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IEEE TRANSACTIONS ON NETWORK AND SERVICE MANAGEMENT, VOL. 11, NO. 2, JUNE 2014
Design of a Probability Density Function
Targeting Energy-Efficient Node Deployment in
Wireless Sensor Networks
Subir Halder and Sipra DasBit, Member, IEEE
Abstract—In wireless sensor networks the issue of preserving
energy requires utmost attention. One primary way of conserving
energy is judicious deployment of sensor nodes within the network area so that the energy flow remains balanced throughout
the network and prevents the problem of occurrence of energy
holes. Firstly, we have analyzed network lifetime, found node
density as the parameter which has significant influence on
network lifetime and derived the desired parameter values for
balanced energy consumption. Then to meet the requirement
of energy balancing, we have proposed a probability density
function (PDF), derived the PDF’s intrinsic characteristics and
shown its suitability to model the network architecture considered
for the work. A node deployment algorithm is also developed
based on this PDF. Performance of the deployment scheme is
evaluated in terms of coverage-connectivity, energy balance and
network lifetime. In qualitative analysis, we have shown the
extent to which our proposed PDF has been able to provide
desired node density derived from the analysis on network
lifetime. Finally, the scheme is compared with three existing
deployment schemes based on various distributions. Simulation
results confirm our scheme’s supremacy over all the existing
schemes in terms of all the three performance metrics.
Index Terms—Energy balance, network lifetime, node deployment, probability density function, wireless sensor network.
I. I NTRODUCTION
HE advent of efficient short range radio communication
and advances in miniaturization of computing devices
have made possible for large-volume commercial production
of wireless sensor nodes as well as large-scale real-world
deployment of the same to form a wireless sensor network
(WSN). Such a network typically suffers from a number of
unavoidable problems such as resource constrained nodes,
random node deployment sometimes in an unattended open
field where it is very difficult to replace/ recharge battery etc.
So the network as a whole must minimize the energy usage
in order to enable untethered and unattended operation for an
extended period of time. Therefore, a critical consideration
in designing such WSNs is conserving energy to maximize
the post deployment network lifetime [1]. The rate of energy
T
Manuscript received May 11, 2013; revised October 10 and December 28,
2013. The associate editor coordinating the review of the paper and approving
it for publication was I.-R. Chen.
S. Halder is with the Department of Computer Science and Engineering, Dr. B. C. Roy Engineering College, Durgapur, India 713206 (e-mail:
[email protected]).
S. DasBit is with the Department of Computer Science and Technology,
Bengal Engineering and Science University, Shibpur, India 711103 (e-mail:
[email protected]).
Digital Object Identifier 10.1109/TNSM.2014.031714.130583
depletion in the network primarily depends on the deployment
nature of the nodes. The nature of deployment, on the other
hand, mainly depends on the application environment [2].
In WSNs, nodes can be deployed either randomly or in
pre-determined manner. In random deployment, nodes are
deployed randomly, generally in an inaccessible terrain. For
example, in the application domain of disaster recovery or
in forest fire detection, nodes are dropped by helicopter
in random manner [3]. On the contrary, in pre-determined
deployment, the locations of the nodes are specified. This
type of deployment is used in applications when sensors are
expensive or when their operations are significantly affected
by their positions. The applications include placing imaging
and video sensors, populating an area with highly precise
seismic sensors, underwater WSN applications, monitoring
manufacturing plants etc.
One important way of conserving energy is by uniform
energy consumption or load distribution throughout the network. Non-uniform dissipation of energy in any part of the
network may stop functioning of that part of the network
leading to a phenomenon known as the energy hole problem
[4]. Sometimes even after the network lifetime is over, due
to energy hole problem a substantial amount of energy still
remains in the nodes leading to significant wastage of energy.
The energy hole problem arises when more data are transmitted by certain nodes of the network than the other nodes
resulting in extra energy dissipation of those nodes [5]. Therefore, if any part of the network is affected by the energy hole
problem, the whole network gets affected badly as uneven
consumption of energy in the network leads to premature
shortening of network lifetime. To avoid this, care should be
taken during node deployment such that energy dissipation of
all the nodes takes place uniformly ensuring load balancing
throughout the network.
A. Motivation
Due to the nature of operation of WSN, nodes near the
sink bear the major share of data forwarding compared to
the nodes in rest of the network. So it is a common problem
that nodes near the sink get drained off more quickly than
the other nodes, thereby, creating energy holes near the sink
resulting in loss of connectivity while most of the nodes are
alive. Further, it is well known that sensor nodes have limited
battery life and it is sometimes infeasible to replenish energy
via battery replacements and therefore prolonging network
c 2014 IEEE
1932-4537/14/$31.00 HALDER and DASBIT: DESIGN OF A PROBABILITY DENSITY FUNCTION TARGETING ENERGY-EFFICIENT NODE DEPLOYMENT IN WIRELESS . . .
lifetime is of utmost importance. In such a network with
energy constraints, in order to prolong network lifetime, nodes
should be deployed with varying densities depending upon its
position from the sink. One important issue that arises in such
energy-constrained networks is to avoid energy holes in order
to improve a network lifetime. To be more specific, we are
interested in finding the answer for the following question-“Is
it possible that all sensor nodes die simultaneously irrespective
of their positions from the sink so that there will be no energy
hole in a WSN, thereby, prolonging network lifetime?” This
motivates us to explore a solution by providing varying node
density in different parts of the network area based upon its
proximity from the sink.
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able to provide desired network lifetime. In addition to
the contribution in [6], here we show the impacts of
routing and medium access control (MAC) protocols on
the performance of the scheme.
The rest of the paper is organized as follows. In Section II,
results from the literature review are presented. The network
model considered for the present work is presented in Section III. Analysis on network lifetime is done in Section IV.
Section V presents the proposed node deployment scheme
along with the proposed PDF based on which the scheme is
developed. In Section VI, the performance of the scheme is
evaluated based on both qualitative and quantitative analyses.
Finally the paper is concluded with some mention about the
future scope of the work in Section VII.
B. Contributions and Organization
One of the solutions to avoid energy hole problem in WSN
while maintaining coverage and connectivity is to deploy
nodes with varying densities based on their positions from the
sink. On the contrary, another promising approach may be to
design a dedicated distribution function which perfectly conforms to such a need of varying densities. We have proposed
a PDF based on which we have developed a node deployment
strategy which not only keeps energy hole problem away from
the network but also ensures enhancement of network lifetime
while maintaining coverage and connectivity of the network.
To the best of our knowledge we are the first to propose such
a tailored-made PDF to be used in node deployment to avoid
energy hole problem. The preliminary version of this work was
reported in [6]. Our proposed PDF based node deployment
strategy is pre-determined in nature. We extend our earlier
work [6] in several aspects. The main contributions of this
paper are as follows:
• Unlike [6] here, we analyze the method of controlling
network lifetime and found node density as a major
parameter which has significant role to control network
lifetime. The main question to be addressed is: How many
nodes per unit area should be deployed in different parts
of the sensor field in order to achieve energy balancing
throughout the network and enhancement of network
lifetime? The desired node densities derived out of the
said analysis, guarantee that all the nodes exhaust their
energy at the same time, and hence, energy balancing is
achieved.
• Based on the analysis, we propose a PDF and derive
the PDF’s intrinsic characteristics e.g. expectation, covariance etc.
• We develop a node deployment algorithm based on the
PDF. It provides the node density to avoid energy hole
problem, thereby, achieves enhanced network lifetime
while maintaining coverage and connectivity of the network.
• Performance of the scheme is evaluated both through
qualitative and quantitative analyses. In qualitative analysis unlike [6], we analyze whether the PDF based scheme
has been able to achieve the desired target set prior to the
designing of PDF, towards energy hole elimination.
• Similar to qualitative analysis, we show through simulation the extent to which our proposed PDF has been
II. L ITERATURE R EVIEW
Many works have been reported so far that deal with the
issue of balancing the load throughout the network with a goal
to reduce the energy hole problem for prolonging network lifetime. All these works have been conducted through different
approaches for achieving this goal. Each type of the above
schemes has their own strengths and limitations. We have
categorized the existing works based on their commonality
in approaches which are presented below.
A. Transmission Paths Based Strategies
In each of the works described below, nodes choose routing
path differently.
1) Data Traffic and Distance Based Strategies: In these
types of strategies, network lifetime is prolonged by making
the nodes choose routing path judiciously considering data
traffic along the path and distance. The distance considered is
either transmission distance between the node and the sink or
the distance between the node and neighbouring nodes towards
sink.
Azad and Kamruzzaman [7] have proposed energy balanced
transmission range regulation policies for maximizing network
lifetime in WSNs with corona based architecture. Firstly,
they have analyzed and found two parameters- ring thickness
and hop size - responsible for energy balancing. They have
proposed a transmission range regulation scheme of each
node and determined the optimal ring thickness and hop size
for maximizing network lifetime. Simulation results show
substantial improvements in terms of network lifetime and
energy usage distribution over existing policies. However,
before implementation of the proposed transmission policies,
it requires significant computation to determine the optimal
ring thickness and hop size. Song et al. [8] have presented
two algorithms viz. centralized and distributed for the same
corona based network architecture. Network lifetime has been
optimized using a proposed decision factor computed by
selecting the right transmission range of nodes in each corona.
They have claimed that the algorithms not only reduce the
complexity for searching right transmission range of node but
also obtain results approximated to the optimal solution.
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IEEE TRANSACTIONS ON NETWORK AND SERVICE MANAGEMENT, VOL. 11, NO. 2, JUNE 2014
2) Probability Based Strategies: These types of strategies
employ probability based routing path selection with a target
to balance energy consumption and enhance network lifetime.
Boukerche et al. [9] have studied the problem of energybalanced data propagation in corona based WSNs for uniform
and non-uniform deployments. The authors have proposed a
density based probabilistic data propagation protocol towards
balancing the energy consumption. In each step a node in
a corona that holds data on-line, calculates the probability
of data delivery either by hop-by-hop or directly to the sink
based on the density information of the neighbouring coronas.
Finally, the authors have shown that the proposed protocol
works well for both uniform and non-uniform network deployments. Powell et al. [10] have proposed a probability
based data propagation algorithm where the nodes compute
off-line the probability of number of times data sent directly
to the sink and data sent to a next hop neighbour. They
have observed that by controlling the ratio of these two,
energy consumption in each layer/slice is balanced but individual node in each slice is not well balanced. It is due to
the unequal number of neighbouring nodes in the adjacent
slices. Jarry et al. [11] have analyzed the data gathering
and network lifetime maximization problems and based on
the analysis, the authors have designed probabilistic on-line
distributed routing strategies for different network structures.
They have formulated the conditions to compute an energybalanced data propagation pattern for WSNs. Cheng et al. [12]
have proposed a spatio-temporal compressive data gathering
mechanism, where nodes send/forward sensory data to the sink
as per a predefined probability. Since the data is forwarded
with predefined probability, only a fraction of the readings
from each node are transmitted to the sink, leading to reduced
traffic and prolonged lifespan.
B. Deployment Based Strategies
The following works on enhancement of network lifetime
are based on judicious node deployment.
Wu et al. [13] have proposed a non-uniform node distribution, where, the number of nodes to be distributed in a layer is
determined based on the minimum number of nodes required
in the upper adjacent layer. They have concluded that only subbalanced energy depletion in the network is possible. Chang
et al. [14] have proposed two node deployment strategies viz.
distance-based and density-based strategies, with an objective
for balanced energy consumption among the nodes. They have
shown that the proposed strategies can efficiently balance
energy consumption of each node and prolong network lifetime. Wang et al. [15] have given an analytical model for the
coverage and network lifetime issues using a 2-D Gaussian
distribution. They have proposed two deployment algorithms
that achieve larger coverage and longer network lifetime using
limited number of sensor nodes. They have concluded that the
proposed algorithms effectively increase network lifetime with
polynomial time complexity. Liu et al. [16] have proposed a
node deployment algorithm for optimizing target surveillance
in WSNs. The proposed deployment algorithm is designed
in such a way that one can obtain predetermined network
lifetime by deploying minimum number of nodes. Through
extensive simulation, the authors have shown that the proposed
algorithm gives close-to-optimal solution.
C. Mobile Sink/Agent Based Strategies
In these types of strategies, load among all the nodes are
distributed in a balanced manner by changing the position of
the sink or using mobile agent so that network lifetime is
enhanced.
Luo and Hubaux [17] have proposed a deployment algorithm by considering mobile sink where the nodes lying nearer
the sink keep on changing resulting in even distribution of load
among the nodes. They have also proposed an appropriate
routing protocol which supports mobility of sink. Ammari
and Das [18] have also provided three different solutions for
eliminating energy hole problem. In one of the three solutions,
the authors have proposed a localized energy aware Voronoi
diagram based data forwarding protocol considering homogeneous nodes and mobile sink. Lin et al. [19] have developed an
energy balancing scheme for network lifetime maximization
in WSNs using mobile agents. They have designed an energy
prediction strategy by means of which mobile agents know
about the remaining energy of all sensor nodes. Accordingly
the mobility of agents is controlled so that the nodes with less
remaining energy can communicate through mobile agent and
avoid long-distance communication, thereby, evading uneven
energy consumption.
In the above, a number of existing works on node deployment are reviewed under different categories. Our scheme
generally belongs to the category of ‘Deployment Based
Strategies.’ The existing works reviewed under this category
either have not used any mathematical model to implement
the node deployment scheme or even if any such model
is used, it has not addressed energy balance and network
lifetime together. Our scheme, on the contrary, presents a node
deployment strategy based on a proposed PDF and addresses
energy balance and network lifetime together.
III. N ETWORK M ODEL
In this section, we describe the network architecture along
with the assumptions. Definitions of coverage and connectivity
in association with sensing and communication model respectively are described next. The energy model is presented at
the end of this section.
A. Architecture
The authors in [20] have proposed a realistic network model
for WSNs where the network area is covered by a set of
concentric circles centered at the sink. They have also provided
an in-depth discussion of its real life implementation. Further,
in [17] it is proved that in case of the said network model,
for enhancement of network lifetime, the best position for a
sink is the center of the circle. Primarily these two works
[17], [20] along with other relevant works [13], [15], [21]
motivate us to consider present network architecture. However,
in this architecture, energy consumption in annuli can be
kept balanced but individual nodes in the annuli may remain
imbalanced [10] and this has been mitigated by considering
q-switch [13] routing policy elaborated in Section VI.B.1.
HALDER and DASBIT: DESIGN OF A PROBABILITY DENSITY FUNCTION TARGETING ENERGY-EFFICIENT NODE DEPLOYMENT IN WIRELESS . . .
207
adding further complexity in theoretical analysis. However,
in simulations to make the assumption realistic, additionally
we consider a real MAC protocol for investigating the impact
of the protocol on network lifetime.
Nr
LN
a
2r
r
L1
L2
Sink
a
Fig. 1.
Layered network area.
We consider a square shaped network area a × a which
is covered by a set of uniform-width coronas or annuli
(Fig. 1) [13], [21]. Each such annulus is designated with
width r as layer. The sink is considered to be located at
the center of the network area and responsible for collecting
data from the sensors nodes. Nodes are placed in different
layers surrounding the sink. A layer is identified as Li where
i = 1, 2, . . . , N . Here i = 1 indicates the layer nearest to
the sink andi = N indicates the layer farthest from the sink
a
.
where N = 2r
We assume that all the sensor nodes are homogeneous with
respect to their initial energy, sensing and communication
ranges while an unlimited amount of energy is set for the sink.
The nodes are static and need to be distributed in each layer
within the network with certain node density. The node density
[5], [13] is the ratio of number of nodes in an area (layer) and
the area (of the layer). It is also assumed that there is no
local coordination among the nodes and therefore, the nearby
nodes report same events to the sink. Periodic data gathering
applications are considered where sensory data generation rate
is proportional to the area (1 square unit) irrespective of the
shape of the network area. Given a unit area, if the data
generation rate is ρ bits per sec, it is meant that this unit
area generates ρ bits/sec of data to be transmitted towards the
sink. So, the data that needs to be reported from a given area
a × a is ρ × a × a bits. The data is collected by the nodes
and sent to the sink through multi-hop communication after a
unit time-interval. A single sink is responsible for gathering
the sensors’ data and controlling network operations. Further,
we have assumed that both the time-intervals between two
successive generations of data per area and two successive
collections of data at the node as 1 sec. During theoretical
analysis, for the sake of simplicity, an ideal MAC layer with
no collision and retransmission is assumed and that does
not lose any generality. The reason for not considering real
MAC in theoretical analysis is the complexity involved in real
MAC, such as need of control packet (TDMA scheduling),
retransmission in case of collision (CSMA/CA) etc. Moreover, some of the overheads such as retransmission are a
real time parameter which needs probabilistic formulation
B. Sensing and Communication Model
1) Sensing Model: We define a unit area to be covered,
if every point in that area is within the sensing range of at
least one active node. Alternatively, if each point in an area
is covered by at least α nodes, it is known as α-coverage.
The nodes perform observation [22] at an angle of 360◦ . The
maximal circular area centered around a node ν that can be
covered by it is defined as its sensing area S(ν). The radius of
S(ν) is called the ν’s sensing range [14] Rs . We assume the
relationship between r and Rs must satisfy the condition r ≤
2Rs [6], [15] for covering the network area (Fig. 1). In case of
1-coverage, node density (λ) = 1/S (ν). Also, the coverage
area C(X) of a set of nodes X is the union of the sensing
areas covered by each node in X i.e. C(X) = ∪∀ν∈X S (ν)
[22].
2) Communication Model: We define a network as connected if any active node can communicate with any other
active node, possibly using intermediate nodes as relays, so
that the information collected by the nodes can be relayed
back to sink [23]. We assume that two nodes can directly
exchange messages if their Euclidean distance is not larger
than the communication range Rc . Further, we assume that
the relationship between r and Rc must satisfy the condition
r ≤ Rc [6], [15] for ensuring connectivity in the network area
(Fig. 1).
Lemma 1: For a given network area a × a, in order to
maintain connectivity of the network, the number
of layers
a
and
to
maintain
(N ) stands in relation with Rc as N = 2R
c
coveragethe number
of layers (N ) stands in relation with Rs
a
.
as N = 4R
s
Proof: If the radius of each layer in the layered architecture
is r, then the distance between the center of the network
area and the periphery of a layer-i is i × r (Fig. 1). Now
for the farthest layer (i.e., layer-N ), the distance between the
center and the periphery of layer-N is N × r and N r = a2
a
(Section III.A) or, N r = 2R
(putting r = Rc ). Therefore, for
c
a given network area a×a, in order to maintain connectivity of
the network
relationship between Rc and N should stand
the
a
as N = 2Rc .
a
a
Similarly, we have N = 2r
= 4R
(replacing r = 2Rs ).
s
Therefore, for a given network area a× a, in order to maintain
coverage of the network
relationship between Rs and N
the
a
should stand as N = 4Rs .
C. Energy Model
We have considered the first order radio model [15] as
the energy model where energy consumption of a node is
dominated by its wireless transmissions and receptions; so
the other energy consumption factors such as for sensing
and processing are neglected. According to this radio model,
energy consumed by a node for transmission and reception is
as follows:
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IEEE TRANSACTIONS ON NETWORK AND SERVICE MANAGEMENT, VOL. 11, NO. 2, JUNE 2014
Energy consumption for transmitting n-bits data over a
distance d is et (n, d) = eelec n + eamp nd2 . Since we have
assumed the transmission range of a node as Rc , the above
can be rewritten by replacing d with Rc :
et (n, Rc ) = eelec n + eamp nRc2 = et n
(1)
where et = eelec + eamp Rc2 and et is energy required to
transmit one bit of data.
Energy consumption for receiving n-bits data is
er (n) = eelec n = er n
(2)
where er is energy required to receive one bit of data and
er = eelec .
IV. A NALYSIS ON N ETWORK L IFETIME
This section presents an analysis on network lifetime with
an objective to find out the parameter(s) which have significant
influence on network lifetime so that the lifetime can be
extended by controlling the parameter values. In presence of
several existing state-of-the-art definitions of network lifetime
[1] and lifetime of a node, the present work considers the
following definitions throughout the paper.
Definition 1. The node lifetime (measured in unit time)
is defined as the time when the initial energy of a node is
completely drained out so that it is neither able to transmit
its own data nor able to forward any data. It is measured
as the ratio of the initial energy of a node and the energy
consumption rate of the node. Considering initial energy of
each sensor node as ε0 and the energy consumption rate of a
node in layer-i as ECRi , the lifetime of a node in layer-i is
ε0
.
LTi = ECR
i
Definition 2. The network lifetime is defined as the time
interval from the beginning of the network operation until
the proportion of dead nodes exceeds a certain threshold,
which may result in loss of coverage of a certain region,
and/or network partitioning [5]. If the total number of nodes
in layer-i is Ti , the lifetime of a layer-i in the network is
ε0 ×Ti
ε0
Ti ×ECRi = ECRi .
It is same as the lifetime of a node in layer-i as defined
previously. If any of the layer’s lifetime terminates which
causes loss of coverage within the layer, it results in termination of network lifetime. Therefore, the network lifetime can
be determined by the
shortest
lifetime of a layer and it is
ε0
.
expressed as min∀i ECR
i
We assume that a node requires minimum energy consumption for transmitting/forwarding data to the sink. Therefore, the
node should transmit data to the sink via the shortest path i.e. a
node in layer-i (Section III.A) requires i hops to transmit data
to the sink. This routing policy is basically a simplified version
of the q-switch, shortest path routing policy proposed in [13]
which is used in our scheme and briefed in Section VI.B.1. In
this analysis the simplification is made by considering a node
in layer-i finds only one node in layer-(i − 1) to forward data
towards sink. Further, we assume that area of a layer-i (Fig. 1)
is Ai = π(2i − 1)r2 , where r is the width of a layer and λi
is the node density of layer-i. Therefore, the total number of
nodes in layer-i is given as Ti = λi × Ai for i = 1, 2, . . . , N .
The nodes of all the layers except the farthest layer from the
sink spend their energy by transmitting their own sensory data,
receiving data from the nodes of adjacent layers farther away
from the sink and forwarding the received data. Nodes in the
farthest layer spend energy only for transmitting their own
data.
Therefore, the data transmission rate of a node (for transmitting its own sensory data) in layer-i is
ρ
ρ × Ai
= .
λi × Ai
λi
(3)
Further, a node in layer-i receives data from the nodes of layer(i + 1) i.e. the nodes in layer-(i + 1) transmit data towards
the nodes of layer-i to forward the same towards the sink. So,
the average data transmission rate of layer-(i + 1) towards a
node in layer-i is
λi+1 π (2i + 1) r2 ρ
λi+1 (2i + 1) ρ
λi+1 Ai+1 ρ
.
=
=
λi Ai
λi π (2i − 1) r2
λi (2i − 1)
The rate of data relayed by a node of layer-i is the sum of the
above quantity for all the farther layers i.e.,
N
h=i+1 λh (2h − 1) ρ
.
(4)
λi (2i − 1)
Therefore, the total data transmission rate per node in layer-i
(mi ) can be obtained from (3) and (4) as follows:
N
ρ
h=i+1 λh (2h−1)ρ
+
for i = 1, 2, . . . , (N − 1)
λ
λi (2i−1)
i
mi =
ρ
for i = N
λN
(5)
The first component of the above expression of mi i.e. ρ/λi
is for transmitting the node’s sensory data and the second
component is for forwarding the outward adjacent layers’ data.
Now the energy consumption rate of a node in layer-i for
transmission is:
For transmitting the node’s own data:
ρ
× et
λi
(6)
where et is energy required to transmit one bit of data.
For transmitting the relay data received from the farther
adjacent layers:
N
h=i+1 λh (2h − 1) ρ
× et .
(7)
λi (2i − 1)
So the energy consumption
rate of each node in layer-i due
Tx
is computed from (6) and (7) as
to transmission ECRi
(8), at the top of the next page.
Similarly, we calculate the energy consumption
rate of each
node in layer-i for receiving ECRiRx data from the farther
layers as follows:
N
h=i+1 λh (2h − 1) ρ
Rx
er
ECRi =
(9)
λi (2i − 1)
where i = 1, 2, . . . , (N − 1) and er is the energy required to
receive one bit of data. Hence the total energy consumption
rate of each node in a layer-i is ECRi = ECRiTx + ECRiRx .
HALDER and DASBIT: DESIGN OF A PROBABILITY DENSITY FUNCTION TARGETING ENERGY-EFFICIENT NODE DEPLOYMENT IN WIRELESS . . .
ECRiTx =
⎧ ⎨
⎩
ρ
λi +
ρ
λN et
N
λh (2h−1)ρ
λi (2i−1)
h=i+1
209
et
for i = 1, 2, . . . , (N − 1)
for i = N
(8)
So, the total energy consumption rate of the nodes of layer-i
is
ECRi =
ECRiTx + ECRiRx
Tx
ECRN
for i = 1, 2, . . . , (N − 1)
.
for i = N
(10)
We know, energy depletion across the network is balanced
[13] when all the nodes of the network exhaust their energy
at the same time. To be more specific, if balanced energy
depletion is attained in the network then all nodes located in
any layer have the same lifetime. Alternatively, all the nodes
exhaust their energy at the same time. Therefore, for energy
balancing, the following condition must be satisfied-ECRi =
ECRi+1 = · · · = ECRN .
Now rewriting (10) with the help of (8) and (9), we have
the equation at the top of the next page.
After simplification and basic transformations, we obtain
(11), at the top of the next page.
Equation (11) implies that the ratio of node density between
two consecutive layers depends on layer number i and the total
number of layers N . Further, the node density in a layer is
uniform but this node density varies in different layers. The
nature of variation is such, that the node density is maximum
at the layer nearest to the sink and it decreases in the layers
farther away form the sink i.e., λ1 > λ2 > · · · > λN .
Now, (11) is a non-linear equation and computation of λi is
fairly complex. However, it suggests that one can compute
λi , for i = 1, 2, . . . , (N − 1), if λN is known. Considering
1-coverage (Section III.B.1), λN = 1/S (ν) where S (ν) =
πRs2 . Moreover, the balanced energy consumption can be
obtained in different layers of the network if the nodes are
distributed in accordance with the desired node density given
in (11).
When the nodes are distributed according to (11) to get
balanced energy consumption, we can ensure that all the nodes
deployed in the sensor field completely deplete their energy
at the same time. Now, from the Definition 1/2, the lifetime
of a node/the network lifetime (LT ) can be expressed as:
ε0
for i = 1, 2, . . . , N . Replacing the denominator
LTi = ECR
i
with the help of (10), (8) and (9) we have LTi in (12), at the
top of the next page.
As we have assumed earlier that the nodes in a layer
report data to the sink in minimum hops, therefore the derived
network lifetime (see (12)) provides the upper bound of the
network lifetime. Also we can say that the upper bound of the
network lifetime is achievable by controlling node density λi
in each layer, as given in (11).
V. P ROBABILITY D ENSITY F UNCTION BASED N ODE
D EPLOYMENT (PDFND)
From the analysis of network lifetime provided in the
previous section, we have found that the ratio of node density
between two consecutive layers depends on layer number
i and the total number of layers N . Further, for balanced
Fig. 2.
3-D graph (surface plot) of the PDF.
energy consumption, the required node density is maximum
in the layer nearest to the sink and it decreases in the layers
farther away from the sink. Considering these observations
and taking guidelines from the analysis, in this section, we
have designed a PDF targeting its implementation in lifetimeenhancing node distribution in WSNs. Also, we have presented
a node deployment algorithm based on the proposed PDF [6].
A. Proposed Probability Density Function [6]
The mathematical domain under consideration is divided
into a number of concentric circles having radii increasing
arithmetically from r to (N × r ) with a difference of r .
In the mathematical domain, if (x, y) be a point and it lies
between circles (i − 1) and i, then the probability density at
that point is
2
k (2i − 1)
2
f x, y; N, i, r =
, ∀ (i − 1)2 r < x2 + y 2 ≤ ir N 2 i4
(13)
where i = 1, 2, . . . , N and k is a constant as follows:
−1
2
52
(2N − 1)
32
2
2
k = N π (r ) 1 + 4 + 4 + · · · +
.
2
3
N4
Fig. 2 is the 3-D graph of the proposed PDF. The characteristics of the PDF show decrease of the functional value with
increase in the value of i implying lower probability and vice
versa.
Theorem 1: The value of constant k is:
−1
2
52
(2N − 1)
32
2
2
k = N π (r ) 1 + 4 + 4 + · · · +
.
2
3
N4
Theorem 2: If the random variables X and Y follow a
proposed PDF with parameters N and i, then the cumulative
distribution function (CDF) of X and Y is given as
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IEEE TRANSACTIONS ON NETWORK AND SERVICE MANAGEMENT, VOL. 11, NO. 2, JUNE 2014
ρ
et +
λi
N
N
λh (2h − 1) ρ
ρ
h=i+2 λh (2h − 1) ρ
(et + er ) =
(et + er )
et +
λi (2i − 1)
λi+1
λi+1 (2i + 1)
h=i+1
λi = λi+1
LTi =
F [X ≤ x, Y ≤ y] =
2
kπ (r )
N2
i j=1
2i + 1
2i − 1
et (2i − 1) + (et + er )
et (2i + 1) + (et +
λi (2i−1)ε0
ρ(2i−1)et +(et +er ) N
h=i+1 λh (2h−1)ρ
λN ε0
ρ(2i−1)et
2
2
2
(2j − 1)
η −i
+
j4
i4
.
2
We choose (x, y) such that 0 ≤ x2 + y 2 ≤ η 2 (r ) , where
i ≤ η ≤ i + 1.
The proofs of Theorem 1 and Theorem 2 are omitted due
to space limitations. We refer the reader to Theorem 1 and
Theorem 2 in [6] for the detail proof.
Theorem 3: If the two random variables X and Y follow
the proposed PDF with parameters N and i, then the expectation of X and Y is given as
2 N 2
3
1
k (r ) 2
+
−
−
.
E [XY ] =
N 2 i=1 i
i3
i2
2i4
Proof: The expectation of the two variables X and Y with
parameters N and i can be given as
E [XY ] = E1 [XY ] + E2 [XY ] + · · · + EN [XY ] =
i
i=1
Ei [XY ]
(14)
where Ei [XY ] is the expectation of X and Y for a given
value of i. Now,
Ei [XY ]
Ei [XY ]
Ei [XY ]
=
=
=
k
N 2 i4
4k
N 2 i4
ir
xy
⎡
⎢
x⎢
⎣ dy dx
(ir )2 − x2
((i − 1) r )2 − x2
4 2
k (r ) 2
3
1
+
−
−
.
N2
i
i3
i2
2i4
⎤
y
⎥
dy ⎥
⎦ dx
0
(15)
So, replacing Ei [XY ] in (14) with the value of (15) we get
4 N k (r ) 2
2
3
1
E [XY ] =
+ 3− 2− 4 .
(16)
N 2 i=1 i
i
i
2i
Theorem 4: If the two random variables X and Y follow a
proposed PDF with parameters N and i, then the covariance
of X and Y is given as the equation at the top of the next
page.
Proof: From the definition of covariance [24] we know that
Cov (X, Y ) = E [XY ] − E [X] E [Y ].
We can find the covariance of the two discrete and random
variables X and Y for a particular value i in the domain. The
N
h=i+1
N
er ) h=i+2
λh (2h − 1)
(11)
λh (2h − 1)
for i = 1, 2, . . . , (N − 1)
(12)
for i = N
covariance of the two discrete and random variables X and Y
for the entire domain is obtained by summing different values
of the parameter i, where i = 1, 2, . . . , N
Cov (X, Y ) =
Cov (X, Y ) =
N
i=1
N
Covi (X, Y )
[Ei [XY ] − Ei [X] Ei [Y ]] .
i=1
We get the expectation of X and Y for a particular value of
i, i.e. Ei [XY ] from (15)
4 2
k (r ) 2
3
1
Ei [XY ] =
+ 3− 2− 4 .
N2
i
i
i
2i
The expectation of X for a particular value of i, i.e. Ei [X]
can be calculated as
k
Ei [X] =
x dy dx
N 2 i4
⎡
⎤
2
2
ir
(ir ) − x
⎢
⎥
4k
⎢
dy ⎥
Ei [X] =
x
⎣
⎦ dx
2
4
N i
2
2
0
((i − 1) r ) − x
3
3
4k (r )
3
1
+
−
Ei [X] =
.
3N 2
i2
i3
i4
From the above equations, the expectation of X in the entire
domain is
3 N 3
1
4k (r ) 3
+ 3− 4 .
E [X] =
(17)
3N 2 i=1 i2
i
i
As our network model is symmetric, so the expectation of
the random variables X and Y in the entire domain is same.
We can say that Ei [X] = Ei [Y ]. The covariance of random
variables X and Y is
N
2
N
Ei [XY ] −
Ei [X]
Cov (X, Y ) =
i=1
Cov (X, Y )
i=1
2
= E [XY ] − [E [X]] .
(18)
In (18), replacing by (16) and (17) we get covariance of
random variables X and Y as the second equation at the top
of the next page.
Although in preliminary version [6] of this work, the PDF
was proposed, the proofs of the Theorem 3 & 4 are provided
in this present work only.
HALDER and DASBIT: DESIGN OF A PROBABILITY DENSITY FUNCTION TARGETING ENERGY-EFFICIENT NODE DEPLOYMENT IN WIRELESS . . .
211
2 N
2
k (r )3 2
3
1
3
1
4 3
+ 3− 2− 4 −
Cov (X, Y ) =
+ 3− 4
r
N 2 i=1
i
i
i
2i
3 i2
i
i
2 N
2
k (r )3 2
3
1
3
1
4 3
+ 3− 2− 4 −
Cov (X, Y ) =
+ 3− 4
r
N 2 i=1
i
i
i
2i
3 i2
i
i
B. Proposed PDF-based Node Deployment
The PDF proposed in the previous section is discrete in
nature. Our objective is to deploy sensor nodes in the layered
network area (Fig. 1) with the proposed PDF. The PDF is
mapped with the node deployment in a layered network area
as follows: the parameter i represents the layer number for
both the proposed PDF and layered network area (see Fig. 1)
where i = 1, 2, . . . , N ; the parameter r of the proposed PDF
corresponds to the width r of the annuli/layer. Therefore, the
relationship between r and Rc is r ≤ Rc , whereas, between
r and Rs is r ≤ 2Rs . The density function is designed as
per the analysis in Section IV. It is a non-uniform one i.e.
the value of PDF is higher for the nodes deployed around the
sink whereas the value is lower as one moves away from the
sink. The PDF of deploying a sensor node at point f (x, y)
located in layer-i is given as follows: k (2i − 1) /N 2 i4 where
i = 1, 2, . . . , N and k is the proportionality constant.
Further, the probability for the nodes deployed within a
layer is k (2i − 1) Ai /N 2 i4 , where Ai is the area of layeri and k is as follows:
−1
N
2
2
2
4
(2i − 1) /i
(19)
k = N πr
i=1
where r is the width of the layer-i.
The area of layer-i is given as
" #
!
$
2
Ai = πi2 r2 − π (i − 1) r2 = π (2i − 1) r2 .
By replacing the value of Ai , the probability (pi ) of deploying
nodes at a layer-i is given as
#
$−1
2
pi = k (2i − 1) πr2 N 2 i4
.
(20)
The number of nodes in a layer-i (Ti ) is equal to the
probability (pi ) of deployment of nodes at layer-i multiplied
by the total number of nodes (Ttotal ) i.e. Ti = pi × Ttotal .
Therefore, the node density of a layer-i (for i = 1, 2, . . . , N )
according to our proposed PDF is given as
N
pi
pi
× Ttotal =
×
λ × Aj
λi =
j=1 j
Ai
Ai
k (2i − 1) πr2 N
λi =
(2j − 1) λj .
(21)
j=1
N 2 i4
The above expression implies that the node density determined
by the proposed PDF is controlled by the parameters i and N
and that conforms to the guideline provided in Section IV.
Further, the node density in a layer is uniform and probability
of deploying nodes is equal, but this node density as well as
probability varies in different layers. The nature of variation
is that the node density is maximum in the layer nearest to
the sink and it decreases in the layers farther away form the
sink. Therefore, we have λ1 > λ2 > · · · > λN .
C. Algorithm for Node Deployment
Algorithm 1 Input: a, r, Ttotal ; Output: λi // area parameter,
width of layer, and total number of nodes to be deployed
compute maximum number of layers N using Lemma 1
compute constant k using Theorem 1
for i = N to 1 do
compute pi using (20)
compute λi using (21)
end for
D. Illustrative Example
Let us consider a 200 × 200 sq unit area where 100 nodes
with Rc = 25 units are deployed employing the proposed
probability density function. The number of layers N is:
%
& %
&
a
200
N=
=
= 4 (using Lemma 1).
2 × Rc
2 × 25
Now replacing the values of N and Rc in (19), the value of
k can be computed as
−1
52
72
32
2
2
k = 4 π (25) 1 + 4 + 4 + 4
2
3
4
k = 0.004 [where r = Rc ].
Now the probability of deploying nodes in each of the
four layers is obtained by replacing k by 0.004, r by
25 and N by 4 in (20). For example, pi is obtained as,
2
pi = 0.49 (2i − 1) /i4 . Probability of deployment of nodes
in layer-1, p1 = 0.49. Similarly p2 = 0.27, p3 = 0.15 and
p4 = 0.09. Using (21), node density in each of the 4 layers
is as follows: in layer-1, λ1 = 0.00024 × 100 = 0.024, in
layer-2, λ2 = 0.0045, in layer-3, λ3 = 0.0015, in layer-4,
λ4 = 0.00065.
We observe that the node density in each layer obtained
from the algorithm conforms to the non-uniform nature of
the PDF. Therefore, it fulfils our objective of deploying more
number of nodes towards the sink and decreasing the number
of nodes as the distance from the sink increases.
Finally, we claim that the proposed deployment is feasible.
As reported in state-of-the-art works [25], [26] on design and
deployment of WSN, air-dropped deployment in a controllable
manner is feasible even in an inaccessible terrain. We propose
to compute the node density in each part (layer/annuli) of the
212
IEEE TRANSACTIONS ON NETWORK AND SERVICE MANAGEMENT, VOL. 11, NO. 2, JUNE 2014
Node density (node/sq. m)
network off-line prior to the actual deployment. At last, the
nodes are to be dropped (e.g. from helicopter) using a point
(sink) as the center following the pre-computed node densities
of the proposed PDF. One important application of PDFND
based node deployment is battlefield surveillance where more
detailed information is needed around the headquarters and
the sink may be placed at the headquarters.
Unlike the preliminary version [6], in the proposed node
deployment (Sections V.B, V.C, and V.D) we have considered
node density instead of number of nodes in each layer as the
parameter of concern for making the scheme energy balanced,
thereby, getting enhanced network lifetime. This is as per the
guideline of the analysis done in Section IV.
Now replacing the value of et and using (5), we have (using
energy model in Section III.C)
N
'
(
λh (2h − 1) ρ
mi et + h=i+1
er = mN eelec + eamp Rc20 .
λi (2i − 1)
After rearranging the above equation, we have the equation at
the top of the next page. Similar to the farthest layer, for the
rest of the layers we can have the equation at the top of the
next page.
Let us assume that the network lifetime under the regulated
transmission policy is LT . The lifetime maximization thus
can be formulated as the following optimization problem:
max LT
Subject to (22)–(26) at the top of the next page.
The objective function is the lifetime of the network. The
constraints (22) and (23) specify the transmission range of the
nodes for balanced energy consumption where the nodes are
located in the farthest layer and in the other layers respectively.
The constraint (24) limits the maximum transmission range of
Achieved node density
1.5
Desired node density
1
0.5
0
-0.5
-1
1
2
3
Layer number
4
5
Node density (node/sq. m)
(a) 5-layer network.
E. Formulating Network Lifetime Maximization as an Optimization Problem
In addition to solving network lifetime maximization by
varying node density in each layer, in this section we show
that the same can be achieved by regulating transmission
range of nodes in each layer. Precisely, we formulate network
lifetime maximization as transmission range regulation based
optimization problem. We assume that the nodes in the farthest
layer have the longest transmission range Rc0 (i = N ) and
the nodes in the other layers (i = 1, 2, . . . , (N − 1)) have
transmission range Rci in decreasing order. As mentioned in
Section IV the nodes in all the layers i (i = 1, 2, . . . , (N − 1))
consume energy for transmitting their own sensory data and
for carrying relay data. However, the nodes in the farthest layer
(i = N ) from the sink do not carry any relay data. So, let us
assume that the energy required to transmit one bit of data in
the farthest layer is et = eelec + eamp Rc20 . Further, in order
to obtain energy balancing the average energy consumption
rate of each layer must be same i.e., ECRi = ECRN
, where
ECRN is energy consumption rate of the nodes of layer-N
under the transmission range regulation policy.
Now using (8), (9), and (10) we get the following:
N
λh (2h − 1) ρ
ρ
ρ (et + er ) =
et + h=i+1
e.
λi
λi (2i − 1)
λN t
-3
2 ×10
-3
11 ×10
Achieved node density
9
Desired node density
7
5
3
1
-1
1
2
3
4
5
6
7
8
9
10
Layer number
(b) 10-layer network.
Fig. 3.
Node densities in each layer for various network sizes.
a node whereas constraint (25) specifies that the total energy
consumed by each node should not exceed its initial energy
reserve.
VI. P ERFORMANCE A NALYSIS
Performance of the present node deployment strategy is
measured based on two parameters such as energy balance and
network lifetime. Both qualitative and quantitative analyses are
presented here.
A. Qualitative Analysis
In this section we have analyzed the performance of the
proposed PDF qualitatively to show in which extent our node
deployment scheme (Section V) is close to fulfill the desired
objective (Section IV).
1) Energy Balance: The network is said to be energy
balanced when the nodes located in any layer have the same
lifetime. Based on this condition we have derived the desired
node density (λi ) of a layer using (11) (see Section IV) as
follows:
λi =
(2i + 1)
λi+1
(2i + 1)
et (2i − 1) + (et + er )
et (2i + 1) + (et +
N
h=i+1
er ) N
h=i+2
λh (2h − 1)
λh (2h − 1)
.
On the other hand, in our node deployment strategy, nodes are
deployed in different layers with varying node density (λi )
HALDER and DASBIT: DESIGN OF A PROBABILITY DENSITY FUNCTION TARGETING ENERGY-EFFICIENT NODE DEPLOYMENT IN WIRELESS . . .
Rc0
1
=
mN eamp
Rc0
1
=
mN eamp
1
Rci =
mi eamp
N
ρ e −
λN t
N
h=i+1 λh (2h − 1) ρ
er − mi eelec
λi (2i − 1)
N
h=i+1 λh (2h − 1) ρ
er − mN eelec
λi (2i − 1)
mi e t +
ρ e −
λN t
λh (2h − 1) ρ
er − mN eelec
λi (2i − 1)
h=i+1
mi e t +
1
Rci =
mi eamp
N
h=i+1 λh (2h − 1) ρ
er − mi eelec
λi (2i − 1)
213
12
12
12
, ∀TN
(22)
, ∀Ti , i < N
(23)
12
(N × r) ≥ Rc0 > Rci > 0 ∀Ti , 1 ≤ i ≤ N
ECRi × LTi − ε0 ≤ 0 ∀i ∈ {1, 2, . . . N }
(24)
(25)
ECRi , LTi ≥ 0 ∀i ∈ {1, 2, . . . N }
which is achieved node density and is given as (Section V.B,
see (21))
λi =
k (2i − 1) πr2 N
(2j − 1) λj .
j=1
N 2 i4
It is observed from both the above equations that node
density among the layers is non-uniform in nature but uniform
within the layers. Moreover, as A1 < A2 < · · · < AN , the
nature of variation of node density is that it is maximum in the
layer nearest to the sink and it decreases in the layers farther
away form the sink.
To see the effectiveness of the proposed scheme, Fig. 3 plots
both the desired and achieved node densities for two different
network sizes. After deriving (see Section IV) the node density
in the farthest layer for both the cases, desired and achieved
node densities of the remaining layers are calculated iteratively
using (11) and (21) respectively. It is clear from the plot, in
all the cases desired and achieved node densities are almost
same and that indicates the proposed scheme has been able to
achieve energy balancing.
2) Network Lifetime: The desired lifetime, according to
(12), is as the equation at the top of the next page. Now
following the method for calculating LTi (Section IV, see (12))
we derive LTi as the equation at the top of the next page. If
we compare LTi (desired) and LTi (achieved), as λi and λi
are found almost same, LTi and LTi are also same.
3) Coverage and Connectivity: In addition to the energy
balance and network lifetime, we have also measured coverage
and connectivity to show the extent of maintaining coverage
and connectivity by the proposed node deployment strategy.
This section formulates necessary constraints to be satisfied
for maintaining coverage and connectivity. It also contains a
couple of Lemmas along with the proofs with an objective to
show the extent of maintaining coverage and connectivity by
the PDF. To measure the coverage, the concept of coverage
(26)
density (Ci ) [27] has been used. If the sensing area S(ν)
(refer to Section III.B.1) of each node is mutually exclusive,
the coverage density of layer-i Ci is defined as
Ci =
Ti × S(ν)
.
Ai
If Ci = 1 i.e., 1-coverage (Section III.B.1), we say that
Ai is covered by minimum number of nodes and coverage
area of each node is mutually exclusive. If Ci > 1 i.e., αcoverage, we say that Ai is covered by more than the minimum
number of nodes and therefore, coverage area of a node is
overlapped with the coverage area of the other nodes in the
area. The sensing accuracy would increase proportionally with
the overlapping of coverage area, thus making the scheme
more robust against sensing failure.
Lemma 2: For a given network area, the proposed PDF
gives the coverage density of a layer-i as Ci = Ti /4 (2i − 1).
Proof: Let us consider Ti numbers of nodes are deployed
in layer-i. The S(ν) of each sensor is calculated as πRs2 where
Rs is the sensing radius of each sensor. So,
C (Ti ) = ∪∀ν∈Ti S(ν) = Ti × S(ν) = Ti × πRs2 .
From the definition of Ci we have
Ci =
Ti × S(ν)
T × πRs2
= i
.
Ai
(2i − 1) πr2
T ×πR2
T
i
s
i
Replacing r by 2Rs , we have Ci = (2i−1)π4R
2 = 4(2i−1) .
s
From the above expression it is observed that, the coverage
density of layer-i depends on the number of nodes deployed in
layer-i and it is inversely proportional to layer number i, which
suits the requirement for energy balancing (Section IV). Also,
one can achieve the desired coverage density by controlling
the number of deployed nodes Ti in various layers within the
network.
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IEEE TRANSACTIONS ON NETWORK AND SERVICE MANAGEMENT, VOL. 11, NO. 2, JUNE 2014
LTi
=
λi (2i−1)ε0
N
[ρ(2i−1)et +(et +er )
LTi =
λN ε0
ρ(2N −1)et
⎧
⎨
⎩
h=i+1
λh (2h−1)ρ]
ε0 (2i−1)λi
ρ(2i−1)et +(et +er ) N
h=i+1 λi (2h−1)ρ
ε0 λN
ρ(2N −1)et
Lemma 3: For a network, if coverage is ensured, connectivity of the network is also ensured.
Proof: We have discussed that when the network area is
covered by minimum number of nodes, then Ci = 1. Now
when the network is covered by minimum number of nodes,
the maximum distance between the two nodes is 2Rs . In
our communication model we have assumed that two nodes
can communicate with each other if the Euclidean distance
between them is less or equal to Rc and we have considered
Rc ≤ 2Rs . As the maximum distance between the nodes is
2Rs and Rc ≤ 2Rs , so we can say that the connectivity is
guaranteed if coverage is ensured.
B. Quantitative Analysis
The effectiveness of the proposed node deployment scheme,
reported in Section V.B is evaluated through simulation.
Moreover, all the theoretical claims made through qualitative
analysis presented in Section VI.A are justified by simulation
results. Simulation results of our scheme PDFND are compared with three existing node deployment schemes namely
non-uniform node distribution strategy (NNDS) [13], node
deployment with Gaussian distribution (NDGD) [15] and node
deployment with Uniform distribution (NDUD) [28].
1) Simulation Environment: The simulation is performed
using MATLAB (version 7.1). We have done qualitative analysis considering simplified q-switch routing [13]. However, in
simulation we have used the same routing protocol with no
simplification for all the three schemes. This routing protocol
is briefed along with the scheme [13] in the next paragraph.
In NNDS the authors have proposed a non-uniform node
distribution strategy for the uniform-width corona model. Here
the nodes are deployed in such a way that the node densities
in a corona increases in geometric proportion with common
ratio q(> 1) from corona (N − 1) to corona 1. We assume
that the number of nodes deployed at the farthest corona and
the common ratio are known a priori. Once the number of
nodes deployed at the farthest corona and the common ratio
are known, nodes for the remaining coronas are computed and
these computed numbers are exponentially increasing function
of the common ratio q. The NNDS uses q-switch routing
where the source node always selects one reachable relay
node with maximum remaining energy in its subsequent inner
layer to forward data. If there is more than one relay node
with the same maximum remaining energy, one of them is
chosen randomly. Once the source node selects the relay node,
it forwards the data of its own as well as those received
from the nodes of adjacent layers farther away from the sink.
The selected relay node repeats this process until the data
for i = 1, 2, . . . , (N − 1)
for i = N
for i = 1, 2, . . . , (N − 1)
for i = N
arrives at a node in layer-1, after which the data is sent to the
sink. Hence, the routing itself takes care of individual node’s
load balancing and that eliminates the problem [10] of annuli
architecture as stated in Section III.A.
In NDGD, the authors have considered that the nodes are
deployed using two dimensional Gaussian distribution and
node density function at point f (xi , yi ) as
−
1
f (x, y) =
e
2πσx σy
(x−xi )2
2
2σx
+
(y−yi )2
2
2σy
,
where σx and σy are the standard deviations for x and y
dimensions. Further, the authors have considered two deployment types: σx = σy and σx = σy . However, during
simulation we have considered σx = σy = σ, which conforms
to a disk model and that is similar to our network model. So,
the probability density function of deploying a sensor node
for point (x, y) is:
1 − x2 +y2 2
e 2σ .
2πσ 2
It is evident from the above equation that any two points in
the disk having same distance from the center-point, have the
same deployment probability.
In NDUD, nodes are uniformly and independently distributed in the layered network area, the probability fa that
a point is covered by sensor nodes is
f (x, y) =
2
fa = 1 − e−λπRs
where Rs is the sensing range of the nodes and λ is the node
density.
We simulate our work both under ideal scenario and realistic scenario. Here, by ideal scenario we mean the scenario considered during theoretical analysis (Section IV and
Section VI.A) i.e., simplified q-switch routing protocol, ideal
MAC layer and the energy consumption only for transmission
and reception. On the other hand, in realistic scenario we
consider q-switch routing protocol and real MAC protocol
which includes idle/sleep schedule of the nodes. Moreover,
unlike ideal scenario, in realistic scenario energy consumption
is considered for idle, sleeping and sensing in addition to
transmission and reception. The real MAC protocol has been
implemented by funneling-MAC [29]. The funneling-MAC is
a hybrid MAC protocol where TDMA (schedule-based) is used
in nodes located within a few hops from the sink whereas
CSMA/CA (contention-based) is used in nodes located far
away from the sink. The sink broadcasts a beacon for nodes
located within a smaller number of hops by controlling the
transmission power of the beacon. The nodes which receive
the beacon are considered as f -nodes and perform TDMA
HALDER and DASBIT: DESIGN OF A PROBABILITY DENSITY FUNCTION TARGETING ENERGY-EFFICIENT NODE DEPLOYMENT IN WIRELESS . . .
TABLE I
PARAMETER VALUES U SED IN S IMULATION
Average residual energy per node (Avg RE per node): It is
defined as the residual energy in a node of a particular layer
after the network lifetime ends. It is evaluated as follows:
Avg RE per node =
while the nodes that do not receive the beacon perform
CSMA/CA. During simulation we have considered the nodes
located within layer-2 use TDMA schedule whereas nodes
beyond layer-2 use CSMA/CA. We have considered energy
consumption rates for sensing, remaining idle and remaining
sleeping are 20%, 5%, 2.5% of the energy consumption rate of
reception respectively. Further, in simulation, all the funnelingMAC implementing parameter (e.g. slot size, superframe size,
moving average factor) values are considered same as in [29].
During implementation of all the schemes, we have deployed 500 and 2000 nodes for network with 5 and 10
layers respectively. For all the schemes, in order to have an
integer number of sensor nodes for each layer, the upper
ceil function is employed. We consider the energy cost to
run the transmitter/receiver radio circuitry per bit processed
(eelec ) as 50 nJ/bit. Also, we consider the energy used by the
transmitter amplifier (eamp ) to achieve an acceptable signal
to noise ratio (30 dB) as 10 pJ/bit/m2 . This setting requires
receiver sensitivity -52 dBm. Furthermore, we consider data
ready delay/measurement delay as 1 sec in our simulation. Our
routing protocol [13] and MAC protocol [29] does not rely
on any synchronization protocol and therefore, the synchronization parameters are not considered during simulation. For
NNDS, the common ratio q is considered as 2 and for NDGD,
standard deviation σ is considered as 70. All the parameters
and their corresponding values used for simulation are listed
in Table I.
Extensive simulation has been performed with a confidence
level of 95% and 5% accuracy. To achieve this, we collect
results in groups of 2,000 observations per group. In one
group, we obtain a mean out of 2,000 observations collected.
We run at least 5 groups to get a minimum of 5 means from
which we calculate the grand mean and estimate the difference
of the grand mean from the true mean with 95% confidence. If
the accuracy obtained is greater than 5%, we run more groups
and collect more observations until the specified 5% accuracy
requirement is achieved.
2) Simulation Metrics: Similar to qualitative analysis, (Section VI.A) energy balance, network lifetime and coverageconnectivity have been considered as performance metrics in
simulation. We define two more parameters namely energy
consumption rate per node and average residual energy per
node for evaluating the extent of energy balance in the
network. Further, we evaluate coverage-connectivity using the
parameter coverage density (Section VI.A.3).
Energy consumption rate per node (ER): It is defined as
energy consumption of a node per unit time.
215
Sum of residual energy of nodes in a layer
.
Number of nodes in the layer
We have conducted two sets of experiments. One set of
experiment is to compare simulation results with analytical
results considering network lifetime as the parameter. In the
other set of experiments, our scheme is compared with three
other competing schemes considering both ideal and realistic
scenarios. In this set of experiments again energy balance,
network lifetime and coverage-connectivity are considered as
performance metrics.
3) Comparison of Results (Analytical vs Simulated): In this
section, the analytical (Section IV) performance of the scheme
in terms of network lifetime is compared with the simulated
(Section V) performance and the results are plotted in Fig. 4.
Here both the set of results consider ideal scenario including
ideal MAC. To be more specific, to make the comparison
platform at par, both set of results are plotted considering
ideal MAC.
We observe from the plot that the nature of graph for the
analytical result is perfectly straight irrespective of network
sizes whereas the simulation result is fairly straight and that
indicates the algorithm has been able to provide almost perfect
energy-balanced network lifetime as desired by the theoretical
analysis. We also observe that the network lifetime decreases
with the increase of network sizes. This is because the data
traffic increases while the network size increases, especially
for the layers nearer the sink. Finally, the most important
observation is, for both the 5-layer (Fig. 4(a)) and 10-layer
(Fig. 4(b)) network sizes, analytical results and the average
of simulation results are almost same. The slight differences
between the analytical and simulated results are due to the
minute variation of desired and achieved node densities (refer
Fig. 3).
4) Comparison of Results (Competing schemes): This section compares our scheme’s performance with the three competing schemes considering both ideal and realistic scenarios.
a) Energy balancing [6]
In this section energy balancing of the scheme is evaluated
in terms of the following two parameters.
(1) ER (Energy consumption rate per node)
Fig. 5 shows the ER for different network sizes. We observe
that in PDFND, for both ideal and realistic scenarios, the
ER for a particular network size is constant for all the
layers and this rate varies with network sizes. Precisely, ER
increases with increase in network size. For example, in case
of ideal scenario, the ER is 1.01 mJ/sec for 5-layer network
whereas for 10-layer network it is 1.19 mJ/sec. Similarly, in
realistic scenario, these values are 1.21 mJ/sec and 1.45 mJ/sec
respectively. On the contrary, in NNDS, NDGD and NDUD
it is observed that the ER varies in different layers for a
given network size. Further, in NNDS, NDGD and NDUD,
irrespective of network size, nodes in layer-1 have maximum
ER and nodes in the farthest layer have the lowest ER.
Therefore, nodes deployed in the layers nearer the sink drain
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IEEE TRANSACTIONS ON NETWORK AND SERVICE MANAGEMENT, VOL. 11, NO. 2, JUNE 2014
9.5
816.23
For λ' (Simulated)
8.5
For λ (Analytical)
7.5
ER (mJ/sec)
Network lifetime (mins)
816.25
816.21
816.19
816.17
P DFND (Ideal)
NNDS (Ideal)
NDGD (Ideal)
NDUD (Ideal)
6.5
5.5
4.5
3.5
2.5
1.5
816.15
0.5
1
2
3
4
Layer number
5
1
2
(a) 5-layer network.
3
4
Layer nember
5
(a) 5-layer network.
683.11
16.5
683.106
For λ' (Simulated)
14.5
For λ (Analytical)
12.5
ER (mJ/sec)
Network lifetime (mins)
P DFND (Real)
NNDS (Real)
NDGD (Real)
NDUD (Real)
683.102
683.098
P DFND (Ideal)
NNDS (Ideal)
NDGD (Ideal)
NDUD (Ideal)
P DFND (Real)
NNDS (Real)
NDGD (Real)
NDUD (Real)
10.5
8.5
6.5
4.5
683.094
2.5
0.5
683.09
1
2
3
4
5
6
7
8
9
1
10
Layer number
(b) 10-layer network.
2
3
4
5
6
Layer number
7
8
9
10
(b) 10-layer network.
Fig. 4. Network lifetime for various network sizes (analytical vs simulated).
Fig. 5.
out their energy much more quickly in comparison to nodes
deployed in layers farther away from the sink. This justifies
our claim that PDFND is relatively more energy balanced
compared to all the competing schemes NNDS, NDGD and
NDUD.
Now for all the schemes, if we compare the results of ideal
and realistic scenario, it is observed that ER (realistic) in all
the cases is higher compared to ER (ideal). The additional
energy usage for realistic scenario is due to the implementation
of MAC protocol. Another important observation in realistic
scenario is that, ER nearer the sink is less compared to ER
away from the sink. As CSMA/CA is used in nodes away
from the sink, unlike TDMA, number of collisions, however
infrequent, is non-zero and this justifies the above result. In
numerical value, for PDFND irrespective of network sizes, the
average increase in ER, in realistic scenario compared to the
ideal scenario are 19% and 20%, for the nodes nearer the sink
and far away from the sink respectively. Similarly these values
are 20% and 22% for NNDS, 22% and 24% for NDGD, 5%
and 10% for NDUD.
(Fig. 6) nodes in each of the two layers viz. layer-5 and layer6, have drained off completely, though the nodes of other
layers in the network have sufficient energy for carrying out
normal network operation, causing the phenomenon known
as energy hole. Similarly, in NDUD, the energy in nodes
of layer-1 has drained off completely though the nodes of
other layers in the network have adequate energy for normal
network operation. So, both NDGD and NDUD suffer from
energy hole problem. In NNDS, the plots upto certain layers
starting from the nearest layer from the sink are relatively flat
compared to the results in rest of the layers and that implies
energy wastage caused by imbalance in energy consumption
among the layers. Therefore, NNDS also suffer from energy
imbalance problem affecting network lifetime. However, the
PDFND plot is almost a straight line indicating that all the
nodes in each layer exhaust energy almost completely ending
the network lifetime. For example, in case of ideal scenario, it
leaves less than 0.2 nJ energy for 5-layer network whereas for
10-layer network it is 0.32 nJ. Similarly, in case of realistic
scenario, it leaves less than 0.25 nJ energy for 5-layer network
whereas for 10-layer network it is 0.29 nJ. Therefore, we can
say that PDFND is energy balanced and utilizes energy, the
scarcest resource, more efficiently than the other deployment
schemes.
(2) Avg RE per node (Average residual energy per node)
Fig. 6 illustrates the comparison considering avg RE per
node as a performance metric. We observe that node deployment using NDGD or NDUD results in relatively abrupt
change in avg RE per node in each layer and this nature
remains independent of network size. For example, in NDGD
Energy consumption rate per node.
HALDER and DASBIT: DESIGN OF A PROBABILITY DENSITY FUNCTION TARGETING ENERGY-EFFICIENT NODE DEPLOYMENT IN WIRELESS . . .
1250
71
P DFND (Ideal)
P DFND (Real)
NNDS (Ideal)
NNDS (Real)
NDGD (Ideal)
NDGD (Real)
NDUD (Ideal)
NDUD (Real)
53
47
41
35
29
23
17
P DFND (Ideal)
NNDS (Ideal)
NDGD (Ideal)
NDUD (Ideal)
1100
Network lifetime (mins)
Avg. RE per node (J)
65
59
11
5
950
650
500
350
200
50
5
4
3
2
Layer number
1
1
2
(a) 5-layer network.
3
4
Layer number
5
(a) 5-layer network.
1050
65
P DFND (Ideal)
P DFND (Real)
NNDS (Ideal)
NNDS (Real)
NDGD (Ideal)
NDGD (Real)
NDUD (Ideal)
NDUD (Real)
53
47
41
35
29
23
P DFND (Ideal)
NNDS (Ideal)
NDGD (Ideal)
NDUD (Ideal)
950
Network lifetime (mins)
59
Avg. RE per node (J)
P DFND (Real)
NNDS (Real)
NDGD (Real)
NDUD (Real)
800
-1
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11
850
750
P DFND (Real)
NNDS (Real)
NDGD (Real)
NDUD (Real)
650
550
450
350
250
150
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-1
50
10
9
8
7
6
5
4
3
2
1
1
Layer number
2
(b) 10-layer network.
Fig. 6.
217
Average residual energy (RE) per node.
3
4
5
6
Layer number
7
8
9
10
(b) 10-layer network.
Fig. 7.
Network lifetime for various network sizes.
30
In this section network lifetime is evaluated for various
network sizes.
The graphs illustrated in Fig. 7 represent the network
lifetime for two different network sizes. For ideal scenario,
it is observed that the network lifetime of PDFND is 18.28%,
48.40%, and 350% more than that of NNDS, NDGD and
NDUD respectively for 5-layer network. For 10-layer network
it is 19.83%, 42.30%, and 380% more than that of NNDS,
NDGD and NDUD respectively. It is also observed that with
increase in network size network lifetime decreases, e.g. for 5layer network it is 816.21 mins whereas for 10-layer network
it is 683.06 mins. This is due to the fact that with increase
in network size, the nodes in the innermost layer need to
relay increased volume of data from the outer layers, thereby,
causing higher energy consumption. Moreover, in PDFND the
flat nature of the plot ensures that in all the layers, network
lifetime terminates in more or less same time as compared
to NNDS, NDGD and NDUD. This ensures that energy in
PDFND is balanced to a greater extent than all the competent
schemes.
Now if we compare the simulation results of network
lifetime, both for ideal and realistic scenarios, network lifetime
is reduced in realistic scenario, as there is additional energy
consumption due to the implementation of MAC protocol.
Further, in realistic scenario, irrespective of network sizes, the
Coverage density
b) Network lifetime [6]
PDFND
25
NDGD
20
NNDS
15
NDUD
10
5
0
1
Fig. 8.
2
3
Layer number
4
5
Coverage density for 5-layered network area.
reduction in network lifetime is less near the sink compared
to other parts of the network. To be more specific, in PDFND
when reduction is 19% near the sink, it is 20% in rest of the
network. Similarly, in NNDS, NDGD and NDUD these values
are 20% & 22%, 22% & 24% and 4% & 10% respectively.
As CSMA/CA is used in the entire network area except near
the sink, due to collision and retransmission additional energy
is consumed compared to near the sink where TDMA is used.
From the above observations, it is also revealed that the impact
caused by inclusion of realistic scenario on network lifetime
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is the highest in NDGD and the least in NDUD.
Although a subset of results on energy balancing and
network lifetime was presented in [6], here the entire result
set is compared (Figs. 5, 6, 7) with one additional competing
scheme NNDS [13]. Moreover, for all the competing schemes
including ours, an additional set of results are plotted using
real MAC.
c) Coverage and connectivity
In Fig. 8, we plot coverage density in all the layers for 5layered network. Our primary observation is that except the
scheme NDUD, in all the other schemes i.e., PDFND, NDGD
and NNDS, coverage density reduces in layers as the distance
of layers from the sink increases fulfilling the objective of
deploying more nodes near the sink. The next observation is
NDGD gives more overlapping sensing coverage in layer-2,
and NDUD in layer-5 but NDGD fails to give any overlapping
sensing coverage (see Section VI.A.3) in layer-5. To be more
specific, in layer-5, coverage density is less than one implying
NDGD’s incapability of providing coverage. On the other
hand, NDUD provides uniform coverage density in all the
layers but that does not provide energy balancing requirement.
However, NNDS provides coverage density as per the requirement of energy balancing. Finally, if we compare PDFND
with the most competent scheme NNDS, we find that in
PDFND coverage density of layers 1, 4, 5 are 74%, 58.36%,
2% more, respectively, than NNDS. However, in layer 2, 3
these values are marginally less (10.66%, 4.86%) than the
NNDS. Therefore, we claim that our scheme PDFND not only
provides coverage density as per the requirement of energy
balancing but also provides higher average coverage density
throughout the network compared to the most competing
scheme NNDS.
VII. C ONCLUSION AND F URTHER W ORK
In this work we have proposed a node deployment scheme
for multi-hop WSNs using a PDF defined by us. The target
of the scheme is to achieve energy balancing and enhancing
network lifetime while maintaining coverage and connectivity.
To start with, we have analyzed network lifetime and identified
node density as a parameter which has significant influence
on network lifetime. Then, theoretical formulation of node
density for balanced energy consumption is presented. Based
on the analysis of network lifetime we have designed a
PDF targeting its implementation in lifetime-enhancing node
distribution in WSNs. Intrinsic characteristics of the PDF
and its suitability for modeling the network architecture of
this work are discussed. A node deployment algorithm is
also developed based on the proposed PDF to implement the
scheme. Further, we have provided theoretical formulation of
coverage-connectivity, energy balancing, network lifetime and
have derived certain constraints, involving some important
network parameters, to be satisfied to achieve the target.
We claim that our scheme successfully achieves the target.
The claims are substantiated by performing both qualitative
and quantitative analyses. Finally, the results of quantitative
analysis are compared with three existing works [13], [15],
[28] on node deployment and that clearly demonstrates our
scheme’s dominance over the existing works.
As a future extension of our work, the deployment strategy
may be made more realistic by considering 3-D environment.
Moreover, the scheme may be analyzed with a target to obtain
optimal node density by considering various QoS parameters.
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Subir Halder received the B.Tech. degree in electronics and communication engineering and the
M.Tech. degree in computer science and engineering from Kalyani Government Engineering College,
Kalyani, India, in 2003 and 2006, respectively. He
is an Assistant Professor of Computer Science and
Engineering at Dr. B. C. Roy Engineering College, Durgapur, India. Currently, he is pursuing the
Ph.D. in computer science and technology at Bengal Engineering and Science University, India. His
current research interests include network modeling
and analysis, performance evaluation and optimization, and wireless sensor
networks. He has published research works in reputed conference proceedings
and journals in his field.
Sipra DasBit is a Professor and Head of the Department of Computer Science and Technology, Bengal
Engineering and Science University, Shibpur, West
Bengal, India. A recipient of the Career Award
for Young Teachers from the All India Council of
Technical Education (AICTE), she has more than 20
years of teaching and research experience. She has
published many research papers in reputed journals
and refereed international conference proceedings.
She also has two books and one book chapter on
mobile computing to her credit. Her current research
interests include wireless sensor networks and mobile computing.