A novel approach for finding optimal number of
cluster head in wireless sensor network
Ravi Ranjan
Subrat Kar
Bharti School of Telecommunication Technology
and Management
Indian Institute of Technology Delhi
Hauz Khas, New Delhi 110016 INDIA
Email: [email protected]
Bharti School of Telecommunication Technology
and Management
Indian Institute of Technology Delhi
Hauz Khas, New Delhi 110016 INDIA
Email: [email protected]
Abstract—Prolonging life time is the most important designing
objectives in wireless sensor networks (WSN). In WSN the total
amount of energy is limited, how to make best use of the limited
resource energy is a very important aspect in research of WSN. In
this paper we provide a method for determining the optimal number of cluster head for homogeneous sensor networks deployed in
different scenario using a reasonable energy consumption model.
In the first scenario nodes are thrown randomly, which can
be modeled using two-dimensional homogeneous spatial Poisson
point process. In the second scenario, nodes are deterministically
placed along the grid. For these two scenario we calculate the
average energy spend in the network in each round according to
LEACH protocol for both single and multi-hop between cluster
head and sink (base station) as a function of the probability of the
node to become a cluster head. Then we find optimal probability
of becoming a cluster head hence the optimal number of cluster
head that would lead to minimize the average energy spends
in the network for each round. Simulation results shows that
optimal probability of becoming a cluster heads that leads to
minimize energy dissipation in the network is not only depend
on the total number of nodes, but also depends on area of the
network A, packet length L and processing energy of nodes.
Index Terms—Wireless sensor networks (WSN), Cluster number, Stochastic geometry, Voronoi cell .
I. I NTRODUCTION
Wireless Sensor Networks are dense networks of low cost,
wireless nodes with limited ability for signal processing that
sense certain phenomena in the area of interest and report
their observations to certain base station for further analysis.
Distributed sensor networks enable a variety of application in
both civilian as well as military domains [1]. An important
application of sensor networks is surveillance of battle-field
or sensitive borders of countries. A simple way to monitor
such areas is to deploy sensors. Deployment could either be
deterministic, i.e., placing a node along grid points, or the
nodes could be deploying randomly. Because of sensor nodes’
self-constraints (generally tiny size, low-energy supply, weak
computation ability, etc.), it is challenging to develop a scalable,robust, and long-lived sensor networks. Much research
effort has focused on this area which results in many new
technologies and methods to address these problems in recent
years. The combination of clustering and data-fusion is one
of the most effective approaches to construct the large-scale
and energy-efficient data gathering sensor networks [2] [3].
Clustering topology is an important technology to prolong
the life-time of the network. LEACH [4] which is the first
clustering protocol has motivated the design of many other
protocols. It is a distributed algorithm for homogeneous sensor
networks where each sensor elects itself as a cluster-head
with some probability and cluster reconfiguration scheme is
used to balance the energy load. Cluster heads aggregate the
packets from there cluster members before forwarding them
to sink. By rotating the cluster head role uniformly among
all nodes, each node tends to expend the same energy over
the time. The LEACH allows only single hop cluster and
direct communication between CH to BS considering energy
consumption only in data collection and transmission. In our
proposed algorithm, the energy consumption is considered at
all phases - in the CH election, aggregation,data routing and
maintenance. Further we consider two type of deployement
scenarios one is random deployment and other is grid deployment and obtain optimal probability of node becoming
a cluster head using resonable energy model.We obtained
numerical result for optimal cluster probability which shows
that optimal values for these scenario will not only depend
on total number of node that was cosidered by leach but also
depends on trasmission range, packet length, circuit dissipation
energy, etc.
II. S YSTEM MODEL
In this paper we study the WSN scenario for homogeneous
sensor network. We have considered sensor nodes deployed in
two ways. In the first scenario (which is more realistic) nodes
located randomly on the plane according to a homogeneous
spatial Poisson point process. In the second scenario nodes
placed deterministically along grid points. We consider that all
nodes are quasi stationary and dispersed into an 2-D square
area of size A = a2 . Hence, the total number of nodes in such
area is also a Poisson random variable N with mean λA, where
A = a2 . Let us consider that p is the probability that a sensor
node becomes a cluster head. Therefore, average number of
CH is np, where p is the total number of nodes in that area. So
in case of random deployment λ0 = λp and λ1 = (1 − p)λ are
the intensity of the corresponding(independent) Poisson point
process. In this case clustering leads to formation of Voronoi
cells with CH being the nuclei of these cells. In the case of
grid, λ0 and λ1 are simply the number of cluster heads and
basic nodes. In this paper we use the same approach as given
in LEACH protocol, in which any node may become a cluster
head with some random probability and the node (not itself a
CH) join the cluster of the closest CH. After the network will
form, CH aggregate all data collected from the member and
transmit the aggregated data to the base station (BS).
A. Node architectures and energy models
A wireless sensor node typically consists of the following
three parts: (a) Sensor component (b) Transceiver component
(c) Signal processing component. The following assumptions
have been made for each of these components:
• For the sensor component: Sensor nodes are assumed
to sense a constant amount of information every round.
Energy consumed in sensing is Esense (L) = γL, where
γ is the power consumed for sensing a bit of data and L
is length of information in bits. In general, the value of
L is constant.
• For transceiver component: A simple model for the radio
hardware energy consumption is used.
L · εf s · d2 ; d < do
Etra (L, d) = L · Eelec +
(1)
L · εmp · d4 ; d ≥ do
•
Erec (L, d) = L · Eelec
(2)
p
where do = (εf s /εmp )
For signal processing component: This component conducts data fusion. The energy spent in aggregating k
steams of L bits row information into a single stream
is determined by
E{Aggr} (k, L) = kδL
(3)
The main energy dissipation of each node includes transmitter
(receiver) electronics and transmit amplifier. WSN application
is extensive which leads to complicated network environment.
Many uncertain factors are possible which will affect the
energy dissipation of the network. So we will use reasonable
energy consumption model to balance the energy load of the
network.
From the Eq.1 the energy dissipation is depend upon size of
each cluster. If the area of the region is fixed, then the size of
each cluster is determined by the number of CH.In our case
np is the number of CH so energy dissipation depends p hence
we have to find optimal value of p that minimize total energy
dissipation in the network.
B. Connectivity and coverage
To provide sensing coverage of region and successful use
of multihop communication for sensor network the condition
for the network connectivity and the area coverage must be
ensured[6]. Let λ be the intensity of poisson process and p
be the reliability probability of each node. The connectedness
probability of nodes and coverage of area is given as follows:
Pconnected and region is covered ≥ 1 − (1/γr)2 · e−πθ
2
pr 2 λ
(4)
We assume that all nodes are reliable i.e. p=1 and that the
sensing range of nodes is r. In a network dimensioning
problem a parameter () is provided such that the connectivity
and coverage probability be at least (1 − ). Therefore, we
require
1
1
log
(5)
λ≥
πθ2 pr2
εγr2
for all γ, θ > 0 : γ + 2θ = 1, We consider r2 < 1 for which
r.h.s of the Eqn.(4) above minimizeed as a function of θ under
the constraints of γ + 2θ = 1. so we can rewrite the
1
1
log
(6)
u(θ) =
πv 2 pr2
ε((1 − 2θ)r)2
It is noted that u(θ) approaches ∞ as θ approaches 0 as well as
1
2 . Hence, there is a point in between where u(θ) is minimized
since u(.) is continuous and lower bound by 0. Hence, the
constrains in(4)reduces to constraint of the form:
λ + θ ≥ u(θ) = a
(7)
where a is completely determined by , p,andr. In the case
of grid , the total number required is exactly λ. Hence the
connectivity requirement for unit area takes a form:
λ0 + λ1 =
1
r2
(8)
C. Data Aggregation model
We used most common model for data aggregation that
assumes a cluster head collects the packets from all the nodes
in its cluster, and after processing and fusion produces a single
packet. It is further assumed that irrespective of the number of
nodes in the cluster,the size of the size of aggregated packet is
constant,i.e., does not depend on number of packets that were
aggregated during data fusion.
III. O PTIMAL C LUSTERING A NALYSIS FOR T WO
D EPLOYMENT S CENARIOS
A. Random Deployed network
WE begin the solution by first considering an actual network, in which the primary energy will dissipates on three
parts: cluster formation process , data routing process and the
maintenance of the network. For LEACH protocol, clustering
process starts at the beginning of each round with some node
electing itself as CH with some random probability. After
becoming a CH it sends control information to the network; the
member nodes receives the information and decides to which
cluster they belong to. In second process the member sends the
collected sensed data to the CH they belong, the CH aggregates
these data and send it to the sink node. In the third process
CH sends real time control information to the network, the
member receive that control information and update its own
information. The third process goes along with the first two
processes. Now we will find energy spend by the network
in three different process for which we need to consider
results from stochastic geometry [5]. When the deployment
is random, each cluster form a Voronoi cell associated with
each CH as nuclei. We first find the expected number of
member nodes in a typical Voronoi cell associated with a
particular CH. We then find expected number of member
outside circle of radius r around a CH for the purpose of
finding average relaying load on a critical node. We follow the
approach used in to determine the expected number of member
associated with CH nodes. Let σ(Π1 ) denotes the sigma
algebra generated by the point process corresponding to the
CH nodes. Since member as well as CH nodes deployed using
a homogeneous point process, we can shift the origin to one
of the CH point and use Campbell’s theorem and Slivnyak’s
theorem to compute the expected number of member node
in a typical Voronoi cell. Let Π0 denotes the Voronoi cell
associated with CH node located at origin , and {xi ∈ Π0 }
denotes the set of all the member points. Then, 1{xi ∈Π0 } is
the indicator function which is one when a member node i
lies in a cell Π0 . Let E[Nv ] be the expected member in cell
Π0 where
X
E[Nv |N = n] ≈ E[Nν ] = E[
1{xi ∈Π0 } ]
{xi ∈Π0 }
Hence, the average length of a member points to its CH may
be expressed as
1
E[Lv | N = n]
= √
(12)
E[Nv | N = n]
2 λ1
As we have described, in each round, the energy spend in the
network can be divided in to three parts,(1) cluster formation
process (2) data routing process (3) network maintenance
process, so according to radio model of first order,
1.In the cluster formation process:
r=
E[ECH1 | N = n] =
E[Lctr ET r (LctrX
, Rmax )
X
+
Lctr ERx +
Lctr EDA
{xi ∈Π0 }
+Lctr ET r (Lctr , Di ) | N = n]
(13)
where, the expected value and mean square of distance
between CH and BS is given by
´ 1p 2
=
xi + yi2 dA
E[Di
|
N
=
n]
=
A A
p
´
´
´
π/4 a/ cos θ
1
2
2
xi + yi dA = 2/A 0
r.rdrdθ = 0.765a
A/2 A
0
´
E[Di2 | N = n] = A A1 x2i + yi2 dA = 23 a2qand maximum
7
0.917 ln( α
)
 
= E E 
value of transmission range is Rmax =
p1 λ
denotes the probability that node do not join any CH.

X
1{xi ∈Π0 } | σ(Π1 )
[EnonCH1 |N = n]
{xi ∈Π0 }
ˆ
2π
ˆ
∞
2
e−λ1 πx λ0 xdxdθ
=
0
0
The event that member point located at (x, θ) belongs to the
Voronoi cell Π0 is equivalent to the event that there is a
member point in a small area xdxdθ located at (x, θ), and
there is no other CH point in a circle of radius r around that
member point. From this, we get
E[Nv ] =
λ0
λ1
(9)
Using similar approach, we can find the expected number of
member node located within a distance of r from CH node as
follows:
P
E[Nv (r) = E[ {xi ∈Π0 } 1{xi ∈Π0 ,|xi |<r} ]
X
= E[E[
1{xi ∈Π0 ,|xi |<r} | σ(Π1 )]]
i ∈Π0 }
´ 2π ´{x
r −λ1 πx2
λ0 xdxdθ
e
0
0
=
which gives
E[Nv (r)] =
; α
= E[ERx + Lctr EDA | N = n]
+E[Lctr ET x (Lctr , |xi |) | N = n]
(14)
2. In the actual data routing process:
X
[ECH2 | N = n] =
E[
ERx (Ldata )
[EnonCH2
P{xi ∈Π0 }
(15)
+
{xi ∈Π0 } Ldata EDA
+ ET x (Ldata , Di ) | N = n]
X
| N = n] = E[
[ERx (Ldata , | |xi |) | N = n]
{xi ∈Π0 }
(16)
And 3. In network maintenance process:
P
E[ECH3 | N = n] =
E[ {xi ∈Π0 } ERx (Lctr )
X
+
Lctr EDA | N = n]
(17)
{xi ∈Π0 }
where D2 is distance between two CH which we can take as
2Rmax
[EnonCH3 | N = n] = E[ET x (Lctr ) + ET x (Lctr , |xi |)] (18)
2
λ0
(1 − e−λ1 πx )
λ1
(10)
These E[Nv (r)] number of critical nodes relay data of
E[Nv ] − E[Nv (r)] member node that are located outside the
circle of radius ‘r0 in the same Voronoi cell. Now total length
of the segments connecting to member node point to the
CH(nuclei) in a Voronoi cell, assume as Lv , where |x| is the
length of the vector x ∈ R2 . Then
E[Lv | N = n] = E[Lv ] =
λ0
]
2λ3/2
(11)
1) Direct communication: In the protocol that uses direct
communication CH sent its aggregated data directly to the base
station. So for total energy in each routing round using above
energy model discribed can be calculated as
1−P
+ CP + DP 2 ] (19)
P
where A = ((4Eelec + 2EDA )Lctr + (2Eelec + EDA )Ldata ),
f s
f s
2
B = (Lctr ( πλ
)+Ldata ( πλ
), C = (Lctr (5f s Rmax
+4Eele +
28
28
4
4
a
)
+
L
(E
+
a
)),
D
=
n(E
data
elec
elec + EDA )
15 mp
15 mp
Since above Equation do not have closed form so we have to
use simulation results for determination of popt .
E[c] = nλa2 [A(1 − P ) + B
E[c] = A· p + B· (1 − p) + C· (1−p)
+ D· p(1 − p)
p
1
√
+E· p 2 + F · p(0.765 np − 1) + G
0.098
0.097
0.096
Total energy spent in a round
2) Multihop communication: For multi hop communication
the aggregated data will be relayed by the other CH. Now for
the connectivity of the inter-cluster overlay, the transmission
range of the CHs should be at least two or more cluster
diameters. Here we have consider the homogeneous sensor
network hence the same radio range R = 4r [7]. Hence the
i
average distance between CHs and BS is h̄ = D
4r .The total
amount of energy sent in the network can be calculated as:
0.095
0.094
0.093
0.092
0.091
(20)
where A = Lctr (2nEelec + 49 mp na4 ), B = Lctr (5nEelec ) +
Ldata (2Eelec ), C = Lctr ( 31 f s a2 ) + Ldata ( 16 f s a2 ),
D
=√ Lctr (n2 (Eelec + EDA )), E
=
(Lctr +
Ldata )( n(0.1.743f s a2 )), F = (Lctr + Ldata )(nEelec ) and
G = Lctr (2nEDA + 13 f s a2 ) + Ldata (nEDA )
0.09
0.089
0.05
0.1
0.15
0.2
0.25
0.3
Probabilty of becoming a CH
0.35
0.4
0.45
Fig. 1. Total energy spent versus the probability of becoming CH for direct
communication for random deployement
B. Grid deployed network
0.11
(1 − p)
+D· p(1−p)+E (22)
p
where A = Lctr (n(3Eelec + 89 mp a4 )) + Ldata (n(Eelec +
4
4
9 mp a )), B = Lctr (5nEelec + Ldata (2nEelec ), C =
1
Lctr ( 3 f s a2 ) + Ldata ( 16 f s a2 ), D = Lctr (n2 (Eelec + EDA )))
and E = Lctr (2nEDA + 13 f s a2 ) + Ldata (nEDA )
2) Multihop communication: To calculate the energy dissipation in multihop case we can consider the network as spatial
coherence region called basic observation area[7].In[7],
the
√
np
average number of√ hop counts is given as h̄ = 2 with np
np
2 (np−1)+1
is even, and h̄ =
otherwise. So the total amount
np
of energy spent can be calculated as:
E[c] = A· p + B· (1 − p) + C· (1−p)
+ D· p(1 − p)
1
√ p
2
+E· p + F · p( np − 1) + G
(23)
where A = Lctr (2nEelec + 49 mp na4 ), B = Lctr (5nEelec ) +
Ldata (2Eelec ), C = Lctr ( 31 f s a2 ) + Ldata ( 61√
f s a2 ), D =
2
Lctr (n (Eelec + EDA )), E = (Lctr + Ldata )( n( 34 f s a2 )),
F = (Lctr + Ldata )(nEelec ) and G = Lctr (2nEDA +
1
2
3 f s a ) + Ldata (nEDA )
IV. S IMULATION RESULTS AND DISCUSSION
In this section, we verify the optimal probability obtained by
stochastic geometry, for direct and multihop communication
in the random and grid scenario in section III into a square
area of length 100 m with 100 node. We found that at the
optimal probability popt is the value at which the energy costs
0.1
0.09
Total energy spent in a round
Now we consider network in which nodes are placed along
grid point with distance r between them. If all nodes are
reliable, this grid will trivially provide connectivity and take
the following form:
1
λ= 2
(21)
r
1) Direct communication: For this case we calculate the
total energy spent in the nework as
E[c] = A· p+B· (1−p)+C·
0
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0.05
0.1
0.15
0.2
0.25
Probability of becoming CH
0.3
0.35
0.4
Fig. 2. Total energy spent versus the probability of becoming CH for multihop communication for random deployement
in the system is minimum via simulation.The value of p can
computed by numerical analysis. The simulation results shows
the total energy spent in a round is minimized at probability
p = 0.5, 0.46 for direct communication and p = 0.36, 0.32 for
multi-hop communication for same area. We also observed
options
Lctr
Ldata
Eelec
EDA
f s
mp
value
20bytes
1000bytes
50nj/bit
50nj/bit/signal
10pj/bit/m2
0.0013pj/bit/m4
TABLE I
S IMULATION PARAMETER
that for same n if area will increase then it leads to increase
in popt which is constant for LEACH.
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Proc. Of IEEE HICSS, January 2000.
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a Bivariate Poisson Process,” Advances in Applied Probability, vol. 28,
no. 4, pp. 965-981, 1996.
[6] V.P. Mhatre, C. Rosenberg, D. Kofman, R. Mazumdar,N. Shroff, ”A minimum cost heterogeneous sensor network with a lifetime constraint,”IEEE
Transactions on Mobile Computing,Vol.4, pp.4-15,june 2005.
[7] L-C Wang, C-W Wang, and C-M Liu,”Optimal Number of Clusters in
Dense Wireless Sensor Networks: A Cross-Layer Approach, ” IEEE
Transaction on vehicular technology, Vol. 58, no.2, feb.2009
0.115
Total energy spent in a round
0.11
0.105
0.1
0.095
0.09
0
0.05
0.1
0.15
0.2
0.25
0.3
Probability of becoming a CH
0.35
0.4
0.45
Fig. 3. Total energy spent versus the probability of becoming CH for direct
communication for grid deployement
0.12
Total energy spent in a round
0.1
0.08
0.06
0.04
0.02
0
0
0.05
0.1
0.15
0.2
0.25
0.3
Probability of becoming CH
0.35
0.4
0.45
Fig. 4. Total energy spent versus the probability of becoming CH for multihop communication for grid deployment
V. C ONCLUSIONS
In this paper we try to find optimal probability of a node
to becoming a cluster head that leads to minimize the overall
energy spent in the network for a more complex and general
scenario. We formulate the optimal way for determining
number of CH for different scenario with the objective of
guaranteed connectivity and minimizing the total energy spent
in the system. We found that the optimal parameter values for
these scenario and complex model will not only depend on n
that was cosidered by leach but also depends on trasmission
range, packet length, circuit dissipation energy, etc.
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