Finding the Number of Clusters Minimizing Energy

Finding the Number of Clusters Minimizing Energy Consumption of Wireless Sensor Networks Hyunsoo Kim and Hee Yong Youn School of Information and Communications Engineering, Sungkyunkwan University, 440-746, Suwon, Korea [email protected], [email protected]

Abstract. Wireless sensor network consisting of a large number of small sensors with low-power can be an effective tool for the collection and integration of data in a variety of environments. Here data exchange between the sensors and base station need to be designed to conserve the limited energy resources of the sensors. Grouping the sensors into clusters has been recognized as an efficient approach for saving the energy of the sensor nodes. In this paper we propose an analytical model based on homogenous spatial Poisson process, which allows the number of clusters in a sensor network minimizing the total energy spent for data exchange. Computer simulation on various size sensor networks with the LEACH algorithm reveals that the proposed model is very accurate. We also compare the proposed model with an earlier one, and it turns out that the proposed model is more accurate. Keywords: Energy consumption, LEACH, number of clusters, Poisson process, wireless sensor network.

1

Introduction

Recent developments in wireless sensor network have motivated the growth of extremely small, low-cost sensors that possess the capability of sensing, signal processing, and wireless communication. The wireless sensor network can be expanded at a cost much lower than conventional wired sensor network. Here each sensor is capable of detecting the condition around it such as temperature, sound, and the presence of other objects. Recently, the design of sensor networks has gained increasing importance due to their potential for civil and military applications such as combat field surveillance, security, and disaster management. The smart dust project at the University of California, Berkeley [7, 8, 12] and WINS project at UCLA [9] are attempting to build extremely small sensors of low-cost allowing autonomous sensing and communication in a cubic millimeter

This research was supported in part by the Ubiquitous Autonomic Computing and Network Project, 21st Century Frontier R&D Program in Korea and the Brain Korea 21 Project in 2005. Corresponding author: Hee Yong Youn.

M. Gavrilova et al. (Eds.): ICCSA 2006, LNCS 3982, pp. 824–833, 2006. c Springer-Verlag Berlin Heidelberg 2006

Finding the Number of Clusters Minimizing Energy Consumption

825

range. The system processes the data gathered from the sensors to monitor the events in an area of interest. Prolonged network lifetime, scalability, and load balancing are important requirements for many wireless sensor network applications. A longer lifetime of sensor networks can be achieved through optimized applications, operating systems, and communication protocols. If the distance between the nodes is small, energy consumption is also small. With the direct communication protocol in a wireless sensor network, each sensor sends its data directly to the base station. When the base station is far away from the sensors, the direct communication protocol will cause each sensor to spend a large amount of power for data transmission [5]. This will quickly drain the battery of the sensors and reduce the network lifetime. With the minimum energy routing protocol, the sensors route data ultimately to the base station through intermediate sensors. Here only the energy of the transmitter is considered while energy consumption of the receivers is neglected in determining the route. The low-energy adaptive clustering hierarchy (LEACH) scheme [9] includes the use of energy-conserving hardware. Here sensors in a cluster detect events and then transmit the collected information to the cluster head. Power consumption required for transmitting data inside a cluster is lower than with other routing protocols such as minimum transmission energy routing protocol and direct communication protocol due to smaller distances to the cluster head than to the base station. There exist various approaches for the evaluation of the routing protocols employed in wireless sensor networks. In this paper our main interest is energy consumption of the sensors in the network. Gupta and Kumar [2,3] have analyzed the capacity of wireless ad-hoc networks and derived the critical power with which a node in a wireless ad-hoc network communicates to form a connected network with probability one. A sensor in a wireless sensor network can directly communicate with only the sensors within its radio range. To enable communication between the sensors not within each other’s communication range, the sensors need to form a cluster with the neighboring sensors. An essential task in sensor clustering is to select a set of cluster heads among the sensors in the network, and cluster the rest of the sensors with the cluster heads. The cluster heads are responsible for coordination among the sensors within their clusters, and communication with the base station. [4] have suggested an approach for cluster formation which ensures the expected number of clusters in LEACH. A model deriving the number of clusters with which the energy required for communication is minimized was developed in [10]. Energy consumption required for communication depends on the distance between the transmitter and receiver. Here, the expected distance between the cluster head and non-cluster head was computed. The distance from a sensor to the base station was not computed but fixed. [11] proposed a method for finding the probability for a sensor to become a cluster head. They directly computed the expected distance from a sensor to the base station with uniform distribution in a bounded region. They assumed that the distance between a cluster head and non-cluster heads in a cluster de-

826

H. Kim and H.Y. Youn

pends only on the density of the sensors distributed with a homogeneous Poisson process in a bounded region. More details are given in Section 2.3. In this paper we develop a model finding the number of cluster heads which allows minimum energy consumption of the entire network using homogeneous spatial Poisson process. [11] modeled the expected distance between a cluster head and other sensors in deciding the number of cluster heads. Here, we additionally include the radio range of the cluster head and the distribution of energy dissipation of the sensors for more accurate prediction of the number. With the proposed model the network can determine, a priori, the number of cluster heads in a region of distributed sensors. The validity of the proposed model is verified by computer simulation, where the clustering is implemented using the LEACH algorithm. It reveals that the number of clusters obtained by the developed model is very close to that of the simulation with which the energy consumption of the network is minimum. We also compare the proposed model with [11], and it consistently produces more accurate value than [11]. The rest of the paper is presented as follows. In Section 2 we review the concept of LEACH, sensor clustering, and previous approaches for modeling the number of clusters. Section 3 presents the proposed model finding the number of clusters which minimizes energy consumption. Section 4 demonstrates the effectiveness of the proposed model by computer simulation and comparison with the earlier one. We provide a conclusion in Section 5.

2 2.1

Preliminaries The LEACH Protocol

LEACH is a self organizing, adaptive clustering protocol that employs a randomization approach to evenly distribute energy consumption among the sensors in the network. In the LEACH protocol, the sensors organize themselves into clusters with one node acting as the local base station or cluster head. If the cluster heads are fixed throughout the network lifetime as in the conventional clustering algorithms, the selected cluster heads would die quickly, ending the useful lifetime of all other nodes belonging to the clusters. In order to circumvent the problem, the LEACH protocol employs randomized rotation of the cluster head such that all sensors have equal probability to be a cluster head in order not to spent the energy of only specific sensors. In addition, it carries out local data fusion to compress the data sent from the cluster heads to the base station. This can reduce energy consumption and increase sensor lifetime. Clusters can be formed based on various properties such as communication range, number and type of sensors, and geographical location. 2.2

Sensor Clustering

Assume that each sensor in the wireless sensor network becomes a cluster head with a probability, p. Each sensor advertises itself as a cluster head to the sensors within its radio range. We call the cluster heads the voluntary cluster heads. For


827

the sensors not within the radio range of the cluster head, the advertisement is forwarded. A sensor receiving an advertisement which does not declare itself as a cluster head joins the cluster of the closest cluster head. A sensor that is neither a cluster head nor has joined any cluster becomes a cluster head; we call the cluster heads the forced cluster heads. Since forwarding of advertisement is limited by the radio range, some sensors may not receive any advertisement within some time duration, t, where t is the time required for data to reach the cluster head from a sensor within the radio range. It then can infer that it is not within radio range of any voluntary cluster head, and hence it becomes a forced cluster head. Moreover, since all the sensors within a cluster are within radio range of the cluster head, the cluster head can transmit the collected information to the base station after every t units of time. This limit on radio range allows the cluster heads to schedule their transmissions. Note that this is a distributed algorithm and does not require clock synchronization between the sensors. The total energy consumption required for the information gathered by the sensors to reach the base station will depend on p and radio range, r. In clustering the sensors, we need to find the value of p that would ensure minimum energy consumption. The basic idea of the derivation of the optimal value of p is to define a function of energy consumption required to transmit data to the base station, and then find the p value minimizing it. The model needs the following assumptions. • The sensors in the wireless sensor network are distributed as per a homogeneous spatial Poisson process of intensity λ in 2-dimensional space. • All sensors transmit data with the same power level, and hence have the same radio range, r. • Data exchanged between two sensors not within each others’ radio range is forwarded by other sensors. • Each sensor uses 1 unit of energy to transmit or receive 1 unit of data. • A routing infrastructure is in place; hence, when a sensor transmits data to another sensor, only the sensors on the routing path forward the data. • The communication environment is contention and error free; hence, sensors do not have to retransmit any data. 2.3

Previous Modeling

Energy consumption required for communication depends on the distance between the transmitter and receiver, i.e., the distance between the sensors. In [11], first, the expected distance from a sensor to the base station is computed. Then the expected distance from the non-cluster head to the cluster head in a Voronoi cell (a cluster) is obtained. Assume that a sensor becomes a cluster head with probability p. Here the cluster-heads and the non-cluster heads are distributed as independent homogeneous spatial Poisson processes of intensity λ1 = pλ and λ0 = (1 − p)λ, respectively. Each non-cluster head joins the cluster of the closest cluster-head to form a Voronoi tessellation [6]. The plane is divided into zones called the Voronoi cells, where each cell corresponds to a Poisson process with

828


intensity λ1 , called nucleus of Voronoi cells. Let Nv be the random variable denoting the number of non-cluster heads in each Voronoi cell, and N be the total number of sensors in a bounded region. If Lv is the total length of all segments connecting the non-cluster heads to the cluster head in a Voronoi cell, then E[Nv |N = n] = λ0 /λ1 ,

3/2

E[Lv |N = n] = λ0 /2λ1 .

(1)

Note that these expectations depend only on the intensities λ0 and λ1 of a Voronoi cell. Using the expectations of the lengths above, an optimal value p minimizing the total energy spent by the network is found.

3 3.1

The Proposed Scheme Motivation

For a wireless sensor network of a large number of energy constrained sensors, it is important to fastly group the sensors into clusters to minimize the energy used for communication. Note that energy consumption is directly proportional to the distance between the cluster heads and non-cluster heads and the distance between the cluster head and base station. We thus focus on the functions of the distances for minimizing the energy spent in the wireless sensor network. Note that the earlier model of Equation (1) depends only on the intensity of the sensor distribution. In this paper, however, the expected distance between the sensors and cluster head is modeled by including p, and the number of sensors, N , and the size of the region for obtaining more accurate estimation on the number of clusters. The number is decided by the probability, p, which minimizes the energy consumed to exchange the data. We next present the proposed model. 3.2

The Proposed Model

In the sensor network the expected distance between the cluster heads to the base station and the expected distance between the sensors to the cluster head in a cluster depend on the number of sensors, the number of clusters, and the size of the region. The expected distance between a sensor and its cluster head decreases while that between a cluster head and the base station increases as the number of clusters increases in a bounded region. An opposite phenomenon is observed when the number of clusters decreases. Therefore, an optimal value of p in terms of energy efficiency needs to be decided by properly taking account the tradeoff between the communication overhead of sensor-to-cluster head and cluster head-to-base station. Figure 1 illustrates this aspect where Figure 1(a) and (b) has 14 clusters and 4 clusters, respectively. Notice that the clusters in Figure 1(a) have relatively sparse links inside the clusters compared to those in Figure 1(b), while there exist more links to the base station. Let S denote a bounded region of a plane and X(S) does the number of sensors contained in S. Then X(S) is a homogeneous spatial Poisson process if it distributes the Poisson postulates, yielding a probability distribution


829

BS

BS

(a) Large number

(b) Small number

Fig. 1. Comparison of the structures having a large and small number of clusters

P {X(S) = n} =

[λA(S)]n e−λA(S) , n!

f or A(S) ≥ 0, n = 0, 1, 2, · · ·

Here λ is a positive constant called the intensity parameter of the process and A(S) represents the area of region S. If region S is a square of side length, M , then the number of sensors in it follows a Poisson distribution with a mean of λA(S), where A(S) is M 2 . Assume that there exist N sensors in the region for a particular realization of the process. If the probability of becoming a cluster head is, p, then N p sensors will become cluster heads on average. Let DB (x, y) be a random variable denoting the distance between a sensor located at (x, y) and the base station. Let PS be the probability of existence of sensors uniformly distributed in region S. Without loss of generality, we assume that the base station is located at the center of the square region (i.e. the origin coordinate). Then, the expected distance from the base station to the sensors is given by DB (x, y) · PS dS E[DB (x, y)|X(S) = N ] =

S M/2

= −M/2

1 x2 + y 2 2 dx dy M −M/2

= 0.3825M

M/2

(2)

Since there exist N p cluster heads on average and the location of a cluster head is independent of those of other cluster heads, the total length of the segments from all the cluster heads to the base station is 0.3825N pM . Since a sensor becomes a cluster head with a probability p, we expect that the cluster head and other sensors are distributed in a cluster as an independent homogeneous spatial Poisson process. Each sensor joins the cluster of the closest cluster head to form a cluster. Let X(C) be the random variable denoting the number of sensors except the cluster head in a cluster. Here, C is the area of a cluster. Let DC be the distance between a sensor and the cluster head in a

830


cluster. Then, according to the results of [1], the expected number of non-cluster heads in a cluster and the expected distance from a sensor to the cluster head (assumed to be at the center of mass of the cluster) in a cluster are given by E[X(C)|X(S) = N ] =

1 − 1, p

(3)

E[DC |X(S) = N ] = x2 + y 2 k(x, y) dA(C) C 2π M/√N pπ Np 2M = r2 2 drdθ = √ , (4) M 3 N pπ 0 0 √ respectively. Here, region C is a circle with radius M/ N pπ. The sensor density of the cluster, k(x, y), is uniform, and it is approximately M 2 /N p. Let EC be the expected total energy used by the sensors in a cluster to transmit one unit of data to their respective cluster head. Since there are N p clusters, the expected value of EC conditioned on X(S) = N is given by, E[DC |X(S) = N ] E[EC |X(S) = N ] = N (1 − p) · r 1 2M 1 − p =N2 · √ √ . 3r π p

(5)

If the total energy spent by the cluster heads to transmit the aggregated information to the base station is denoted by EB , then E[DB |X(S) = N ] E[EB |X(S) = N ] = N p · r 0.3825N pM = . r

(6)

Let ET be the total energy consumption with the condition of X(S) = N in the network. Then E[ET |X(S) = N ] = E[EC |X(S) = N ] + E[EB |X(S) = N ].

(7)

Taking the expectation of Equation (7), the total energy consumption of the network is obtained. E[ET ] = E[E[ET |X(S) = N ]] 0.3825pM 2M 1 − p , = E[X(S)1/2 ] · √ √ + E[X(S)] · 3r π p r

(8)

where E[·] is expectation of a homogeneous Poisson process. E[ET ] will have a minimum value for a value of p, which is obtained by the first derivative of Equation (8); 2c2 p3/2 − c1 (p + 1) = 0, √ where c1 = 2M · E[(X(S))1/2 ]/3 π and c2 = 0.3825M · E[X(S)].

(9)


831

Equation (9) has three roots, two of which are imaginary. The second derivative of Equation (8) is positive and log concave for the only real root of Equation (9), and hence the real root minimizes the total energy consumption, E[ET ]. The only real root of Equation (9) is as follows. p=

4

0.1050(c41 + 24c21 c22 ) 0.0833c21 + 2 6 2 4 2 2 2 6 8 1/3 c2 c2 (2c1 + 72c1 c2 + 432c21 c42 + 83.1384c 1 c1 c2 + 27c2 ) 0.0661 + 2 (2c61 + 72c41 c22 + 432c21 c42 + 83.1384c21 c21 c62 + 27c82 )1/3 . (10) c2

Performance Evaluation

In order to validate the proposed model, we simulate a network of sensors distributed as a homogeneous Poisson process with various spatial densities in a region. We employ the LEACH algorithm to generate a cluster hierarchy and find how much energy the network spends with the p value obtained using the developed model. For various intensities λ = (0.01, 0.03, 0.05) of sensors in a bounded region, we first compute the probability for a sensor to be a cluster head by Equation (9). Table 1 lists the probability obtained using the developed model and the corresponding energy consumption of Equation (8) for different intensities and radio range of a sensor, r. Here, M is set to 100, and thus there exist 100, 300, and 500 sensors if λ = 0.01, 0.03, and 0.05, respectively. Table 1. The p probability and corresponding energy consumption with the proposed model

λ 0.01 0.03 0.05

p 0.147 0.099 0.083

ET r=2 699.02 1500.39 2131.84

r=1 1398.03 3000.77 4263.68

r=3 466.01 1000.26 1421.23

Simulation(LEACH ) Proposed model

Energy Consumption

3.3

3.2

3.1

3

0.12

0.13

0.14

0.15

0.16

p

Fig. 2. The total energy consumptions from the proposed model and simulation

832

H. Kim and H.Y. Youn Table 2. The p probability and corresponding energy consumption with [11]

λ 0.01 0.03 0.05

p 0.181 0.121 0.101

r=1 1674.76 3615.65 5147.63

ET r=2 837.381 1807.83 2573.81

r=3 558.25 1205.32 1715.88

Table 3. Comparison of the proposed model with an earlier model [11]

Model The proposed model [11] Percentage of energy saving

100 0.057324 0.059178 3.133

No. of Nodes 300 0.1092644 0.197071 2.247

500 0.299371 0.310891 3.706

Note that the validity of the p value estimated by the proposed analytical model can be verified only through actual implementation of the clustering. Figure 2 shows the energy consumed by the entire network as p changes, obtained using the LEACH algorithm which is one of the most popular clustering algorithms for sensor network, when λ = 0.01 and r = 1. Recall that the p value predicted by the proposed model in this case is 0.147, while the optimal value of p obtained from the simulation turns out to be 0.145. The proposed model thus can be said to be very accurate for deciding the p value. We obtain similar results as this for different cases of λ and r values. We also compare the proposed model and that of [11]. In order to show the relative effectiveness of the proposed model, the probability and the corresponding energy consumption of [11] are obtained and listed in Table 2. Observe from the tables that the probabilities of the proposed model are smaller than [11]. Table 3 summarizes the results of comparison. Here the p values decided from the proposed model and [11] are applied to the LEACH algorithm for obtaining the energy consumed by the network. Notice that the proposed model consistently provides better p values than [11] regardless of the sensor density.

5

Conclusion

The sensors becoming the cluster heads spend relatively more energy than other sensors because they have to receive information from other sensors within their cluster, aggregate the information, and then transmit them to the base station. Hence, they run out of their energy faster than other sensors. To solve this problem, the re-clustering needs to be done periodically or the cluster heads trigger re-clustering when their energy level falls below a certain threshold. We have developed a model deciding the probability a sensor becomes a cluster head,


833

which minimizes the energy spent by the network for communication. Here the sensors are distributed in a bounded region with homogeneous spatial Poisson process. The validity of the proposed model was verified by computer simulation, where the clustering is implemented using the LEACH algorithm. It revealed that the number of clusters obtained by the developed model is very close to that of the simulation with which the energy consumption of the network is minimum. We also compared the proposed model with [11], and it consistently produces more accurate value than [11]. Here we assumed that the base station is located at the center of the bounded region. We will develop another model where the location of base station can be arbitrary. We will also expand the model for including other factors such as the shape of the bounded region.

References 1. S.G. Foss and S.A. Zuyev, “On a Voronoi Aggregative Process Related to a Bivariate Poisson Process ,” Advances in Applied Probability, Vol. 28, no. 4, 965-981, 1996. 2. P. Gupta and P.R. Kumar, “The capacity of wireless networks,” IEEE Transaction on Information Theory, Vol. IT-46, No. 2, 388-404, March, 2000. 3. P. Gupta and P.R. Kumar, “Critical power for asymptotic connectivity in wireless networks,” Stochastic Analysis, Control, Optimization and Applications: A Volume in Honor of W. H. Fleming, 547-566, 1998. 4. W.B. Heinzelman, A.P. Chandrakasan and H. Balakrishnan, “An ApplicationSpecific Protocol Architecture for Wireless Microsensor Networks”, IEEE Trans. on Wireless Communications, Vol. 1, No. 4, 660-670, 2002. 5. W.R. Heinzelman, A. Cahandrakasan and H. Balakrishnan, “Energy-efficient communication protocol for wireless sensor networks, in the Proceedings of the 33rd Hawaii International Conference on System Sciences, Hawaii, 2000. 6. T. Meng and R. Volkan, “Distributed Network Protocols for Wireless Communication”, In Proc. IEEEE ISCAS, 1998. 7. V. Hsu, J.M. Kahn, and K.S.J. Pister, “Wireless Communications for Smart Dust”, Electronics Reaserch Laboratory Technical Memorandum M98/2, Feb. 1998. 8. J.M. Kahn, R.H. Katz and K.S.J. Pister, “Next Century Challenges: Mobile Networking for Samrt Dust,” in the 5th Annual ACM/IEEE International Conference on Mobile Computing and Networking (MobiCom 99), 271-278, Aug. 1999. 9. G.J. Pottie and W.J. Kaiser, “ Wireless integrated network sensors,” Communications of the ACM, Vol 43, No. 5, 51-58, May, 2000. 10. T. Rappaport, “Wireless Communications: Principles & Practice”, Englewood Cliffs, NJ: Prentice-Hall, 1996. 11. S. Bandyopadhyay and E.J. Coyle, “An energy efficient hierarchical clustering alogorithm for wireless sensor networks,” IEEE INFOCOM 2003 - The Conference on Computer Communications, vol. 22, no. 1, 1713-1723, March 2003. 12. B. Warneke, M. Last, B. Liebowitz, Kristofer and S.J. Pister, “Smart Dust: Communication with a Cubic-Millimeter Computer,” Computer Magazine, Vol. 34, No 1, 44-51, Jan. 2001.