2012 2nd IEEE International Conference on Parallel, Distributed and Grid Computing

Mathematical Modelling of Packet Transmission through Cluster Head from Unequal Clusters in WSN

Raju Duttat, Shishir Gupta\ Mukul Kumar Das3

IDept of Mathematics, Narula Institute of Technology, Kolkata 700109,West Bengal, India rdutta80@gmaiLcom

2Dept of Mathematics, Indian School of Mines, Dhanbad 826004, 1harkhand,India shishir _ism@yahoo.com

3Dept of Electronics Engineering, Indian School of Mines, Dhanbad 826004, 1harkhand,India mkdasl2@gmail.com

Abstract- The network lifetime always a critical issue in sensor networks because the sensor nodes are characterized by restricted and non-replaceable energy supply. Thus maximizing lifetime of network by minimizing energy consumption poses a challenge in design of protocols. Therefore, proper organization of clustering and orientation of nodes within the cluster becomes one of the important issue to extend the lifetime of the whole sensor network through cluster head. We investigate the problem of energy consumption in ClusterHead (CH) rotation (i.e. re- clustering) in wireless sensor networks. The selection criteria of the CH are based on the residual energy of nodes within the clusters and minimum average distance from the Base Station. The total energy in delivering a packet from the sensor node to other nodes has been mathematically derived. Moreover, the expected number of packet retransmissions and the effect of it on the network is also discussed in our proposed energy modeL In this paper we applied the approach for producing energy-aware unequal clusters with optimal selection of cluster head and discussed several aspects of the network mathematically and statistically. This work presents an analysis of its design and implementation aspects. The simulation results demonstrate that our approach of re-c1ustering in terms of energy consumption and lifetime parameters.

Keywords- Wireless Sensor Network; Unequal Cluster; Energy­Aware Clusters; Packet Transmissions.

I. INTRODUCTION Cluster schemes are hierarchical. When a source node wants to send a packet to another network node that is not in the same cluster, the node uses a reactive routing protocol in order to discover the route. Cluster schemes has many advantages .For a given number of sampling units, cluster sampling is more convenient and less costly. The advantages of cluster sampling are

1. Within a cluster, all normal nodes send their data to the CH. The resulting absence of flooding scheme, multiple route which is energy saving.

2. The backbone network consists only of the CHs, which are fewer in number then all nodes in the entire network and therefore simpler.

3. the change of nodes within the cluster affects only that cluster but not the entire network .

4. collection of data for neighbouring elements is easier, cheaper, faster and operationally more convenient then observing units spread over a region.

S. It is less costly then simple random sampling due to the saving of time in journeys, identification, contacts etc.

6. When a sampling frame of elements may not be readily available.

Energy conservation is always a challenging issues in wireless sensor networks. Lifetime of wireless network based on battery power which has limited energy source. For increasing the lifetime of such network and nodes, it is important that one has to fmd the techniques either increasing the battery power or an alternative source of energy for the nodes. One of the methods for increasing the lifetime of nodes is by adjusting the transmission power of sensor node during transmission. However, adjusting the transmission power is not always sufficient to improve the battery power of sensor nodes and optimize the energy consumption. Now-a-days to increase in the network life time interference plays an important role for minimizing the energy consumption. In wireless networks when a node is transmitting information then other nodes within the transmission range may receive the packets and experience interference from the transmitting node. So the combination of received signal strength from different transmitting node is interference. Due to interference, the quality of service of wireless network to a great extent causing collision of packets, packet loss, retransmission frequency. Decreasing interference level may save the node power by minimizing collision, retransmission and congestion. In this paper we consider transmission and reception power of nodes

978-1-4673-2925-5/12/$31.00 ©2012 IEEE 515

2012 2nd IEEE International Conference on Parallel, Distributed and Grid Computing

for a good quality routing path for delivering the packets from source to destination. Our protocol minimizes the total energy consumption of routing path from source to destination and balances load among the nodes. We propose our protocol with shortest path routing.

II. RELATED WORK Power Consumption and Maximizing Network Lifetime during Communication of Sensor Node in WSN has been discussed in [1]. In [2] Efficient Clustering Techniques to Optimize the System Lifetime in Wireless Sensor Network. Interference is widely studied in wireless network including wireless mesh network, adhoc network and also sensor network. In [3] topology control of wireless networks is achieved using link level interference. In [4] sender computes the potential interference of receivers and adjusts their transmission ranges to reduce the receiver interference using global information. [5][6] describe topology control using nodal interference or link interference. In [7][8][9][10] the route having less interference from source to destination was found, in order to avoid the areas with more interference. These protocols only consider the interference incurred in the nodes, not the energy issue. The energy consumption pattern in an integrated interference-aware and confidentiality-enhanced multipath routing scheme for continuous data streams on Wi-Fi based multi-hop wireless ad hoc networks has also been proposed [11]. In [12] and [13] dynamic virtual carrier sensing and interference aware routing protocol to select the optimum path based on two criteria, shortest path and interference of nodes, have been proposed. The sensor network is a wireless collection of portable devices that offers are a variety of services, including area and wildlife monitoration, etc. [14]. The wireless sensor networks (WSNs) have long been an attractive option to the researchers and scientists for its ease in deployment and maintenance. The battery-operated sensors are often deployed in unattended hostile region, which makes the power source of the sensors difficult to recharge. However, in the research there is number of relevant energy preserving techniques available, which tends to extend the network lifetime. A cluster-based approach is introduced in [15] using PSO, which was found to function better than LEACH-C (Low-Energy Adaptive Clustering Hierarchy-Centralized) [16] protocol in terms of energy efficiency. Another improved PSO has been proposed in [17] for improving the performance of the optimization technique. In [18] another research work evaluates a routing optimization method on the basis of graph theory and PSO. Further, the authors in [19-21] have used PSO tooptimize the location of the sensors with an objective to enhance the network coverage and connectivity. In this paper, we apply an algorithm to select the cluster head (CH) from unequal clusters in order to reduce the overall energy consumption during packet transmission to sink. Further, we analyze the effect of link failure probability on transmission of packets and derive total power consumption of a node.

III. NETWORK MODEL AND ASSUMPTIONS

The assumptions made to describe our network scenario (fig. 1).

� The sensor network is assumed to be a circular geographic region with the sink S, positioned at coordinate (xo Yo), and radius Rs.

� The sensors are uniformly deployed in the sensing area As. Moreover, the number of sensor nodes is distributed according to 2-dimensional Poisson point process with p as the expected density of Aca .

� The cluster covers circular area with its cluster head at the center 0 with radius Rca. � There are total k clusters in the sensor network.

Further, owing to the uniform node deployment strategy, we can compute an approximation for the cluster radius, Rca:

2 2 KxAca =As �KxnxRca =nxRs

k k where K = I K. and Rca = I Rca'

i=l I i=l I

(1)

� The base station (or sink) periodically sends a request to the cluster head of unequal cluster size to upload samples collected by the sensors (fig. 1). On receiving the request, the cluster head broadcasts a data­gathering-signal to all its cluster members.

� In our contribution we have applied unequal clustered sensor network, where the nodes are stationery. The basic aim is to find optimized position for cluster head from the base station, i.e. the distance is as close as possible to the base station. Such localization for cluster head would ultimately minimize the average distance covered by the sensors to transmit data to the cluster head and to the base station.

IV. CLUSTER HEAD SELECTION ALGORITHM AND POWER CONSUMPTION SCHEMES

The normal nodes in a cluster only transmit and relay their data to their CH. In addition to transmitting their data, the CHs are also receiving data from the normal nodes and transferring these data. The CHs therefore consume more power then the normal nodes, and when the CHs run out of energy the cluster will break down. To avoid this situation and keep network healthy we have implemented a Cluster Selection Efficient Protocol (CSEP), where clusters will be selected in the basis of few assumption. � Here we consider a dynamic clustered network where

clusters are unequal. � In each cluster the CH is selected by cluster head

routing protocol. � In each cluster a node will be selected as a CH whose

energy level is high then others node.

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2012 2nd IEEE International Conference on Parallel, Distributed and Grid Computing

The CH will be selected from the cluster if the Variance within the cluster is less. The CH will be selected from the cluster if the

distance from the base station is less. But if the closer cluster is unable to find CH then next closer distance cluster have opportunity to select CH.

Yaxis

, I

i • • \' ..... . --_ .....

;Xl l.6ata

& Uploading

request

X axis

Fig I: Network Model with data transmission from Sensor node and uploading request from BS

V. UNEQUAL CLUSTER SAMPLING Estimator of Mean and its Variance In many practical situations cluster size vary. Suppose there are N clusters. Let the ith cluster consists of Mi elements (i =1 ,2 , . . . . . . . N) and £ M i = M 0 The Cluster population mean per element Y. is

1

Proof: To prove that Y n is not unbiased, we can write

[ £ Yi 1 � Yi - - .Thus Y n is a biased E ( Y n ) =E _I-n- = � = Y n "' Y

estimator of population mean Y . The bias of the estimator is .

b COy (-Y' M ,) fi gIVen y B (- ) = E (- ) _ V = _ I , 1 or a Y n Y n M population in which Mi 's do not appreciably vary from one cluster to another, the bias may not be materially significant. If Mi and Yi. are un correlated, the bias is zero and Y n is an

unbiased estimator in this case.

VI. CLUSTER SAMPLING FOR PROPORTION Suppose it is required to estimate the proportions of elements belongs to a specified classes when the population consists N clusters, each of size M and a random sample of n clusters is selected. Suppose the M elements in any cluster can be classified into two classes. Assuming that y i j = I if the jth elements of the ith cluster belongs to the class

= 0, otherwise.

It can be easily seen that p' = � is the true proportion in the 1 M

ith cluster, ai being the number of elements in the ith cluster belongs to the specified class. An unbiased estimator of the

population proportion

n

N N L ai L PI ·s P = -1 __ =_1_1 NM N

given by

� = p Where Pi ,the proportion of elements is n

defined by belongs to the specified class in the ith cluster of the sample. The sampling variance of Pc is given by N Mi N 2: 2: y " 2: M ' Y i=lj=1 IJ i=1 1 I, Y = N -'-=-'''''N---

2:M 2:M i=1 1 i=1 1

N 2: M ' Y i=1 1 I,

MO

where Y i. is the

mean per element of the ith cluster. We may also defined the N_ L Y pooled mean Y N = � N

N _ n .� (p i - P) 2

V(Pc )= __ I=1 - �S2 n N- I

-n b

(4)

Where S� is the variance between cluster proportions and is

given by

2 N (P'_P'12 N MSb =2: I J = __ PQ,

i N-l N-I where Q = I-P (5)

Let a random sampling, without replacement of 'n' clusters be for large N, we have S 2 == S � + S : drawn and all elements of the clusters surveyed.

n - variance S� is given by � PI Q 1

= PQ and the within

Th .

f Y . . b L Y 1 (2) N en estimator 0 IS given y Y n = � n Hence the intracluster correlation coefficient p can be written

Lemma: Show that simple arithmetic mean given by relation (2) is not an unbiased estimator. The bias and sampling as variance of the estimator are given by

the range of p is given by

(_)_ COV(yi ,Mi ) and (_ ) _ (1- f) 2 where B Yn - M V Yn -

n sb

N -I (3)

517

M -1 _____ ( _ 0

< p < M -l MoCM -I)- -/' N M i (M i-I )(Yi - vy

S b = I M CM -1 XN -1)

(N -l)s �2 where NS 2

2012 2nd IEEE International Conference on Parallel, Distributed and Grid Computing

therefore, the sampling variance, in terms of the intracluster correlation coefficient can be expressed as

vO'c ) = � - f})PQ [I + (M -I)p] (6) N -I nM

An estimation of total number of units belonging to the

(11)

VIII. ENERGY CONSUMPTIOM MODEL specified class, can be obtained by multiplying by Pc by NM The radio energy dissipated by a sensor node is mainly in form and the expression for its sampling variance is N 2 M 2 times that given by equation (6). If Simple Random Sampling (SRS) of nM elements could be taken, the variance of the sampling proportions p would be obtained by binomial theory and is .

b NPQ gIVen y v . (I» = (I - f )---,------==-------,-bIn nM (NM -I) The efficiency of cluster sampling as compared to SRS, without replacement, can be obtained as

V( Pd (MN -I)[I+(M -I)p] V bin (P)

And for large N

N - 1

V( P c 2 = M[I+(M -I)p] V bin (P)

(7)

the factor shows the relative change in the variance caused by the use of clusters. If the cluster sizes Mi are variable, the estimate �

Pc I a i is

I M i a ratio estimate.lts variance is given by approximately by the

� M 2 (p . _ P) 2 formula (6), � N - n .!..:i ="-'1'----_' __ ' __ _ V( Pc) = ----=- -nM N - I

(8)

if this sample is compared with a simple random sample of n M elements, we fmd as a generalization of (7), by

V( 1\) ervIN - 1) [1 + (M - 1) p] where M = I M i Vbin(P) N-I N

VII. EXPECTED NUMBER OF RETRANSMISSION ATTEMPTS

(9)

Initially we assume that an aggregation tree exists in every cluster with the CH as the root. Moreover, within the cluster, transmission of data to the CH follows the path in the aggregation tree. We assume that there are h hops or links between the source node and CH. Also, every link (between two sensors) possesses link failure probability (p LFP ) . Clearly, for a tree of m links, the number of transmissions required for one successful sending of data to CH is also m . Now, the probability of m successful transmission for one successful end-to-end data delivery (towards CH) IS

of electronics and amplifier energy. Here Ee1ec is the energy

dissipated per bit to run the radio electronics (E 1 ) largely e ec depends on how efficiently the signal is encoded, modulated and filtered and �amp is the energy expended to run the power amplifier for transmitting a bit over unit distance. However, the energy dissipation rate in the radio amplifier (c ) is directly amp proportional to d..l , where d is the distance between the source (member of the cluster) and destination node Cluster Head and A is the path loss component. Therefore, the expected value of

k

L PI Where x-::;'p-::;'Rs' P = �and PI = K;

K rrR 2 Cal To transmit an m-bit packet over a distance d the energy used by a sensor node is given by:

e = {mXEelec+mxcampXd;2 T 4 m x Eelec + m x camp X D

(12)

In equation (12) Vo is the threshold distance and d, denotes the

distance between the (i+Iyh node and ilh node in the cluster and D is the beyond threshold distance where the signal strength is affected by multi-path fading between the leader and the base station . In our approach, we have used both the free space (distance2 power loss) and the multipath fading (distance4 power loss) channel modes. In our model, we assume, that inter-nodal distances are small compared to distance between the nodes and the Base Station (BS). Thus for communication among sensors we take n = 2, and that between the leader and BS, we take n = 4, in equation (12). The energy spent in the receiving the packet is, eR = m x Ee,ec Finally the total energy expanded (E NODE ) by a sensor node

(1- P LFP yn . Also, the probability of at least one failed transmission, leading to failed data delivery, is (1 -(1 -PLFP)m) . Let random variable Y which denotes the number of successful data delivery attempts. Further, (f.1 - 1) failures followed by

to support transmission -reception operation as well as in other and is energy consumption (E oliler ) can be determined by the one successful attempt satisfy geometric distribution

given by: p[Y = �]= II - (1 - PLFP)m J (�-I)X (1 - PLFP)m Therefore, the expected number of attempts leading successful delivery of data is:

(10) following equation:

to a

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2012 2nd IEEE International Conference on Parallel, Distributed and Grid Computing

2p2( Jp2 -x6 -2YO) + (13)

=AX mEelec+mEampX; (P-XO 2PXO(Y�-Jp2_x�) +

YO(2� + Yo( �P -x6 -Yo)J + (1-A)mEelec + Eother

where other energy conswnption due to the environmental noise. Finally, the total energy in delivering a packet can be expressed as following : �OTAL= h x ENODE x Ef!.tJ

2�( �p2-x6 -2yo J +

'x mE +�O xP (p x 2P\l(Y20 _�pLx02 ) + " I "�amp -3 - 0 =hx e ec

+(I-A)mEelec + Eother x I (I-PLFt

Yo(2X6 +YO( 3�p2_X6 -yo JJ

where hops or links between the source node and CH

IX. RESULTS

(14)

Extensive simulation done by using Matlab for the distribution of sensors in two dimension and. The results of the distribution of sensors are shown in different figures.

X 10'0 Power consumption vs Cluster density 3.6 ,----,--------,------,-----,-----,------r----,-----,------,----,

E Q)

3. 4

-;;; 3. 2 >-'" Q) = '0 3 c .� a. � 2. 8 '" c o u � 26 o "-

2. 2'------'----------'------'------'------'------'------'------'------'------" o 10 20 30 40 50 60 70 80 90 100 Cluster density

Fig 1 :The power conswnption increases as cluster density mcreases

Lambda vs Power consumption X 10'0 4. 5,------,-------,----,--------r---,-------,--,------,a

4 E 3. 5 � '" >-

(f) 3 Q)

= '0 2. 5 c .� a. E 2 ::J '" c 8 1. 5

0. 2 0. 4 0.6 0. 8 1. 2 1. 4 1.6 Lambda

Fig 2: Power consumption reduces according to the increase of Lambda.

X. CONCLUSION In this paper we have considered unequal clustering techniques to minimize interference, transmission power and reception power of nodes to derive a good quality routing path for delivering the packets from source to destination based on our proposed algorithm. Our protocol optimizes the total energy consumption of routing path from source to destination and balances load among the nodes. For a given source to destination pair in the network, there may exist more than one routing paths. But according to our proposed model only the optimum route will be considered based on our algorithm for cluster and cluster head selection. Fig 1 shows that the power consumption of the system of unequal clusters has increased as the density of the cluster increases. In fig 2 introduced parameter lambda reduces the power consumption according to the different values of lambda. We compared our protocol with simple random sampling. From expression (9) we conclude that the sampling variance Pc of unequal clustering gives better

result if this sample is compared with a simple random sample of n M elements and if intracluster correlation coefficient pis small. Expression (13) shows the energy expanded by a sensor node to support transmission -reception of data and (14) gives the total energy consumption for delivering a data packet. At present we are carrying further studies in this area.

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2012 2nd IEEE International Conference on Parallel, Distributed and Grid Computing

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