0 & \mbox{otherwise}\\
 \end{array} \right.
 \label{eq14} 
 \end{equation}
@@ -371,27 +367,27 @@ U_{p,t} = \left \{
 \begin{equation} \label{eq:ip2r}
 \left \{
 \begin{array}{ll}
-\min \sum_{p \in P} (w_{\theta,t} \Theta_{p,t} + w_{u,t} U_{p,t})&\\
+\min \sum_{p \in P} (w_{\theta} \Theta_{p} + w_{U} U_{p})&\\
 \textrm{subject to :}&\\
-\sum_{j \in J}  \alpha_{jp} X_{j,t} - \Theta_{p,t}+ U_{p,t} =1, &\forall p \in P, \forall t \in T\\
+\sum_{j \in J}  \alpha_{jp} X_{j} - \Theta_{p}+ U_{p} =1, &\forall p \in P\\
 %\label{c1} 
-\sum_{t \in T} X_{j,t} \leq \frac{RE_j}{e_t} &\forall j \in J \\
+%\sum_{t \in T} X_{j,t} \leq \frac{RE_j}{e_t} &\forall j \in J \\
 %\label{c2}
-\Theta_{p,t}\in \mathbb{N} , &\forall p \in P, \forall t \in T\\
-U_{p,t} \in \{0,1\}, &\forall p \in P, \forall t \in T \\
-X_{j,t} \in \{0,1\}, &\forall j \in J, \forall t \in T
+\Theta_{p}\in \mathbb{N} , &\forall p \in P\\
+U_{p} \in \{0,1\}, &\forall p \in P \\
+X_{j} \in \{0,1\}, &\forall j \in J
 \end{array}
 \right.
 \end{equation}
 \begin{itemize}
-\item $X_{j,t}$ : indicating whether or not sensor $j$ is active in round $t$(1 if yes and 0 if not)
-\item $\Theta_{p,t}$ : {\it overcoverage}, the number of sensors minus one that are covering point $p$ in round $t$
-\item $U_{p,t}$ : {\it undercoverage}, indicating whether or not point $p$ is being covered (1 if not covered and 0 if covered) in round $t$
+\item $X_{j}$ : indicating whether or not sensor $j$ is active in the round (1 if yes and 0 if not)
+\item $\Theta_{p}$ : {\it overcoverage}, the number of sensors minus one that are covering point $p$
+\item $U_{p}$ : {\it undercoverage}, indicating whether or not point $p$ is being covered (1 if not covered and 0 if covered) 
 \end{itemize}
-The first group of constraints indicates that some point $p$ should be covered by at least one sensor in every round $t$ and, if it is not always the case, overcoverage and undercoverage variables help balance the restriction equation by taking positive values. Second group of contraints ensures for each sensor that the amount of energy consumed during its activation periods will be less than or equal to its remaining energy.
-There are two main objectives. We limit overcoverage of principle points in order to activate a minimum number of sensors and we prevent that parts of the subregion are not monitored by minimizing undercoverage. The weights $w_{\theta,t}$ and $w_{u,t}$ must be properly chosen so as to guarantee that the maximum number of points are covered during each round. 
+The first group of constraints indicates that some point $p$ should be covered by at least one sensor and, if it is not always the case, overcoverage and undercoverage variables help balance the restriction equation by taking positive values. Second group of contraints ensures for each sensor that the amount of energy consumed during its activation periods will be less than or equal to its remaining energy.
+There are two main objectives. We limit overcoverage of primary points in order to activate a minimum number of sensors and we prevent that parts of the subregion are not monitored by minimizing 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. 
  
-%In equation \eqref{eq15}, there are two main objectives: the first one using  the Overcoverage parameter to minimize the number of active sensor nodes in the produced final solution vector $X$ which leads to improve the life time of wireless sensor network. The second goal by using the  Undercoverage parameter  to maximize the coverage in the region by means of covering each principle point in $SSET^k$.The two objectives are achieved at the same time. The constraint which represented in equation \eqref{eq16} refer to the coverage function for each principle point $P_p$ in $SSET^k$ , where each $P_p$ should be covered by
+%In equation \eqref{eq15}, there are two main objectives: the first one using  the Overcoverage parameter to minimize the number of active sensor nodes in the produced final solution vector $X$ which leads to improve the life time of wireless sensor network. The second goal by using the  Undercoverage parameter  to maximize the coverage in the region by means of covering each primary point in $SSET^k$.The two objectives are achieved at the same time. The constraint which represented in equation \eqref{eq16} refer to the coverage function for each primary point $P_p$ in $SSET^k$ , where each $P_p$ should be covered by
 %at least one sensor node in $A^k$. The objective function in \eqref{eq15} involving two main objectives to be optimized simultaneously, where optimal decisions need to be taken in the presence of trade-offs between the two conflicting main objectives in \eqref{eq15} and this refer to that our coverage optimization problem is a multi-objective optimization problem and we can use the BPSO to solve it. The concept of Overcoverage and Undercoverage inspired from ~\cite{Fernan12} but we use it with our model as stated in subsection \ref{Sensing Coverage Model} with some modification to be applied later by BPSO.
 %\subsection{Notations and assumptions}
 
@@ -418,15 +414,15 @@ There are two main objectives. We limit overcoverage of principle points in orde
 \section{\uppercase{Simulation Results}}
 \label{exp}
 In this section, we conducted a series of simulations to evaluate the efficiency of our approach
-based on the discrete event simulator OMNeT++ (http://www.omnetpp.org/).we conduct simulations for six
+based on the discrete event simulator OMNeT++ (http://www.omnetpp.org/). We conduct simulations for six
 different densities varying from 50 to 300 nodes. Experimental results were obtained from randomly generated
 networks in which nodes are deployed over a $ 50\times25(m2) $sensing field. For each network deployment, we
-assume that the deployed nodes can fully cover the sensing field with the given sensing range. 100 simulation runs are performed with different network topologies. The results presented hereafter are the average of these 100 runs.Simulation ends when there is at least one active node has no connectivity with the network.Our proposed coverage protocol use the Radio energy dissipation model that defined by~\cite{HeinzelmanCB02} as energy consumption model by each wireless sensor node for transmitting and receiving the packets in the network.The energy of each node in the network is initialized randomly within the range 24-60 joules, and each sensor will consumes 0.2 watts during the period of sensing which it is 60 seconds.Each active node will consumes 12 joules during sensing phase and each sleep node will consumes 0.002 joules.Each sensor node will not participate in the next round if it's remaining energy less than 12 joules. In all experiments the parameters are given by $R_s = 5m $ , $ W_{\Theta} =1$ and $W_{\Psi} = P^2$.
-We evaluate the efficiency of our approach using some performance metrics such as:coverage ratio, number of
-active nodes ratio, energy saving ratio, number of rounds, network lifetime and execution time of our approach.Coverage ratio measures how much area of a sensor field is covered. In our case, the coverage ratio is regarded as the number of principle points covered among the set of all principle points within the field.In our simulation the sensing field is sub divided into two subregions each one equal to $ 25\times25(m2)  $ of the sensing field.
+assume that the deployed nodes can fully cover the sensing field with the given sensing range. 100 simulation runs are performed with different network topologies. The results presented hereafter are the average of these 100 runs. Simulation ends when there is at least one active node has no connectivity with the network. Our proposed coverage protocol use the Radio energy dissipation model that defined by~\cite{HeinzelmanCB02} as energy consumption model by each wireless sensor node for transmitting and receiving the packets in the network. The energy of each node in the network is initialized randomly within the range 24-60 joules, and each sensor will consumes 0.2 watts during the sensing period  of 60 seconds. Each active node will consumes 12 joules during sensing phase and each sleep node will consume 0.002 joules. Each sensor node will not participate in the next round if it's remaining energy less than 12 joules. In all experiments the parameters are given by $R_s = 5m $ , $ w_{\Theta} =1$ and $w_{U} = |P|$.
+We evaluate the efficiency of our approach using some performance metrics such as : coverage ratio, number of
+active nodes ratio, energy saving ratio, number of rounds, network lifetime and execution time of our approach. Coverage ratio measures how much area of a sensor field is covered. In our case, the coverage ratio is regarded as the number of primary points covered among the set of all primary points within the field. In our simulation the sensing field is subdivided into two subregions each one equal to $ 25\times25(m2)  $ of the sensing field.
 
 \subsection{The impact of the Number of Rounds on Coverage Ratio:} 
-In this experiment, we study the impact of the number of rounds on the coverage ratio and for different sizes for sensor network.For each Sensor network size we will take the average of coverage ratio per round and for 100 simulation.Fig. 3 show the impact of the number of rounds on coverage ratio for different network sizes and for two subregions.
+In this experiment, we study the impact of the number of rounds on the coverage ratio and for different sizes of sensor network. Fig. \ref{fig3} shows the impact of the number of rounds on coverage ratio for different network sizes and for two subregions.
 
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