Design of Heterogeneous Sensor Networks with Lifetime and ...

IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. 1, NO. 3, JUNE 2012

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Design of Heterogeneous Sensor Networks with Lifetime and Coverage Considerations Moslem Noori, Student Member, IEEE, and Masoud Ardakani, Senior Member, IEEE

Abstract—Heterogeneous wireless sensor networks use more than one type of nodes to improve network lifetime and scalability. In this paper, we focus on the design of a random network consisting of two types of nodes: ordinary ones and powerful ones. Ordinary nodes transmit their sensed information through powerful nodes to the data sink while powerful nodes are able to directly communicate to the sink. First, a joint lifetime-coverage analysis is presented which studies the effect of random sensors’ locations on the lifetime and coverage. Then, the optimal number of each type of nodes that minimizes the network cost, while guaranteeing a desired coverage during a given time period, is found. Index Terms—Wireless sensor network, heterogeneous network, lifetime, coverage.

I. I NTRODUCTION

H

ETEROGENEOUS wireless sensor networks (HWSNs) consist of more than one type of nodes to allow for a more efficient network design. For example, by network clustering and assigning powerful nodes as cluster heads (CHs), the network lifetime can be prolonged [1]. To this end, different architectures have been proposed to design HWSNs. Energy efficiency usually has the highest priority in the design of wireless sensor networks due to the limited energy resources at the sensors [2]–[6]. However, it is not the only criteria and other network features (e.g. coverage and connectivity), which represent the network performance, are also taken into account. This results in a joint network design approach which simultaneously optimizes the network energy efficiency and other performance measures. In [2], authors propose a node placement design for a deterministic HWSN with lifetime and connectivity considerations. Later, the authors extended their work to random networks [3]. Lee et al. investigate design of an HWSN based on its coverage and implementation cost [6]. The assumed coverage model is based on the sum of the sensing area of all nodes within the network. The coverage of HWSNs has also been studied in the literature without involving the network energy efficiency. For example, using results from geometric integration, [7] studies the coverage of a random HWSN in a general case where each sensor can have a different sensing area. However, the coverage is not studied over an arbitrary duration of time. That is, death of nodes and its effect on coverage is not considered. In this work, we aim at design of efficient random HWSNs with taking network coverage over a given time duration into

Manuscript received January 24, 2012. The associate editor coordinating the review of this letter and approving it for publication was N. Mehta. The authors are with the Department of Electrical and Computer Engineering, University of Alberta (e-mail: {moslem, ardakani}@ece.ualberta.ca). Digital Object Identifier 10.1109/WCL.2012.031512.120063

consideration. Here, the heterogeneity of the network refers to having different types of sensors in the network. More specifically, we assume that the network is composed of two types of nodes, ordinary nodes and powerful nodes, all are randomly spread over a large area. This random deployment requires considering the stochastic nature of the covered area as well as the random distance between ordinary and powerful nodes influencing the lifetime of ordinary nodes. Assuming the same sensing area for nodes of each type, we use some existing results from geometric probability to find an accurate, yet simple, joint model for coverage and lifetime. Unlike existing work, our model takes the dynamics of the network, i.e. nodes death effect, into account. We then use this joint model to propose a joint lifetime-coverage network design. To talk more specifically, we first apply results from the coverage theory [8] to find a simple condition on the density of nodes for achieving a desired initial coverage over the area. Second, we transform studying lifetime of ordinary nodes to a coverage problem in terms of the density of the powerful nodes. This study reveals the influence of the number of powerful nodes on the network coverage at the end of the desired period. Then, we formulate the network design as an integer program whose goal is to minimize the network cost while the coverage of the network over a desired time duration is insured. We analytically solve this integer program in a special case and propose an analytical close-to-optimal solution for the general case. II. S YSTEM M ODEL We assume N ordinary and M powerful nodes are deployed uniformly over a large area R with size R. Thereafter, we refer to the powerful nodes as cluster heads (CHs) and to the ordinary sensors as nodes. Each node monitors the area and each T hours sends data directly (i.e. single-hop) to its closest CH who forwards all data from its associated cluster to the sink. Since network clustering is used for large-scale networks, we assume R is large. For nodes, we assume an initial energy of Ei Joules and a sensing range of RS1 meters. CHs may or may not sense the area. When they are assumed to sense, a sensing range of RS2 is considered for them. Further, CHs have no constraint on their energy resource. For simplicity, we assume that the nodes do not apply sleep scheduling. However, when the nodes’ density is high, sleep scheduling can be used to enhance the network lifetime. Notice that implementing sleep scheduling can be complex and not feasible in all cases [9]. The consumed energy for transmission of a data packet is modeled as [1] (1) e(d) = kdγ + c

c 2012 IEEE 2162-2337/12$31.00 

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IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. 1, NO. 3, JUNE 2012

where k and c represent the loss coefficient and the overhead energy for one packet transmission, respectively. Also, d denotes the transmission distance and γ is the path loss exponent. By a Cg coverage over the network lifetime Tf , we mean any arbitrary point in R is covered with probability at least Cg during Tf . To ensure a Cg coverage over Tf , a proper choice of M and N must be made addressed in Section IV.

Ordinary node Powerful node Dm RS

1

Fig. 1.

A Randomly deployed HWSN.

III. A NALYSIS OF THE N ETWORK C OVERAGE In the following, we first study the initial network coverage. Then, we address how the number of CHs affects the death of nodes. The composition of these two results enables us to present an expression for the network final coverage after Tf . Our main focus is on the case where only nodes sense the area. The case where CHs also sense is briefly discussed afterwards. Using results from coverage process theory [8], it is seen that the mean and the variance of Cn (the probability of covering an arbitrary point in R by ordinary nodes) are approximately (2) μCn = 1 − e−πλ1 ,  1   2  8πR S1 −2πλ1 2 eλ1 B(x) − 1 x dx − πλ1 (3) e = σC n R 0 where



B(x) =

√ π − 2x 1 − x2 − 2 arcsin(x) 0

0 ≤ x ≤ 1, x > 1.

2 σC n

Ci = Cn = 1 − e−πλ1 .

(5)

To find the final coverage, Cf , we need to find the number of dead nodes in the network after Tf . We accomplish this by finding the consumed energy by a node, Ec , during Tf . Clearly, Ec is equal to the number of packets sent to CH multiplied by the consumed energy per packet transmission. The sensor achieves Tf if Ec < Ei . Using (1), one can determine Dm , the maximum distance between a node and a CH allowing the node to function up to Tf , as (Figure 1)  Dm =

Ei T c − kTf k

 γ1 .

(6)

Thus, if there exists no CH closer than Dm to a node, the node will die before Tf . Therefore, the number of operating nodes after Tf is related to the area of the network covered by M random disks centered at CHs with radius Dm . Again, using coverage process theory, we can find the probability, C  , that a random sensor has a CH close enough and lives until N D2 Tf . For large R, defining λ = Rm , we have 

C  = 1 − e−πλ .



Cf = C  · Ci = (1 − e−πλ1 )(1 − e−πλ ).

(7)

When the sensing area of a node is small compared to

(8)

Now, we discuss the case where CHs also monitor the area. Assuming a sensing range of RS2 for CHs, when R → ∞, CHs cover an arbitrary point with probability CCH = 1 − e−πλ2

(9)

MR2S2

where λ2 = R . Any point in the network is either covered by nodes or CHs. Thus, Ci = 1 − e−π(λ1 +λ2 ) .

(10)

Since the communication range of sensors is usually greater than their sensing range [10], we assume RS2 < Dm . At Tf , a point is covered either by live nodes, or CHs or both. Hence, Cf = C  Cn + CCH − CCH Cn .

(4) and λ1 = . It can be seen that if R → ∞, then → 0. Thus, the initial coverage, Ci , approaches the following deterministic value (since its variance tends to 0) N R2S1 R

R, the boundary effect is negligible. Thus, due to the independence of the nodes and CHs placement in R,

(11)

In (11), the first term is the part of the area which is covered by live nodes. Also, the third term represents coverage by nodes and CHs. Please notice that since RS2 < Dm , all nodes that are located in the coverage range of CHs stay live up to Tf . Hence, 

Cf = 1 − e−πλ2 + (e−πλ2 − e−πλ )(1 − e−πλ1 ).

(12)

IV. N ETWORK D ESIGN The results of the previous section can be used to predict the coverage of a network with known parameters. In this section, we go one step further and use these results to design an HWSN. Specifically, we adjust the number of nodes and CHs in the network to reduce the implementation cost while maintaining a guaranteed coverage Cg during the network lifetime. For this purpose, we assign a cost to each type of sensors and formulate the problem as an integer program. For brevity, we focus on the case where CHs do not sense, but a similar design approach can be taken when both type of sensors do area monitoring. Assume that each node costs Hn and each CH costs HCH such that Hn < HCH . The optimization constraint is Cf = Ci C  ≥ Cg . Thus, using (5) and (7), one can formulate minimizing the network cost problem as min

N,M

subject to

N Hn + M HCH  1−e

πN R2 S − R 1

N >0 ,

2 πM Dm  1 − e− R ≥ Cg

M > 0.

(13) (14) (15)

NOORI and ARDAKANI: DESIGN OF HETEROGENEOUS SENSOR NETWORKS WITH LIFETIME AND COVERAGE CONSIDERATIONS

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extra node compared to the actual optimal integer solution. This cost penalty is easily negligible for large networks. is an integer, the optimal Now, we claim that when HHCH n integer value for M is either M ∗ = Mr∗  or M ∗ = Mr∗ . Consequently, the optimal value of N is  ∗ 1  1 − e−βM ln N∗ = . (18) α 1 − e−βM ∗ − Cg

Fig. 2.

Graphical illustration of the optimization problem.

This nonlinear integer program can be solved using conventional methods for integer programs. Here, we first relax the problem so that M and N can be non-integer. The solution of the relaxed problem is then used to find an analytical closeto-optimal solution for general case and an optimal solution for a special case of the original integer program problem. A graphical illustration for the relaxed problem is depicted in Figure 2. The shaded area above the curve (1 − e−αN )(1 − e−βM ) = Cg is the feasible region of the problem where πR2

πD2

S1 m α = R and β = R . We use C to refer to the feasible region. To find the optimal solution of the relaxed problem, we have to slip the line N Hn + M HCH = H, which we call it , toward the feasible region. Here H represents the total network cost. The optimal solution of the relaxed problem, i.e. (Mr∗ , Nr∗ ), is the point where  is tangent to C. The optimal solution of the integer program is the first point with integer elements that  passes while it slips in the shaded area. Noticing that in the relaxed problem, the constraint in (14) is met with equality, N can be found as a function of M as follows

N = f (M ) =

1  1 − e−βM ln . α 1 − e−βM − Cg

(16)

Using the above equation, it is easy to show that C is a convex region. By replacing (16) in (13) and then taking its derivative, the optimal value of M for relaxed problem is Mr∗ =

1 ln β (2 +

1 γ



2 1 γ

− Cg )2 − 4(1 − Cg ) (17) αHCH . The optimal relaxed value of N , Nr∗ , where γ = βH n Cg ∗ is then found by replacing Mr in (16). Thus, the total cost for the relaxed problem is Nr∗ Hn + Mr∗ HCH = Hr∗ . A close-to-optimal solution for the network design optimization problem is (Mr∗ , Nr∗ ). Since any integer solution for the optimization problem has a cost more than Hr∗ , the proposed close-to-optimal solution has at most one extra CH and one − Cg ) −

(2 +

To prove this claim, assume that the optimal integer point is P1 = (M1 , N1 ) such that M1 > Mr∗ . Thus, N1 Hn + M1 HCH = H1 is the minimum network implementation cost. Now, consider P2 = (Mr∗ , N1 + (M1 − Mr∗ ) HHCH ). It is n easy to verify that P2 lies on the line N Hn + M HCH = H1 (the same line as P1 ) and consequently has the same implementation cost as P1 . Furthermore, it can be shown that P2 falls in C. To show this, recall that P3 = (Mr∗ , N3 ), such that N3 Hn + Mr∗ HCH = H1 , is in C because Hr∗ ≤ H1 . On the other hand since P1 is also in C, due to the convexity of C, any other point between P1 and P3 , including P2 , falls in C. Since P2 has equal cost as P1 , P2 is also an optimal point. This results in the conclusion that if M ∗ > Mr∗ , then M ∗ = Mr∗ is one optimal solution. Similarly, it can be shown that when HCH ∗ ∗ Hn is integer, M = Mr  is one optimal value for M if ∗ ∗ M < Mr . V. S IMULATION R ESULTS In this section, we verify the accuracy of our proposed coverage analysis for two different area shapes (circle and square). We also present an example on the optimal network design. First, we focus on the case where only nodes participate in area monitoring. We consider cases I, II, III, and IV. For cases I and II, RS1 = 10 m, but the node density N = 10−2 node/m2 and δ = 5 × 10−3 node/m2 , is δ = R respectively. In Case III and IV, the node density is δ = 10−2 but RS1 = 8m and RS1 = 4m respectively to study the effect of the sensing range on coverage. Based on our analysis, these 2 parameters lead to Ci = 1 − e−πδRS1 = 0.957, Ci = 0.792, Ci = 0.866 and Ci = 0.395 respectively. The numerical results are presented in Table I. As it can be seen, the analysis provides an accurate estimation of the network coverage where the accuracy improves by increasing the area size. Now, we study the accuracy of our estimate of the network coverage after Tf = 5000 hours. To this end, consider the M aforementioned cases where δ  = R = 10−4 and Ei = 3 J. The transmission parameters in (1) are γ = 2, k = 10−7 and c = 5 × 10−5 for a packet with 1000 bits [1]. Also, each node transmits a packet to its nearest CH every hour. As a consequence, Dm = 74.16m.These parameters result in 2  2 Cf = (1 − e−πδRS1 )(1 − e−πδ Dm ) = 0.789, Cf = 0.651, Cf = 0.712, and Cf = 0.325 for all four cases. The simulation results are presented in Table I. Now, let us assume that CHs are also able to monitor the area and their sensing range is RS2 = 15m. Letting RS1 = 10m, δ = 10−2 , δ  = 10−4 , and Ei = 3 J, according to our analysis, the initial coverage of the area should be Ci = 1 − e−π(λ1 +λ2 ) = 0.960. The simulation results for the network initial coverage and the final coverage for Tf = 4000,

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TABLE I P ROBABILITY OF INITIAL AND FINAL AREA COVERAGE . Area (m2 )

Initial coverage

Square: 2.5 × 105 Circle: 2.5 × 105 Square: 1 × 106 Circle: 1 × 106 Square: 2.25 × 106 Circle: 2.25 × 106 Analytical result

Final coverage

Case I

Case II

Case III

Case IV

Case I

Case II

Case III

Case IV

0.953 0.954 0.955 0.955 0.957 0.957 0.957

0.785 0.787 0.787 0.788 0.789 0.790 0.792

0.861 0.862 0.863 0.864 0.864 0.865 0.866

0.392 0.392 0.393 0.393 0.394 0.394 0.395

0.771 0.773 0.783 0.785 0.789 0.789 0.789

0.627 0.631 0.644 0.646 0.650 0.651 0.651

0.686 0.692 0.704 0.707 0.712 0.712 0.712

0.307 0.310 0.315 0.316 0.318 0.319 0.325

TABLE II P ROBABILITY OF COVERAGE WHEN CH S ALSO SENSE . Area (m2 )

Initial coverage

Square: 2.5 × 105 Circle: 2.5 × 105 Square: 1 × 106 Circle: 1 × 106 Square: 2.25 × 106 Circle: 2.25 × 106 Analytical result

0.956 0.956 0.958 0.958 0.959 0.959 0.960

Tf = 5000 and Tf = 6000 are presented in Table II, verifying the accuracy of our analysis. Notice that these parameters result in Dm = 83.67m, Dm = 74.16m, and Dm = 67.08m respectively. Finally, we present an example for the network optimal design, when CHs do not participate in area monitoring. Assume that the area is a square whose sides are 1000 m. We desire Cg = 0.75 after Tf = 5000 hours. Also, each CH costs 10 times more than each ordinary node. The transmission parameters and sensing range of nodes are RS1 = 10m and RS2 = 15m. Following our proposed analytical method in Section IV, we find M ∗ = 248 and N ∗ = 5164. Searching over other possible values of M and N verifies that the solution is indeed optimal and no other network setup has a lower implementation cost satisfying (14).

VI. C ONCLUSION For an HWSN with ordinary nodes in charge of sensing the environment and powerful nodes serving as CHs, we presented a joint lifetime-coverage analysis using previous results from the theory of coverage processes. We showed that the network coverage at any given time can be expressed as a function of the initial coverage (determined from the density of ordinary nodes) and the density of CHs. Extension to the case where CHs would also sense the area was briefly discussed as well. We used our results to optimize the number of each type of nodes in the network for minimum cost and guaranteed coverage over a given time duration.

Tf = 4000 0.831 0.837 0.850 0.853 0.857 0.859 0.854

Final coverage Tf = 5000 Tf = 6000 0.772 0.720 0.776 0.725 0.791 0.739 0.794 0.742 0.798 0.746 0.800 0.748 0.790 0.727

VII. ACKNOWLEDGMENTS The authors would like to thank Natural Sciences and Engineering Research Council of Canada (NSERC) and Alberta Innovates for supporting our research. R EFERENCES [1] W. Heinzelman, A. Chandrakasan, and H. Balakrishnan, “An application-specific protocol architecture for wireless microsensor networks,” IEEE Trans. Wireless Commun., vol. 1, no. 4, pp. 660–670, Oct. 2002. [2] Q. Wang, K. Xu, G. Takahara, and H. Hassanein, “Device placement for heterogeneous wireless sensor networks: minimum cost with lifetime constraints,” IEEE Trans. Wireless Commun., vol. 6, no. 7, pp. 2444– 2453, July 2007. [3] K. Xu, H. Hassanein, G. Takahara, and Q. Wang, “Relay node deployment strategies in heterogeneous wireless sensor networks,” IEEE Trans. Mobile Comput., vol. 9, no. 2, pp. 145–159, Feb. 2010. [4] G. Shirazi and L. Lampe, “Lifetime maximization in UWB sensor networks for event detection,” IEEE Trans. Signal Process., vol. 59, no. 9, pp. 4411–4423, Sep. 2011. [5] A. Behzadan, A. Anpalagan, and B. Ma, “Prolonging network lifetime via nodal energy balancing in heterogeneous wireless sensor networks,” in Proc. 2011 IEEE International Conference on Communications, pp. 1 –5. [6] J.-J. Lee, B. Krishnamachari, and C.-C. Kuo, “Impact of heterogeneous deployment on lifetime sensing coverage in sensor networks,” in Proc. 2004 IEEE Conference on Sensor and Ad Hoc Communications and Networks, pp. 367–376. [7] L. Lazos and R. Poovendran, “Coverage in heterogeneous sensor networks,” in Proc. 2006 International Symposium on Modeling and Optimization in Mobile, Ad Hoc and Wireless Networks, pp. 1–10. [8] P. Hall, Introduction to the Theory of Coverage Processess. Wiley, 1988. [9] J. Deng, Y. S. Han, W. Heinzelman, and P. Varshney, “Balancedenergy sleep scheduling scheme for high density cluster-based sensor networks,” in Proc. 2004 Workshop on Applications and Services in Wireless Networks, pp. 99–108. [10] R. Olfati-Saber and N. Sandell, “Distributed tracking in sensor networks with limited sensing range,” in Proc. 2008 American Control Conference, pp. 3157–3162.