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\begin{document}
%
% paper title
% can use linebreaks \\ within to get better formatting as desired
\title{Coverage and Lifetime Optimization \\
in Heterogeneous Energy Wireless Sensor Networks}

\author{\IEEEauthorblockN{Ali Kadhum Idrees, Karine Deschinkel, Michel Salomon, 
and Rapha\"el Couturier}
\IEEEauthorblockA{FEMTO-ST Institute, UMR 6174 CNRS \\
University of Franche-Comt\'e  \\
Belfort, France\\
Email: ali.idness@edu.univ-fcomte.fr, $\lbrace$karine.deschinkel, michel.salomon, 
raphael.couturier$\rbrace$@univ-fcomte.fr}}

\maketitle

\begin{abstract}
One of  the fundamental challenges in Wireless  Sensor Networks (WSNs)
is the coverage preservation and the extension of the network lifetime
continuously  and  effectively  when  monitoring a  certain  area  (or
region) 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 first  divided into
subregions using  a divide-and-conquer method and  then the scheduling
of sensor node  activity is planned for each  subregion.  The proposed
scheduling  considers rounds  during which  a small  number  of nodes,
remaining active  for sensing, is  selected to ensure  coverage.  Each
round consists  in four phases:  (i)~Information Exchange, (ii)~Leader
Election, (iii)~Decision,  and (iv)~Sensing.  The  decision process is
carried  out  by a  leader  node,  which  solves an  integer  program.
Simulation  results show that  the proposed  approach can  prolong the
network lifetime and improve the coverage performance.
\end{abstract}

\begin{IEEEkeywords}
Wireless   Sensor   Networks,   Area   Coverage,   Network   lifetime,
Optimization, Scheduling.
\end{IEEEkeywords}
%\keywords{Area Coverage, Network lifetime, Optimization, Distributed Protocol}
 
\IEEEpeerreviewmaketitle

\section{Introduction}

%\indent  The fast  developments  in the  low-cost  sensor devices  and
%wireless  communications have  allowed  the emergence  the WSNs.   WSN
%includes  a large  number  of small,  limited-power  sensors that  can
%sense, process  and transmit data over a  wireless communication. They
%communicate   with   each    other   by   using   multi-hop   wireless
%communications, cooperate  together to  monitor the area  of interest,
%and the  measured data can be  reported to a  monitoring center called
%sink  for analysis it~\cite{Sudip03}.  There are  several applications
%used  the WSN  including  health, home,  environmental, military,  and
%industrial applications~\cite{Akyildiz02}. The coverage problem is one
%of the fundamental challenges  in WSNs~\cite{Nayak04} that consists in
%monitoring    efficiently    and     continuously    the    area    of
%interest. Thelimited  energy of sensors represents  the main challenge
%in the  WSNs design~\cite{Sudip03}, where  it is difficult  to replace
%and/or  recharge their  batteries  because the  the  area of  interest
%nature  (such  as  hostile  environments)  and the  cost.  So,  it  is
%necessary  that  a WSN  deployed  with  high  density because  spatial
%redundancy  can then  be exploited  to  increase the  lifetime of  the
%network. However, turn on all the sensor nodes, which monitor the same
%region at the same time leads to decrease the lifetime of the network.

Recent   years  have  witnessed   significant  advances   in  wireless
communications and embedded micro-sensing MEMS technologies which have
led to the emergence of Wireless  Sensor Networks (WSNs) as one of the
most promising  technologies \cite{Akyildiz02}. In  fact, they present
huge   potential  in   several  domains   ranging  from   health  care
applications to military applications. A sensor network is composed of
a  large  number of  tiny  sensing devices  deployed  in  a region  of
interest.  Each  device  has  processing  and  wireless  communication
capabilities, which enable it to sense its environment, to compute, to
store information  and to  deliver report messages  to a  base station
\cite{Sudip03}.  One of  the main design issues in  WSNs is to prolong
the network  lifetime, while  achieving acceptable quality  of service
for  applications. Indeed,  sensors  nodes have  limited resources  in
terms of memory, energy and computational power.

Since sensor nodes have limited battery life and since it is impossible to
replace batteries,  especially in remote and  hostile environments, it
is desirable that  a WSN should be deployed  with high density because
spatial redundancy can  then be exploited to increase  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. To
extend the lifetime of the network, the main idea is to take advantage
of the overlapping sensing regions of some sensor nodes to save energy
by turning  off some of them during  the sensing phase~\cite{Misra05}.
Obviously, the deactivation of nodes  is only relevant if the coverage
of the monitored  area is not affected. In  this paper, we concentrate
on  the area coverage  problem \cite{Nayak04},  with the  objective of
maximizing the network lifetime  by using an adaptive scheduling.  The
area of interest is divided into subregions and an activity scheduling
for sensor nodes is planned for each subregion.  In fact, the nodes in
a subregion  can be seen  as a cluster  where each node  sends sensing
data  to  the  cluster  head  or  the  sink  node.   Furthermore,  the
activities in a subregion/cluster can continue even if another cluster
stops due to too many  node failures.  Our scheduling scheme considers
rounds,  where a  round  starts  with a  discovery  phase to  exchange
information between sensors of the  subregion, in order to choose in a
suitable manner a sensor node  to carry out a coverage strategy.  This
coverage strategy  involves the solving  of an integer  program, which
provides the  activation of the sensors  for the sensing  phase of the
current round.

The remainder of the paper is organized as follows. The next section
% Section~\ref{rw}
reviews the related work in the field.  Section~\ref{pd} is devoted to
the    scheduling     strategy    for    energy-efficient    coverage.
Section~\ref{cp} gives  the coverage model formulation,  which is used
to schedule  the activation  of sensors.  Section~\ref{exp}  shows the
simulation results obtained using the discrete event simulator OMNeT++
\cite{varga}. They  fully demonstrate  the usefulness of  the proposed
approach.  Finally,  we give  concluding remarks and  some suggestions
for future works in Section~\ref{sec:conclusion}.

\section{Related works}
\label{rw}
\indent In this section, we only review some recent works dealing with
the coverage lifetime maximization  problem, where the objective is to
optimally  schedule  sensors'  activities  in  order  to  extend  WSNs
lifetime.

In \cite{chin2007}, the author proposed a novel distributed heuristic,
called Distributed Energy-efficient  Scheduling for k-coverage (DESK),
which  ensures  that  the  energy  consumption among  the  sensors  is
balanced and the lifetime  maximized while the coverage requirement is
maintained.   This  heuristic works  in  rounds,  requires only  1-hop
neighbor information,  and each sensor  decides its status  (active or
sleep)   based   on  the   perimeter   coverage   model  proposed   in
\cite{Huang:2003:CPW:941350.941367}.    More    recently,   Shibo   et
al. \cite{Shibo}  expressed the coverage  problem as a  minimum weight
submodular  set cover  problem  and proposed  a Distributed  Truncated
Greedy Algorithm (DTGA) to solve it. They take in particular advantage
from  both temporal and  spatial correlations  between data  sensed by
different sensors.

The  works  presented  in  \cite{Bang,  Zhixin, Zhang}  focus  on  the
definition  of  coverage-aware,  distributed  energy-efficient  and
distributed clustering  methods respectively.  They aim  to extend the
network  lifetime  while ensuring  the  coverage.   S.   Misra et  al.
\cite{Misra05} proposed a localized algorithm which conserves energy and
coverage  by  activating  the  subset  of  sensors  with  the  minimum
overlapping area. It preserves  the network connectivity thanks to the
formation of the  network backbone. J.~A.~Torkestani \cite{Torkestani}
designed a Learning  Automata-based Energy-Efficient Coverage protocol
(LAEEC)  to construct  a Degree-constrained  Connected  Dominating Set
(DCDS)  in   WSNs.   He  showed   that  the  correct  choice   of  the
degree-constraint  of DCDS  balances the  network load  on  the active
nodes and leads to enhance the coverage and network lifetime.
 
The main  contribution of our approach addresses  three main questions
to build a scheduling strategy.\\
%\begin{itemize}
%\item 
{\indent \bf  How must the  phases for information  exchange, decision
  and sensing be  planned over time?}  Our algorithm  divides the timeline into rounds.  Each round contains 4 phases: Information Exchange,
Leader Election, Decision, and Sensing.

%\item 
{\bf What are the rules to decide which node has to be turned on
  or off?}  Our algorithm tends to limit the overcoverage of points of
  interest  to avoid  turning on  too many 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  a decision?}  A  leader node should
make such  a decision. Our work  does not consider only  one leader to
compute and to  broadcast the scheduling decision to  all the sensors.
When  the network  size increases,  the network  is divided  into many
subregions and the decision is made by a leader in each subregion.
%\end{itemize}

\section{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 provision.
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 areas  called subregions and
then  our coverage  protocol  will be  implemented  in each  subregion
simultaneously.   Our protocol  works in  rounds fashion  as  shown in
figure~\ref{fig1}.

\begin{figure}[ht!]
\centering
\includegraphics[width=85mm]{FirstModel.eps} % 70mm
\caption{Multi-round coverage protocol}
\label{fig1}
\end{figure} 

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 the sensing task.  This protocol is
more reliable  against an unexpected node failure  because it works
in rounds.   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 round starts, 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  their
information (including their residual energy) at the beginning of each
round.  However,   the  pre-sensing  phases   (INFO  Exchange,  Leader
Election,  Decision) are energy  consuming for  some nodes,  even when
they do not  join the network to monitor the  area. Below, we describe
each phase in more details.

\subsection{Information exchange phase}

Each sensor node $j$ sends  its position, remaining energy $RE_j$, and
the number of local neighbours  $NBR_j$ to all wireless sensor nodes in
its subregion by using an INFO  packet and then listens 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:}

%The working phase works in rounding fashion. Each round include 3 steps described as follow :

\subsection{Leader election phase}
This  step includes 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  round.  All the  sensor  nodes cooperate  to
select 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  in order  of priority  are: larger
number  of neighbours,  larger remaining  energy, and  then in  case of
equality, larger index.

\subsection{Decision phase}
The  WSNL will  solve an  integer  program (see  section~\ref{cp})  to
select which sensors will be  activated in the following sensing phase
to cover  the subregion.  WSNL will send  Active-Sleep packet  to each
sensor in the subregion based on the 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{Sensing phase}
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 asleep)  for sensing task is
the same  for all wireless sensor  nodes in the  network.  Each sensor
will receive  an Active-Sleep  packet from WSNL  informing it  to stay
awake or to go to sleep  for a time  equal to  the period of  sensing until
starting a new round.

%\subsection{Sensing coverage model}
%\label{pd}

%\noindent We try to produce an adaptive scheduling which allows sensors to operate alternatively so as to prolong the network lifetime. For convenience, the notations and assumptions are described first.
%The wireless sensor node use the  binary disk sensing model by which each sensor node will has a certain sensing range is reserved within a circular disk called radius $R_s$.
\indent We consider a boolean  disk coverage model which is the most
widely used sensor coverage model in the literature. Each sensor has a
constant sensing range $R_s$. All  space points within a disk centered
at  the sensor with  the radius  of the  sensing range  is said  to be
covered by this sensor. We also assume that the communication range $R_c \geq 2R_s$ ~\cite{Zhang05}. 



%\begin{figure}[h!]
%\centering
%\begin{tabular}{cc}
%%\includegraphics[scale=0.25]{fig1.pdf}\\ %&amp; \includegraphics[scale=0.10]{1.pdf} \\
%%(A) Figure 1 &amp; (B) Figure 2
%\end{tabular}
%\caption{Unit Circle in radians. }
%\label{fig:cluster1}
%\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 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.

\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.
%\begin{figure}[h!]
%\centering
%\begin{tabular}{cc}
%%\includegraphics[scale=0.25]{fig2.pdf}\\ %&amp; \includegraphics[scale=0.10]{1.pdf} \\
%%(A) Figure 1 &amp; (B) Figure 2
%\end{tabular}
%\caption{Wireless Sensor Node Area Coverage Model.}
%\label{fig:cluster2}
%\end{figure}
By  knowing the  position (point  center: ($p_x,p_y$))  of  a wireless
sensor node  and its $R_s$,  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.

\indent  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{fig2}) as follows:\\
$(p_x,p_y)$ = point center of wireless sensor node\\  
$X_1=(p_x,p_y)$ \\ 
$X_2=( p_x + R_s * (1), p_y + R_s * (0) )$\\           
$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_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_{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})) $.

 \begin{figure}[h!]
%\centering
% \begin{multicols}{6}
\centering
%\includegraphics[scale=0.10]{fig21.pdf}\\~ ~ ~(a)
%\includegraphics[scale=0.10]{fig22.pdf}\\~ ~ ~(b)
\includegraphics[scale=0.25]{principles13.eps}
%\includegraphics[scale=0.10]{fig25.pdf}\\~ ~ ~(d)
%\includegraphics[scale=0.10]{fig26.pdf}\\~ ~ ~(e)
%\includegraphics[scale=0.10]{fig27.pdf}\\~ ~ ~(f)
%\end{multicols} 
\caption{Sensor node represented by 13 primary points}
\label{fig2}
\end{figure}

\section{Coverage problem formulation}
\label{cp}


\indent   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  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 $X_{j}$,  which  determine  the
activation of  sensor $j$ in the  sensing phase of 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 &amp; \mbox{if the primary point $p$ is covered} \\
 &amp; \mbox{by sensor node $j$}, \\
  0 &amp; \mbox{otherwise.}\\
\end{array} \right.
%\label{eq12} 
\end{equation}
The number of active sensors that cover the primary point $p$ is equal
to $\sum_{j \in J} \alpha_{jp} * X_{j}$ where:
\begin{equation}
X_{j} = \left \{ 
\begin{array}{l l}
  1&amp; \mbox{if sensor $j$  is active,} \\
  0 &amp;  \mbox{otherwise.}\\
\end{array} \right.
%\label{eq11} 
\end{equation}
We define the Overcoverage variable $\Theta_{p}$ as:
\begin{equation}
 \Theta_{p} = \left \{ 
\begin{array}{l l}
  0 &amp; \mbox{if the primary point}\\
    &amp; \mbox{$p$ is not covered,}\\
  \left( \sum_{j \in J} \alpha_{jp} * X_{j} \right)- 1 &amp; \mbox{otherwise.}\\
\end{array} \right.
\label{eq13} 
\end{equation}
\noindent More precisely, $\Theta_{p}$ 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
by:
\begin{equation}
U_{p} = \left \{ 
\begin{array}{l l}
  1 &amp;\mbox{if the primary point $p$ is not covered,} \\
  0 &amp; \mbox{otherwise.}\\
\end{array} \right.
\label{eq14} 
\end{equation}

\noindent Our coverage optimization problem can then be formulated as follows\\
\begin{equation} \label{eq:ip2r}
\left \{
\begin{array}{ll}
\min \sum_{p \in P} (w_{\theta} \Theta_{p} + w_{U} U_{p})&amp;\\
\textrm{subject to :}&amp;\\
\sum_{j \in J}  \alpha_{jp} X_{j} - \Theta_{p}+ U_{p} =1, &amp;\forall p \in P\\
%\label{c1} 
%\sum_{t \in T} X_{j,t} \leq \frac{RE_j}{e_t} &amp;\forall j \in J \\
%\label{c2}
\Theta_{p}\in \mathbb{N} , &amp;\forall p \in P\\
U_{p} \in \{0,1\}, &amp;\forall p \in P \\
X_{j} \in \{0,1\}, &amp;\forall j \in J
\end{array}
\right.
\end{equation}
\begin{itemize}
\item $X_{j}$  : indicates whether or  not the sensor  $j$ is actively
  sensing in the round (1 if yes and 0 if not);
\item $\Theta_{p}$  : {\it overcoverage}, the number  of sensors minus
  one that are covering the primary point $p$;
\item $U_{p}$  : {\it undercoverage},  indicates whether or  not the primary point
  $p$ is being covered (1 if not covered and 0 if covered).
\end{itemize}

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  positive values.  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.
 
\section{Simulation results}
\label{exp}

In this section, we conducted  a series of simulations to evaluate the
efficiency  and the relevance of  our approach,  using the  discrete event
simulator  OMNeT++  \cite{varga}. We  performed  simulations for  five
different densities varying from 50 to 250~nodes. Experimental results
were  obtained from  randomly generated  networks in  which  nodes are
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 fully cover the sensing
  field with the given sensing range.
10~simulation  runs  are performed  with
different  network  topologies for  each  node  density.  The  results
presented hereafter  are the  average of these  10 runs.  A simulation
ends  when  all the  nodes  are dead  or  the  sensor network  becomes
disconnected (some nodes may not be  able to send, to a base station, an
event they sense).

Our proposed coverage protocol uses the radio energy dissipation model
defined by~\cite{HeinzelmanCB02} as  energy consumption model for each
wireless  sensor node  when  transmitting or  receiving packets.   The
energy of  each node in a  network is initialized  randomly within the
range 24-60~joules, and each sensor node will consume 0.2 watts during
the sensing period, which will last 60 seconds. Thus, an
active  node will  consume  12~joules during the sensing  phase, while  a
sleeping  node will  use  0.002  joules.  Each  sensor  node will  not
participate in the next round if its remaining energy is less than 12
joules.  In  all  experiments,  the  parameters  are  set  as  follows:
$R_s=5~m$, $w_{\Theta}=1$, and $w_{U}=|P^2|$.

We  evaluate the  efficiency of  our approach by using  some performance
metrics such as: coverage ratio,  number of active nodes ratio, energy
saving  ratio, energy consumption,  network lifetime,  execution time,
and number of stopped simulation runs.  Our approach called strategy~2
(with two leaders)  works with two subregions, each  one having a size
of $(25 \times 25)~m^2$.  Our strategy will be compared with two other
approaches. The first one,  called strategy~1 (with one leader), works
as strategy~2, but considers only one region of $(50 \times 25)$ $m^2$
with only  one leader.  The  other approach, called  Simple Heuristic,
consists in uniformly dividing   the region into squares  of $(5 \times
5)~m^2$.   During the  decision phase,  in  each square,  a sensor  is
randomly  chosen, it  will remain  turned  on for  the coming  sensing
phase.

\subsection{The impact of the number of rounds on the coverage ratio} 

In this experiment, the coverage ratio measures how much the 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.  Figure~\ref{fig3} shows  the impact  of the
number of rounds on the  average coverage ratio for 150 deployed nodes
for the  three approaches.  It can be  seen that the  three approaches
give  similar  coverage  ratios  during  the first  rounds.  From  the
9th~round the  coverage ratio  decreases continuously with  the simple
heuristic, while the two other strategies provide superior coverage to
$90\%$ for five more rounds.  Coverage ratio decreases when the number
of rounds increases  due to dead nodes. Although  some nodes are dead,
thanks to  strategy~1 or~2,  other nodes are  preserved to  ensure the
coverage. Moreover, when  we have a dense sensor  network, it leads to
maintain the full coverage for a larger number of rounds. Strategy~2 is
slightly more efficient than strategy 1, because strategy~2 subdivides
the region into 2~subregions and  if one of the two subregions becomes
disconnected,  the coverage   may  be  still  ensured   in  the  remaining
subregion.

\parskip 0pt 
\begin{figure}[h!]
\centering
\includegraphics[scale=0.37]{CR1.eps} %\\~ ~ ~(a)
\caption{The impact of the number of rounds on the coverage ratio for 150 deployed nodes}
\label{fig3}
\end{figure} 

\subsection{The impact of the number of rounds on the active sensors ratio} 

It is important to have as few active nodes as possible in each round,
in  order to  minimize  the communication  overhead  and maximize  the
network lifetime.  This point is  assessed through the  Active Sensors
Ratio (ASR), which is defined as follows:
\begin{equation*}
\scriptsize
\mbox{ASR}(\%) = \frac{\mbox{Number of active sensors 
during the current sensing phase}}{\mbox{Total number of sensors in the network
for the region}} \times 100.
\end{equation*}
Figure~\ref{fig4} shows  the average active nodes ratio versus rounds
for 150 deployed nodes.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.37]{ASR1.eps} %\\~ ~ ~(a)
\caption{The impact of the number of rounds on the active sensors ratio for 150 deployed nodes }
\label{fig4}
\end{figure} 

The  results presented  in figure~\ref{fig4}  show the  superiority of
both proposed  strategies, the strategy  with two leaders and  the one
with a  single leader,  in comparison with  the simple  heuristic. The
strategy with one leader uses less active nodes than the strategy with
two leaders until the last  rounds, because it uses central control on
the whole sensing field.  The  advantage of the strategy~2 approach is
that even if a network is disconnected in one subregion, the other one
usually  continues  the optimization  process,  and  this extends  the
lifetime of the network.

\subsection{Impact of the number of rounds on the energy saving ratio} 

In this experiment, we consider a performance metric linked to energy.
This metric, called Energy Saving Ratio (ESR), is defined by:
\begin{equation*}
\scriptsize
\mbox{ESR}(\%) = \frac{\mbox{Number of alive sensors during this round}}
{\mbox{Total number of sensors in the network for the region}} \times 100.
\end{equation*}
The  longer the ratio  is,  the more  redundant sensor  nodes are
switched off, and consequently  the longer the  network may  live.
Figure~\ref{fig5} shows the average  Energy Saving Ratio versus rounds
for all three approaches and for 150 deployed nodes.

\begin{figure}[h!]
%\centering
% \begin{multicols}{6}
\centering
\includegraphics[scale=0.37]{ESR1.eps} %\\~ ~ ~(a)
\caption{The impact of the number of rounds on the energy saving ratio for 150 deployed nodes}
\label{fig5}
\end{figure} 

The simulation  results show that our strategies  allow to efficiently
save energy by  turning off some sensors during  the sensing phase. As
expected, the strategy with one leader is usually slightly better than
the second  strategy, because the  global optimization permits  to turn
off more  sensors. Indeed,  when there are  two subregions  more nodes
remain awake  near the border shared  by them. Note that  again as the
number of  rounds increases  the two leaders'  strategy becomes  the most
performing one, since it takes longer  to have the two subregion networks
simultaneously disconnected.

\subsection{The percentage of stopped simulation runs}

We  will now  study  the percentage  of  simulations, which  stopped due  to
network  disconnections per round  for each  of the  three approaches.
Figure~\ref{fig6} illustrates the percentage of stopped simulation
runs per  round for 150 deployed  nodes.  It can be  observed that the
simple heuristic is  the approach, which  stops first because  the nodes
are   randomly chosen.   Among  the  two   proposed  strategies,  the
centralized  one  first  exhibits  network  disconnections.   Thus,  as
explained previously, in case  of the strategy with several subregions
the  optimization effectively  continues as  long  as a  network in  a
subregion   is  still   connected.   This   longer   partial  coverage
optimization participates in extending the network lifetime.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.36]{SR1.eps} 
\caption{The percentage of stopped simulation runs compared to the number of rounds for 150 deployed nodes }
\label{fig6}
\end{figure} 

\subsection{The energy consumption}

In this experiment, we study the effect of the multi-hop communication
protocol  on the  performance of  the  strategy with  two leaders  and
compare  it  with  the  other  two  approaches.   The  average  energy
consumption  resulting  from  wireless  communications  is  calculated
by taking into account the  energy spent by  all the nodes when  transmitting and
receiving  packets during  the network  lifetime. This  average value,
which  is obtained  for 10~simulation  runs,  is then  divided by  the
average number of rounds to define a metric allowing a fair comparison
between networks having different densities.

Figure~\ref{fig7} illustrates the energy consumption for the different
network  sizes and  the three  approaches. The  results show  that the
strategy  with  two  leaders  is  the  most  competitive  from  the energy
consumption point  of view.  A  centralized method, like  the strategy
with  one  leader, has  a  high energy  consumption  due  to  many
communications.   In fact,  a distributed  method greatly  reduces the
number  of communications thanks  to the  partitioning of  the initial
network in several independent subnetworks. Let us notice that even if
a  centralized  method  consumes  far  more  energy  than  the  simple
heuristic, since the energy cost of communications during a round is a
small  part   of  the   energy  spent  in   the  sensing   phase,  the
communications have a small impact on the network lifetime.

\begin{figure}[h!]
\centering
\includegraphics[scale=0.37]{EC1.eps} 
\caption{The energy consumption}
\label{fig7}
\end{figure} 

\subsection{The impact of the number of sensors on execution time}

A  sensor  node has  limited  energy  resources  and computing  power,
therefore it is important that the proposed algorithm has the shortest
possible execution  time. The energy of  a sensor node  must be mainly
used   for  the  sensing   phase,  not   for  the   pre-sensing  ones.   
Table~\ref{table1} gives the average  execution times  in seconds
on a laptop of the decision phase (solving of the optimization problem)
during one  round.  They  are given for  the different  approaches and
various numbers of sensors.  The lack of any optimization explains why
the heuristic has very  low execution times.  Conversely, the strategy
with  one  leader, which  requires  to  solve  an optimization  problem
considering  all  the  nodes  presents  redhibitory  execution  times.
Moreover, increasing the network size by 50~nodes   multiplies the time
by  almost a  factor of  10. The  strategy with  two leaders  has more
suitable 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.

\begin{table}[ht]
\caption{EXECUTION TIME(S) VS. NUMBER OF SENSORS}
% title of Table
\centering

% used for centering table
\begin{tabular}{|c|c|c|c|}
% centered columns (4 columns)
      \hline
%inserts double horizontal lines
Sensors number &amp; Strategy~2 &amp; Strategy~1  &amp; Simple heuristic \\ [0.5ex]
 &amp; (with two leaders) &amp; (with one leader) &amp; \\ [0.5ex]
%Case &amp; Strategy (with Two Leaders) &amp; Strategy (with One Leader) &amp; Simple Heuristic \\ [0.5ex]
% inserts table
%heading
\hline
% inserts single horizontal line
50 &amp; 0.097 &amp; 0.189 &amp; 0.001 \\
% inserting body of the table
\hline
100 &amp; 0.419 &amp; 1.972 &amp; 0.0032 \\
\hline
150 &amp; 1.295 &amp; 13.098 &amp; 0.0032 \\
\hline
200 &amp; 4.54 &amp; 169.469 &amp; 0.0046 \\
\hline
250 &amp; 12.252 &amp; 1581.163 &amp; 0.0056 \\
% [1ex] adds vertical space
\hline
%inserts single line
\end{tabular}
\label{table1}
% is used to refer this table in the text
\end{table}

\subsection{The network lifetime}

Finally, we  have defined the network  lifetime as the  time until all
nodes  have  been drained  of  their  energy  or each  sensor  network
monitoring an area has become disconnected.  In figure~\ref{fig8}, the
network  lifetime for different  network sizes  and for  both strategy
with two leaders  and the simple heuristic is  illustrated.  We do not
consider anymore the centralized strategy with one leader, because, as
shown  above, this strategy  results in  execution times  that quickly
become unsuitable for a sensor network.

\begin{figure}[h!]
%\centering
% \begin{multicols}{6}
\centering
\includegraphics[scale=0.37]{LT1.eps} %\\~ ~ ~(a)
\caption{The network lifetime }
\label{fig8}
\end{figure} 

As  highlighted by figure~\ref{fig8},  the network  lifetime obviously
increases when  the size of  the network increases, with  our approach
that leads to  the larger lifetime improvement.  By  choosing the best
suited nodes, for  each round, to cover the region  of interest and by
letting the other ones sleep in order to be used later in next rounds,
our  strategy efficiently prolonges  the network  lifetime. Comparison
shows that  the larger the sensor  number is, the  more our strategies
outperform the simple heuristic.   Strategy~2, which uses two leaders,
is the best  one because it is robust to  network disconnection in one
subregion. It also means that  distributing the algorithm in each node
and  subdividing the  sensing field  into many  subregions,  which are
managed independently and simultaneously,  is the most relevant way to
maximize the lifetime of a network.

\section{Conclusion and future work}
\label{sec:conclusion}

In this paper,  we have addressed the problem of  the coverage and the
lifetime optimization in WSNs. To cope with this problem, the field of
sensing  is  divided into  smaller  subregions  using  the concept  of
divide-and-conquer method,  and then a  multi-rounds coverage protocol
will optimize  coverage and  lifetime performances in  each subregion.
The  proposed  protocol  combines  two efficient  techniques:  network
leader election  and sensor activity scheduling,  where the challenges
include how to select the  most efficient leader in each subregion and
the best  representative active  nodes. Results from  simulations show
the relevance of the proposed  protocol in terms of lifetime, coverage
ratio,  active  sensors  ratio,  energy  saving,  energy  consumption,
execution  time, and  the number  of  stopped simulation  runs due  to
network  disconnection.  Indeed,  when  dealing with  large and  dense
wireless  sensor networks,  a  distributed approach  like  the one  we
propose  allows   to  reduce  the   difficulty  of  a   single  global
optimization problem by partitioning  it in many smaller problems, one
per subregion,  that can  be solved more  easily.  In future  work, we
plan to  study a  coverage protocol which  computes all  active sensor
schedules in only one step for many rounds,  using optimization  methods
such as  swarms optimization or evolutionary algorithms.
% use section* for acknowledgement
%\section*{Acknowledgment}

\bibliographystyle{IEEEtran}
\bibliography{bare_conf}

\end{document}


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