This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2008 proceedings.
Path coverage property of randomly deployed sensor networks with finite communication ranges Junko Harada, Shigeo Shioda Graduate School of Engineering, Chiba University, 1-33 Yayoi, Inage, Chiba 263-8522, Japan Telephone/Fax: +81-32-290-3237, E-mail:
[email protected]
Hiroshi Saito NTT Network Inovation Laboratories, 3-9-11, Midori-cho, Musashino, Tokyo, 180-8585, Japan E-mail:
[email protected]
Abstract— We analyze the path coverage property of a sensor network, where a large number of sensors are randomly deployed. We characterize the path coverage in terms of three metrics: fraction of coverage, probability of complete coverage, and probability of partial coverage. We derive the expressions for the three coverage metrics as functions of the sensor density, the sensing range of a sensor, and the communication range of a sensor. Based on the derived expressions, we study the impact of the communication range on the path coverage and, for example, show that a sensor should have a long wireless-communication range to obtain large detectability of intruding objects (partial coverage probability of the trajectories of objects) with modest density of sensors.
I. I Coverage is an important performance index of a sensor network because it represents how well the field of interest is monitored by sensors, or how effective a sensor network is in detecting objects intruding the field of watch. Knowing the fundamental coverage property of a sensor network helps to better construct sensor networks. For example, it could tell us how densely sensors should be deployed to detect an intruding object with a given probability or to trace a given fraction of trajectory of the object. The aim of this work is to analyze the coverage property of a sensor network, where a large number of sensors are randomly deployed. In particular, we focus on the pathcoverage property, where “path” corresponds to the trajectory of a moving object in the field. Note that there exist various requirements on the path coverage depending on applications. Some applications require low degree of coverage to detect a moving object at once. Other applications may require higher degree of coverage to trace the whole trajectory of the moving objects. In this work, we characterize the path coverage in three metrics; the fraction of a path covered by sensors (fraction of coverage), the probability of the whole path being completely covered (probability of complete coverage), and the probability of the path being partially covered (probability of partial coverage). We derive the analytical formulas for the above-mentioned metrics in terms of the density of sensors, the communication range of a sensor, and the sensing range of a sensor. The coverage problem of a sensor network has been extensively studied in recent years. For example, Meguerdichian et
al. [1], [2] proposed an algorithm to discover the minimum exposure path, which in turn is used to characterize the worst coverage performance of a sensor network. Clouqueur et al. [3] studied the best placement of sensors that maximizes the detection probability of an object intruding a given field. These studies considered structured sensor networks where the location of each sensor is known. Liu et al. [4] studied the stochastic coverage problem for a randomly distributed sensor network. They derive the expression for the fraction of the field covered by sensors and for the ability of the network to detect moving objects. Ram et al. [5] studied the path coverage of randomly deployed sensor networks, and Manohar et al. [6] extended their results to cases where sensor locations form a non-homogeneous random sensor network. Lazos and Poovendran [7] studied the stochastic coverage problem of a sensor network in a very general setting where each sensor need not to have identical sensing capabilities and the sensing area of a sensor has any arbitrary shape. These studies assumed that each sensor has a long wireless-communication range so that they can report sensing data directly to a gateway (base station). In other words, the connectivity of sensors to gateways was not concerned. The contribution of this work is to derive analytical expressions for the three coverage metrics taking into account of the connectivity of sensors to the base station. The derived formulas are further used to provide the fundamental relationship between the communication range and various coverage metrics. As a result, for example, we find that a sensor should have a long wireless-communication range to obtain large probability of partial coverage with modest density of sensors. The rest of this article is organized as follows. In Section II, we state our sensor network model and some assumptions used in the work. In Section III, we present some related background on the coverage theory that we used for deriving analytical coverage expressions. In Section IV, we derive analytical expressions for path coverage of sensor networks when each sensor has an infinite-communication range. In Section V, we study the path coverage when each sensor has a finite-communication range and thus some sensors do not have connectivity to the gateway. In Section VI, we validate derived analytical expressions for path coverage via simulation. In
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Inactive Sensor
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rs
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Fig. 2.
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– Probability of complete coverage pc (l) : the probability of the whole path of length l being completely covered by active sensors. The probability of complete coverage corresponds to the probability that an object moving along the path is always detected within the field of watch.
Sensing model
Section VII, we present our conclusion.
– Probability of partial coverage p pc (l) : the probability of the path of length l being partially covered by one of active sensors. The probability of partial coverage corresponds to the detectability of an object moving along the path
II. N M A. Sensor location We assume that the locations of sensors are uniformly and independently distributed in the field of watch. Under the uniform and random distribution of sensors, the locations of sensors can be modeled by a stationary two-dimensional Poisson process. In this article, we denote the density of sensors as λ. The number of sensors located in a region of area A, NA , follows a Poisson distribution, that is (λA)n −λA P[NA = n] = e , n!
Segment.
III. C P In this section, we present some related background on coverage process that we use in Sections IV and V for deriving analytical expressions for coverage. For the detail, please see [11], [12]. A. Definition
E[NA ] = λA.
(1)
B. Sensing model We assume that each sensor has an identical sensing range, r s . A sensor can detect all events within the sensing range with probability 1, but it cannot detect any events at all outside the sensing range. This simplified sensing model is usually called “Boolean sensing model” [4], [8], [9], [10]. We also assume that each sensor has an identical communication range, rw . A sensor can communicate with all sensors or gateways (base stations) within its communication range (Fig. 1). A gateway has a wired connection to the data-storage server. A sensor can report its sensing data to the data server if it is connected directly or indirectly (via other sensors) to a gateway. A sensor is said to be “active” if it has a (direct or indirect) connection to a gateway; a sensor is inactive if it is not active. A sensing region covered by an active sensor (circular region of radius r s ) is called active sensing area. C. Metrics for coverage In this article, we characterize the coverage of a sensor network by the following three metrics. – Fraction of coverage f p : the fraction of the path covered by active sensors. The fraction of coverage, f p , corresponds to the fraction of time which a moving object is detected by active sensors.
The part of the path covered by an active sensing area is called segment. Figure 2 shows an example of the segment. We denote the beginning of the ith segment as ξi and the length of the ith segment along the path as ηi . If we let x + y denote the point that is y away from the point x1 , then ξi + ηi is the end point of the ith segment. The sequence of segments C = {(ξi , ξi + ηi )} is called coverage process [11]. As used in [11], we use the term “clump” to denote a connected set of segments and use the term “spacing” to denote a gap between successive two clumps (Fig. 3). We let Z j denote the length along the path of the jth clump and Y j denote the length along the path of the jth spacing (the gap between jth clump and ( j + 1)th clump). We also use the term “cycle” to denote the connected set of adjacent clump and spacing; for example, the jth cycle is the connected set of the jth clump and jth spacing (Fig. 3). X j denote the length of the jth cycle (X j = Y j + Z j ). The coverage process on the straight line is analytically tractable when sensors are randomly deployed. In the following subsections, we show some basic results concerning the coverage process on the straight line. If the path is a straight line and sensors are randomly deployed, {Xi }, {Yi }, and {Zi } are independent and identically distributed, so in the followings we omit the subscript i. 1 We assume that the path has the direction. Thus, the distance y might be minus.
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2nd clump
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From (3), we can compute the expectation or variance of the cycle length. For example, the expectation of the cycle length is given by eλπrs . 2λr s 2
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1 eλπrs − 1 = . α 2λr s 2
E[Z] = E[X] −
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It was shown in [11] that the distribution of the clump length is close to exponential for large values of λ. More precisely, lim P[Z/E[Z] ≤ x] = 1 − e−x .
λ→∞
E[ N (t )] 2rs Ot Fig. 4.
Region where sensors starting segments in [0, t] exist.
B. Distribution of segment length When the path is a straight line, the beginning points of segments, {ξi }, follow a Poisson process. Let N(t) denote the number of beginning points of segments within [0, t]. Since N(t) is equal to the number of sensors existing in the hatched region in Fig. 4, the expectation of N(t) is E[N(t)] = 2r s λt. The density of the segment-beginning points, α, is given by α = E[N(1)] = 2r s λ. When the path is a straight line, the lengths of segments, {ηi }, are identically and independently distributed. The distridef bution function of the length of segment, G(t) = P[η ≤ t], is given by: t < 0, √02 r s −(t/2)2 G(t) = (2) 0 ≤ t ≤ 2r s , 1− rs 1 t > 2r s . C. Cycle length and clump length Although the segment-beginning points follow a Poisson process, the clump-beginning points does not. To see this, observe that, if a segment-beginning point starts a clump, then the adjacent segment-beginning point is unlikely to become a clump-beginning point. The sequence of the clump beginning points, however, makes a renewal process. The LaplaceStieltjes transform of the cycle length (renewal interval), def ζ(s) = E[e−sX ], is given by ∞ −1 t 1 exp −st − α {1 − G(x)}dx dt . ζ(s) = 1 − s+α 0 0 (3)
Thus, P[Z ≤ x] ≈ 1 − e−x/E[Z] ≈ 1 − exp{−2λr s e−λπrs x}. 2
(6)
IV. P (rw = ∞) In this section, we study the path coverage property of a random sensor network when rw = ∞; that is, all sensors are active (the network is fully connected). Saito et al. proposed architecture for sensor networks where each sensor has a wide wireless-communication range so that all sensors can directly communicate with base stations [13]. Such an architecture is equivalent to letting rw = ∞ with respect to the connectivity of the network. A. Fraction of coverage Consider an open path of length l, which is labeled as S in the following. Let C(t) denote the expectation of the sum of covered region on S in the interval [0, t] on S where one end of S is located at origin. C(t) should satisfy the following differential equation; dC(t) = p(t), dt where p(t) is the probability the point located at t is covered by at least one active sensing area. Since sensors are randomly deployed, p(t) does not have dependence on t and thus p(t) = p, def
where p = 1 − e−λπrs . This argument yields that C(t) = pt. The coverage of S is then given by 2
f p = C(l)/l = p.
(7)
The above mentioned argument is also applicable to any arbitrary closed paths. Thus, the fraction of coverage should be equal to p regardless of the length or shape of the path.
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Fig. 5.
Region where sensors partially covering the path exist
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Fig. 6. Active sensor ratio vs. sensor density (rw = 1).
B. Probability of complete coverage First consider a segment of straight line of length l. Such a segment is completely covered if and only if one end of the segment exist in a clump and the residual length of the clump is longer than l. Thus, we have ∞ P[Z > u] du pc (l) = p E[Z] l ∞ 2 = 2λr s e−λπrs P[Z > u]du. l
As explained in Section III, if the density of sensors is high, the distribution of the clump length, P[Z ≤ u], can be approximated by the exponential distribution (6). The exponential approximation yields ∞ 2 2 exp{−2λr s e−λπrs u}du, pc (l) ≈ 2λr s e−λπrs (8) l −λπr2s l}. = exp{−2λr s e It is not possible to get the exact expression for complete coverage probability for paths with arbitrary shapes. By regarding the path of length l as a segment of straight line of length l, however, we could approximately evaluate the complete coverage probability by (8). We will numerically show in Section VI that (8) gives accurate estimates of the complete coverage probabilities of paths with various shapes. C. Probability of partial coverage A segment of straight line of length l is covered by some active sensing area if and only if at least one senor exists in the hatching region in Fig. 5. Thus, it follows from (1) that p pc (l) = 1 − e−λ(2rs l+πrs ) . 2
From the above results, we could also get approximate formulas for the partial coverage probabilities of paths with arbitrary shapes as follows:
closed path, 1 − e−2λrs l (9) p pc (l) ≈ 2 1 − e−λ(2rs l+πrs ) open path. In Section VI, we show the accuracy of the approximate formulas (9) by simulation. Remark 4.1: For a straight line, some of results explained here have been obtained in existing works including [5].
Fig. 7.
Shape of a random path.
V. P (rw < ∞) Next we consider the case where rw < ∞. Here, we derive the approximate formulae for the three coverage metrics using the notion of the critical density. According to the percolation theory, there exists a critical density λc , where a phase transition occurs with respect to the size of the largest cluster, in which sensors can communicate with each other [14], [4]. If the density of sensors is below the critical density, all clusters are finite in size almost surely and most of clusters are not connected to a gateway if few gateways are located. In such a case, most of sensors are inactive. In contrast, if the density of sensors exceeds the critical density, a unique unbounded cluster emerges. Although the emergence of the unbounded cluster does not ensure that all the sensors are connected and thus active, we have confirmed through simulation that the ratio of active sensors rapidly tends from 0 to 1 around the critical density as shown in Fig. 6 where rw = 1. Thus, here we assume that if the sensor density is below λc , all sensors are inactive, while if the sensor density exceeds λc , then all sensors are active. Under this assumption, the three metrics for path coverage have the following expressions:
p λ ≥ λc (rw ), (10) fp = 0 otherwise,
2 exp{−2λr s e−λπrs l} λ ≥ λc (rw ), pc (l) ≈ (11) 0 otherwise, λ ≥ λc (rw ) (closed path), 1 − e−2λrs l −λ(2r s l+πr2s ) p pc (l) ≈ (12) λ ≥ λc (rw ) (open path), 1 − e 0 otherwise, where, in order to remind the dependence of λc on rw , we use the notation λc (rw ). It was numerically shown in [4] that the critical density for two-dimensional area is given by λc (rw ) = 0.353(2/rw )2 .
(13)
The accuracy of (10), (11), and (12) are shown in Section VI. Based on (10), (11), and (12), we are able to investigate the fundamental relationships between the communication range and the three coverage metrics. To see this, we define def
λ f r (q, rw ) = min{λ; f p ≥ q},
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def
1 Theory Line
Fraction of Coverage
where λ f r (q, rw ) is the sensor density required for making the fraction of coverage larger than q when the communication def range is equal to rw . We refer λ f r (q) = λ f r (q, ∞) to as the minimum required sensor density for making the fraction of coverage larger than q because λ f r (q) ≤ λ f r (q, rw ) for all rw . Likewise, we define the followings:
Probability of Complete Coverage
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λc (q, rw ; l) = min{λ; pc (l) ≥ q},
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Using variable λ f r (q, rw ), we further define for λ ≥ λc (q),
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Fig. 9. Probability of complete coverage and the sensor density (rw = ∞).
def
rw: f r (q, λ) = min{rw ; λ f r (q, rw ) = λ},
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rw:c (q; l) = rw:c (q, λc (q); l), def
rw:pc (q, λ; l) = min{rw ; λ pc (q, rw ; l) = λ},
Theory Line Random
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def
Probability of Partial Coverage
Probability of Partial Coverage
where rw: f r (q, λ) is the communication range, above which the fraction of coverage is larger than q when the sensor def density is equal to λ. We refer rw: f r (q) = rw: f r (q, λc (q)) to as the sufficient communication range for making the fraction of coverage larger than q because rw: f r (q) ≥ rw: f r (q, λ) for all λ ≥ λc (q). Likewise we define
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Fig. 10. Probability of partial coverage and the sensor density (rw = ∞: open path).
Theory Circle Rectangle
Fig. 11. Probability of partial coverage and the sensor density (rw = ∞: closed path).
def
rw:pc (q; l) = rw:pc (q, λc (q); l). Based on rw: f r (q), rw:c (q; l), and rw:pc (q; l), in Section VI, we will numerically investigate the impact of the communication range on the three coverage metrics. VI. N R A. Simulation Condition In this section, we validate our theoretical results via simulation experiments. In the experiments, we randomly deployed a variable number of sensors in a square region of 1000×1000. All sensors had a circular sensing area with radius r s = 1 (that is, the unit of distance was r s ). We evaluated the coverage of paths with one of four shapes (straight line, circle, rectangle, and random) in terms of the three metrics (fraction of coverage, probability of complete coverage, and the probability of partial coverage) and compared the results with theoretical ones. For reference, in Fig. 7, we depict a random-shape path used in the simulation. B. Dependence of the coverage metrics on the path shape First, we investigated the dependence of the coverage metrics on the shape of the path when rw = ∞. 1) Fraction of coverage: In Fig. 8, we show the fraction of coverage as a function of the sensor density λ when rw = ∞. We observe that the simulation results verify formula (7) and that the fraction of coverage does not depend on the shape of the path.
2) Probability of complete coverage: In Fig. 9, we show the probability of complete coverage of the path of length 100 as a function of the sensor density λ when rw = ∞. We also observe that formula (8) agrees with the simulation results and that the probability of complete coverage does not largely depend on the shape of the path. 3) Probability of partial coverage: In Figs. 10 and Fig. 11, we respectively show the probability of partial coverage of open path and closed path of length 100 as a function of sensor density when rw = ∞. We observe that, as with other coverage metrics, formula (9) agrees with the simulation results and that the complete coverage probability does not largely depend on the shape of the path. C. Path coverage metrics when rw < ∞ Next, we investigated the path coverage metrics when rw < ∞ and evaluated the accuracy of (10), (11), and (12). 1) Fraction of coverage: In Fig. 12, we show the fraction of coverage of a straight line with length 100 when rw = 2, rw = 4 and rw = 10. We observe that formula (10) agrees with the simulation results, which verify the accuracy of (10). 2) Probability of complete coverage: In Fig. 13, we show the complete coverage probability of a straight line when rw = 1 and rw = 2. Note that formula (11) makes very small differences between cases rw = 1, rw = 2, and rw = ∞, and these differences are not seen in Fig. 13. The simulation results show that the difference in the communication range has very small impact on the complete coverage probability,
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Fig. 12. Fraction of coverage and the sensor density (rw < ∞).
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Fig. 15. Sufficient communication range and minimum required sensor density in terms of fraction of coverage.
Fig. 14. Fraction of coverage and the sensor density (rw < ∞).
VII. C
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In this work, we analyze the path coverage of a randomly deployed sensor network in terms of the fraction of coverage, probability of complete coverage, and probability of partial coverage, taking into account of the connectivity of sensors to gateways (base stations). Through the numerical experiments, we gain several insights on the influence of the communication range as well as the sensor density on the path coverage property. For example, in terms of the coverage, detecting an intruding object at once requires much less density of sensors than that for completely tracing the path. To utilize this benefit in terms of the coverage, however, we should deploy sensors that have long wireless-communication ranges; otherwise, we should also deploy sensors with high density in order to ensure Sufficient Communication Range
Finally, we analyze the impact of the communication rage on the path coverage. Figure 15 shows the sufficient communication range rw: f r (q) and the minimum required sensor density λ f r (q) when the fraction of coverage is equal to q. The figure shows that rw: f r (q) is a decreasing function of q while λ f r (q) is an increasing function of q. This implies that, as q becomes larger, deploying sensors more densely becomes more important while making the communication range longer becomes less important. The figure also shows that letting rw = 4r s is sufficient in order to make the fraction of coverage larger than 0.2. Figure 16 shows the sufficient communication range rw:c (q; 100) and the minimum required sensor density λc (q; 100) in terms of the complete coverage probability of a path with length 100. The figure shows that rw:c (q; 100) and λc (q; 100) are respectively decreasing and increasing functions of q. The figure also shows that letting rw = r s is sufficient in terms of the complete coverage probability, while sensors should be deployed with high density (larger than 1). That is, in order to completely trace the path with some positive probability, deploying sensors with high density is much more important than taking the communication range longer. Figure 17 shows the sufficient communication range rw:pc (q; 100) and the minimum required sensor density λ pc (q; 100) in terms of the partial coverage probability of a path with length 100. In contrast to Fig. 16, Fig. 17 reveals that the sufficient communication range is much longer than the sensing range (rw:pc (q; 100) > 10r s ), while the sensor density can be taken at a low value (less than 0.02). Note that partially covering the path is equivalent to detecting the moving object at least once. That is, in terms of the detectability of moving objects, the communication range is much more important than the sensor density. To look this finding more closely, in Fig. 18, we plot the sensor density required for 90-percent detectability, λ pc (q, rw ; 100), as a function of the communication range. We observe that, when rw is equal to the sufficient communication range rw:pc (0.9; 100) (= 11.16), the sensor density required for 90-percent detectability is 0.011. When rw = 2 (= 2r s ), however, the sensor density for 90-percent detectability is 0.353, which is about thirty times as large as that when rw = rw:pc (0.9; 100). For reference, we illustrate an example of deployment of sensors with density 0.353 (0.011) in Fig. 19 (Fig. 20), where circles represent sensing ranges of deploying sensors. This example also illustrates that the communication range has large impact on the detectability of intruding objects.
Probability of Complete Coverage
Fraction of Coverage
1
and formula (11) gives good approximation for the complete coverage probability. 3) Probability of partial coverage: In Fig. 14, we show the partial coverage probability of a straight line with length 100 when rw = 4, rw = 10, and rw = 20. We observe that the partial coverage probability has some dependence on the communication range and formula (12) agrees with the simulation results.
Probability of Partial Coverage
Fig. 17. Sufficient communication range and minimum required sensor density in terms of partial coverage probability.
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Fig. 18.
Sensor density required for 90-percent detectability.
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[8] D. Tian and N.D. Georganas, “A coverage-preserving node scheduling scheme for large wireless sensor networks,” in First ACM International Workshop on Wireless Sensor Networks and Applications, 2002, pp. 32– 41. [9] F. Ye, G. Zhong, S. Lu, and L. Zhang, “Peas: A robust energy conserving protocol for long-lived sensor networks,” in Proc. IEEE ICDCS, 2003. [10] S. Shakkottai, R. Srikant, and N. Shroff, “Unreliable sensor grids: Coverage, connectivity and diameter,” in Proc. IEEE INFOCOM, 2003. [11] P. Hall, Introduction to the Theory of Coverage Processes, John Wiley & Sons, 1988. [12] D. Stoyan, W.S. Kendall, and J. Mecke, Stochastic Geometry and Its Applications (2nd Edition), John Wiley & Sons, 1995. [13] H. Saito, M. Umehira, and M. Morikura, “Considerations of global ubiquitous network infrastructure,” IEICE Trans. Commun., vol. Vol. J88-B, no. 11, 2005. [14] O. Dousse, P. Thiran, and M. Hasler, “Connectivity in ad-hoc and hybrid networks,” in Proc. IEEE INFOCOM, 2002. [15] H. Saito and S. Shioda, “Application of insensitivity analysis of coverage processes to wireless sensor networks,” submitted for publication.
Density 0.011.
the connectivity to the gateway even for the detection of intruding object. Meanwhile, for tracing the whole path, each sensor does not need to have long wireless-communication range. To cover the whole path, a large number of sensors should be deployed with high density, which ensures the connectivity at the same time if the wireless-communication range of each sensor is comparable to the sensing range. Although in this paper we assume that each sensor has identical sensing capabilities, this assumption can be relaxed. More detailed analysis when sensors have various sensing ranges will be reported elsewhere [15]. R [1] S. Meguerdichian, F. Koushanfar, G. Qu, and M. Potkonjak, “Exposure in wireless ad-hoc sensor networks,” in Mobile Computing and Networking, 2001, pp. 139–150. [2] S. Meguerdichian, F. Koushanfar, M. Potkonjak, and M. B. Srivastava, “Coverage problems in wireless ad-hoc sensor networks,” in Proc. IEEE INFOCOM, 2001. [3] T. Clouqueur, V. Phipatanasuphorn, P. Ramanathan, and K. K. Saluja, “Sensor deployment strategy for target detection,” in First ACM International Workshop on Wireless Sensor Networks and Applications, 2002. [4] B. Liu and D. Towsley, “A study on the coverage of large-scale sensor networks,” in First IEEE International Conference on Mobile Ad-hoc and Sensor Systems, 2004. [5] S. Sundhar Ram, D. Manjunath, S.K. Iyer, and D. Yogeshwaran, “On the path coverage properties of random sensor networks,” IEEE Trans. Mobile Computing, vol. 6, no. 5, pp. 494–506, 2007. [6] P. Manohar, S. Sundhar Ram, and D. Manjunath, “On the path coverage by a non homogeneous sensor field,” in Proc. IEEE Globecom, 2006. [7] L. Lazos and R. Poovendran, “Stochastic coverage in heterogeneous sensor networks,” ACM Trans. Sensor Networks, vol. 2, no. 3, pp. 325– 358, 2006.
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