The Intrusion Detection in Mobile Sensor Network Gabriel Y. Keung
The Hong Kong University of Science and Technology, Department of Computer Science and Engineering, Hong Kong, China
[email protected]
Bo Li
The Hong Kong University of Science and Technology, Department of Computer Science and Engineering, Hong Kong, China
[email protected]
Qian Zhang
The Hong Kong University of Science and Technology, Department of Computer Science and Engineering, Hong Kong, China
[email protected]
ABSTRACT
1.
Intrusion detection is an important problem in sensor networks. Prior works in static sensor environments show that constructing sensor barriers with random sensor deployment can be effective for intrusion detection. In response to the recent surge of interest in mobile sensor applications. This paper studies the intrusion detection problem in a mobile sensor network, where it is believed that mobile sensors can improve barrier coverage. Specifically, we focus on providing k-barrier coverage against moving intruders. This problem becomes particularly challenging given that the trajectories of sensors and intruders need to be captured. We first demonstrate that this problem is similar to the classical kinetic theory of gas molecules in physics. We then derive the inherent relationship between barrier coverage performance and a set of crucial system parameters including sensor density, sensing range, sensor and intruder mobility. We examine the correlations and sensitivity from the system parameters, and we derive the minimum number of mobile sensors that needs to be deployed in order to maintain the k-barrier coverage for a mobile sensor network. Finally, we show that the coverage performance can be improved by an order of magnitude with the same number of sensors when compared with that of the static sensor environment.
Recently there has been an increased interest in the deployment of mobile sensors for intrusion detection, partly motivated by the demand of border surveillances. For instance, the American Border Patrol operation [1] tested sensormounted UAV in conjunction with the existing static sensor network (or so-called “virtual fence”) along the American/Mexican borders. Under such a scenario, mobile sensors are not only able to cover larger areas, but also act as a backup when a “coverage hole” appears on the virtual fence due to ground sensor faults or use as a follow-up verification due to the false positives generated from ground sensors. Therefore, a mobile sensor network (MSN) can potentially improve the intruder detection by overcoming the coverage and physical sensing limitations in a stationary wireless sensor network (WSN), which is the focus in this paper. In this paper, we are primarily interested in the barrier coverage by considering both moving intruders and mobile sensors. Specifically, we extend the notion of k-barrier coverage, which firstly introduced in [9] in stationary WSNs, in which a sensor network deployed over a belt region is said to provide k-barrier coverage if every path that crosses the width of the belt is covered by at least k distinct sensors. This barrier coverage definition is different from the general coverage formulation [19] in that the k-barrier coverage does not capture simultaneous coverage from multiple sensors; instead it provides the cumulative coverage from distinct sensors. Different from our problem, the k-barrier coverage in a WSN is mainly determined by the initial network configuration. By specifying the deployment strategy and knowing sensing characteristics of sensors, we can derive the kbarrier coverage performance of a WSN and the performance remains unchanged over time. Currently, barrier coverage studies in MSNs mainly focus on designing sensor reposition algorithms to achieve coverage requirements [15]. Our focus in this paper is completely different from all prior works. We formulate the dynamic aspects of the k-barrier coverage that depend on both sensor and intruder mobility, and examine the k-barrier coverage performance under different network parameters and configurations. We now briefly describe the k-barrier coverage provided by an MSN, and the related research problems. Problem Statement: Given an initial sensor deployment over a belt region and a sensor mobility pattern, intruders are assumed to cross from one parallel boundary of the region to the other. We define the k-barrier coverage for an MSN if every intruder path that crosses the width
Categories and Subject Descriptors G.3 [Probability and Statistics]: Stochastic Processes; C.2.1 [Computer-Communication Networks]: Network Architecture and Design—performance measures
General Terms Theory, Performance
Keywords Mobile Sensor Network, Mobility, Barrier Coverage
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. MobiHoc’10, September 20–24, 2010, Chicago, Illinois, USA. Copyright 2010 ACM 978-1-4503-0183-1/10/09 ...$10.00.
INTRODUCTION
intruder
intruder
sensor
sensor
sensor sensor sensor
2R
W
sensor sensor
Belt Area
Figure 1: Example of intrusion detection problem in mobile sensor network. of the belt is cumulatively detected by at least k mobile sensors along its traveling path. We are interested in computing the probability Pr(Λ ≥ k) for a given MSN and its network parameters to satisfy k-barrier coverage, where Λ is the cumulative coverage count by mobile sensors. Given the random nature of both intruders and sensors, it is not feasible to develop a deterministic algorithm to determine whether an MSN can provide k-barrier coverage; instead, we focus on obtaining the probability of k-barrier coverage by considering all possible sensor coverage from any trajectories of sensors and intruders. It is particularly challenging to model such a stochastic problem for multiple moving objects. Furthermore, we are also interested in determining the minimum number of mobile sensors that needs to be deployed in order to provide the k-barrier coverage in MSNs. It turns out that our problem is very similar to the problem in classical kinetic theory of gas molecules in physics, which allows us to establish and derive the inherently dynamic relationship between moving intruders and mobile sensors. The objectives of this paper are: 1) identify and characterize the dynamic aspects of the k-barrier coverage under the sensor and intruder mobility, 2) to compute the probability of k-barrier coverage in a MSN, 3) to investigate the probability of k-barrier coverage sensitivity under different network parameters and system configurations, and 4) to examine the average travel distance by an intruder between successive sensor coverage and its distribution. Summary of Results: We study the k-barrier coverage problem in mobile sensor networks, while the existing works have only considered k-barrier coverage in stationary wireless sensor networks. We derive the dynamic relationship for the k-barrier coverage performance from the density of mobile sensors, sensing range, sensor and intruder mobility. We analyze the correlations and sensitivity between the parameters. We compare the coverage performance in term of k-barrier coverage probability with that from stationary wireless sensor networks, which demonstrates coverage performance significant gain can be obtained. Organization: The rest of this paper is organized as follows. Section 2 summarizes the intrusion detection studies in a wireless sensor network, and the motivations to extend the study to a mobile sensor network. Section 3 describes the network and mobility model, as well as definitions of the intrusion detection problem in a mobile sensor network. Section 4 formulates the intrusion detection problem. Section 5 examines the k-barrier coverage sensitivity under different network parameters and configurations. Section 6 reviews the related works, and finally Section 7 concludes the paper.
2.
we next present the motivation for our study in MSNs and the key challenges. Finally, we introduce the gas kinetic theory and mean free path from physics to facilitate the study of our problem.
PRELIMINARY AND OVERVIEW We first outline the intrusion detection problem in WSNs,
2.1
Intrusion Detection Problem in a Stationary WSN
There has been significant interest in border surveillance using WSNs. One typical example is the operation of American Border Patrol, which places sensors along the American/Mexican borders (called an electronic “virtual fence”) [1]. Ground sensors and infrared sensors are deployed for detecting ground vibrations from foot steps and identifying warm body movement respectively. This illustrates that constructing sensor barriers can be effective for intrusion detection. The concept of barrier coverage in WSN was firstly proposed in [9], in which it defines that k-barrier coverage is to detect any intruders crossing the belt area (border) along any paths by at least k distinct sensors. The work presented a centralized algorithm to determine whether a network provides k-barrier coverage. An efficient distributed algorithm was proposed in [12] to construct sensor barriers on irregular shaped area without any constraints on crossing paths. The local barrier coverage was introduced in [4], in which it generalized the notion of k-barrier coverage to L-local kbarrier coverage. This can guarantee the detection of all crossing paths whose trajectory is confined to a slice with length L of the belt region. A follow-up work proposed a concept called quality of barrier coverage in [5], which was used to measure the localized quality of k-barrier coverage. A distributed algorithm was also developed to measure the quality and identify regions to be repaired. All prior works consider a stationary wireless sensor network, in which sensors remain stationary after the initial deployment. The k-barrier coverage of such a network is thus primarily determined by the initial network configuration and sensor deployment. Once the deployment strategy and sensing characteristics of the sensor are known, the kbarrier coverage can be computed and remains unchanged over time.
2.2
Motivation and Overview
It is well known that node mobility improves the spatial coverage of sensor networks [11], [16]. For intrusion detection, there were several works [2], [15], that considered designing sensor reposition algorithms for maintaining a barrier coverage service in an MSN. Sensor mobility can also be exploited to improve the intrusion detection. By leveraging the barrier coverage concept in a stationary wireless sensor network, in this paper, we consider how to provide barrier coverage service over a belt region for intrusion detection in an MSN. The key difficulty is caused by the sensor mobility and intruder movement. With continuous sensor movement, previously uncovered regions can become covered as sensors move into the region; similarly, previously covered regions can become uncovered when sensors move away. As a result, the regions covered by mobile sensors change over time. Probabilistically there is a higher probability that intruders will be inside a covered area in an MSN over a given period of time. Furthermore, in a stationary WSN, an undetected or uncovered intruder can remain undetected or uncovered if the intruder moves
along an uncovered path. In an MSN, however, an intruder is more likely to be detected since moving sensors patrol around a belt area. Therefore, mobile sensors, while providing a time-varying barrier coverage, can potentially significantly improve the intrusion detection performance. However, there are several challenges for intrusion detection in a MSN: How do we describe and define the k-barrier coverage in an MSN over a time period? What is the proper time duration that an intruder is covered by mobile sensors? What is the relationship between intrusion detection probability and sensor and intruder mobility? Finally, how to quantify the k-barrier coverage performance gain with the deployment of mobile sensors?
2.3
The Kinetic Theory of Gas Molecules
Given the stochastic nature of the intrusion detection problem in an MSN, it is extremely complicated to capture all possible trajectories of sensors and intruders. Fortunately, as it turns out that our problem is similar to a problem in classical kinetic theory of gas molecules in physics, which describes the behavior of gas molecules in macroscopic terms and has a well developed theoretical framework that we can leverage for our study. Specifically, the theoretical framework for mean free path of moving electrons in air molecules [13] is very suitable for our study. In physics, the mean free path is the average travel distance by a particle (e.g. an electron) between successive impacts with other particles such as air molecules. The basic quantity of interest is the time for a collision between an electron and a molecule, and the distance traveled by the electron until the next collision. The mean free path of an electron can be calculated by deriving the Boltzmann constant, the temperature, the pressure, and the diameter of air molecules [13]. This essentially requires to calculate the position of molecules until they cross the trajectory of an electron. Given that both air molecules and electron particles are moving objects, this scenario is almost identical for mobile intrusion detection, in which a mobile sensor needs to “collide” with a moving intruder for detection. Intuitively, in order to apply the mean free path theory, we can treat a mobile sensor as an air molecule, and an intruder as an electron. The average travel distance of an intruder between successive coverage by mobile sensors can then be derived from the kinetic theory. This will allow us to further compute the probability that a MSN meets with certain k-barrier coverage performance. To apply the kinetic theory to the intrusion detection problem, we need to make several assumptions: (1) the spatial separations between sensors (molecules) are large comparing to the sensing region (molecular size); (2) sensors are initially located randomly and move with a constant speed; (3) sensors (molecules) should obey the Newton’s laws of motion, where sensors undergo elastic collisions with the region boundaries (walls), but otherwise exert no forces on each other.
3.
NETWORK AND MOBILITY MODEL
In this section, we describe the basic system assumptions, and in particular the mobility models for mobile sensors and moving intruders. We then extend the notion of k-barrier coverage in MSNs. We consider an MSN to consist of N (A) mobile sensors initially placed inside a two dimensional geographical region
Figure 2: k-barrier coverage of intruder traveling path A shown in Figure 1. The region can be in any convex shapes under the proposed formulation. To keep it mathematically tractable, we assume A is a long and narrow belt-like region, which has four boundaries. Two of them are parallel, and the other two are orthogonal to the parallel pair. We assume the width of the area to be W and the length to be |A|/W , where |A| represents the area of the region. For the initial configuration (at time t = 0), we assume sensors are independently deployed with random uniform distribution. The random deployment can be the result of certain deployment strategies. For example, sensors may be air-dropped or launched via artillery in battlefields or unfriendly environments. Under this assumption, the sensor location can be modeled by a stationary two-dimensional Poisson process. Denote the density of Poisson process as nA . The number of sensors located in the region A, N (A), follows a Poisson distribution of parameter nA · |A|. e−nA |A| (nA |A|)k (1) k! We define Pr(Y ) to be the probability that event Y occurs, and Pr(Y ) = 1 − Pr(Y ). We use E(X) to denote the expected value of a random variable X. Pr(N (A) = k) =
3.1
Sensing Model
We assume that each sensor has a sensing range, R. A sensor can only sense the environment and detect events within its sensing region, which is the disk of radius R centered at the sensor. An intruder, treated as a point, is any object that is subjected to sensor detection as it crosses the barrier. It is said to be covered or detected by a sensor if it has been located inside the sensing region of the sensor. This definition is usually referred as a binary or disc-based sensing model [19]. In the intrusion detection formulation in this paper, we essentially define a probabilistic coverage (see Section 4). Ideally, a probabilistic sensing model such as the one in [8] could be more appropriate. For simplification and mathematically tractability, we adopt the disc-based sensing model in this work.
3.2
Mobility Model
In an MSN, depending on the mobile platform and application scenario, sensors can choose from a wide variety of mobility strategies, from passive movement to highly coordinated and complicated motion. Sensors deployed in the air, ocean or on wild animals move passively according to
external forces such as air, ocean currents or wild animal movement patterns; simple robots may have a limited set of mobility patterns, and advanced robots can navigate in a more complicated itinerary. In this work, we consider the following sensor mobility model. We assume that sensors move independently of each other and without any coordination between them. The movement of a sensor is characterized by its speed and direction. A sensor randomly chooses a direction θ ∈ [0, 2π) according to the distribution with probability density function PΘ (θ). The speed of the sensor is randomly chosen from max a range vm ∈ [0, vm ], according to a probability density max function of PVm (vm ) and vm is the maximum sensor speed. A sensor travels to the boundary of the area A with the chosen speed and direction. Once the boundary is reached, the sensor bounds back, by choosing another angular direction and continues the process. We refer the above model as the random direction mobility model [3]. The random direction mobility model, along with the random way point mobility model, is perhaps the most widely used synthetic mobility model in mobile communications. It was introduced to overcome density waves in the average number of neighbors produced by the random way point mobility model [14]. A density wave is a cluster of sensors in one part of the simulation area. In the case of the random way point mobility model, this clustering occurs near the center of the simulation area. In other words, the probability that a mobile sensor selects a new destination located at the center of the simulation area, or a destination requiring to travel through the middle of the simulation area, is high. Thus, the mobile sensors appear to converge, disperse and converge again for generating density waves. In order to alleviate such behavior, the random direction mobility model was chosen to describe the sensor movement pattern. Intruder movement is assumed to follow a crossing path, which is defined as a path (line segment) crossing from one parallel boundary to another. A crossing path is orthogonal if its length equals to the width of the area W . This is denoted as an orthogonal crossing line. In order to apply the gas kinetic theory, we assume the velocity of an intruder is a constant vi . As it will become clear in Section 4, from our formulation of this paper, the intruder movement is not explicitly restricted to any specific mobility model; instead the most relevant parameter is the length of the path. For the mathematical tractability, it is assumed that the intruder mobility is independent of the sensor mobility. In reality, however, there could be spatial and temporal correlations on the mobility pattern, which is not captured in this formulation.
3.3
Coverage Measurement
We define the k-barrier coverage for MSNs in this subsection. Specifically we extend the definition of k-barrier coverage in a WSN [9]. We first define the k-barrier coverage for an intruder traveling path, if the path crossed the width of the belt is cumulatively detected (covered) by at least k mobile sensors. For examples in Figure 2, k-barrier coverage of the intruder traveling path (i) is given by k ≤ 3; while the k-barrier coverage of the intruder traveling path (ii) is given by k ≤ 1. It is worth noticing that the definition of k-barrier coverage is different from the general coverage definition [19], in which the k-barrier coverage does not nec-
Figure 3: The difference between the intersections between sensor traveling path and intruder traveling path and the coverage for the intruder path. essarily guarantee the simultaneous coverage from multiple sensors, but only cumulatively. We next further define the k-barrier coverage for a mobile sensor network if every intruder path that crosses the width of the belt is cumulatively detected (covered) by at least k mobile sensors along its traveling path. The performance metrics Pr(Λ ≥ k) is the probability that a MSN satisfies this k-barrier coverage definition, where Λ is the cumulative coverage count by mobile sensors for any intruder paths. We refer it as probability of k-barrier coverage for an MSN. We also define uncovered distance as the travel distance of an intruder between successive sensor coverage. For example from Figure 2, there are two uncovered distances for intruder path (i), between the successive coverage by sensors A and C, and between the successive coverage by sensors C and B. Finally we define the frequency of coverage or called coverage rate as number of sensor coverage per unit time, which can be calculated by the intruder speed over the uncovered distance. The inverse of coverage rate is the uncovered time duration between successive sensor coverage.
4.
INTRUSION DETECTION PROBLEM IN AN MSN
In this section, we formulate the intrusion detection problem in a MSN. We first identify and characterize dynamic aspects of the k-barrier coverage, which depend on both sensor and intruder movement. Figure 3 shows the cumulative coverage area of mobile sensors during a time duration τ . The cumulative coverage area depends on the sensor speed vm , the sensing range R and the time duration τ . Given that sensors and intruders are constantly moving, a sensor traveling path may intersect with an intruder’s traveling paths. Such an intersection, however, does not necessarily imply the sensor coverage on the intruder path. Figure 3 illustrates the difference between the intersection and the coverage of an intruder path. Three sensors (A, B and C) move inside the area with the grey marked trajectories, while an intruder moves on a solid black line across the area. Since the trajectory of C does not intersect with the intruder path from time t = 0 to t2 , the intruder is not covered by C. The trajectory of A intersects with the intruder path, A covers the intruder at time t = t1 . Although the trajectory of B also intersects with the intruder path, B does not cover the intruder since B moves far away from the intruder at time t = t2 .
Figure 4: An effective coverage region with sensing range R at time t = τ . As we discussed earlier that this problem is similar to a problem in classical kinetic theory of gas molecules in physics, specifically, the mean free path theory. We can treat a mobile sensor as an air molecule, and an intruder as an electron. The average travel distance of an intruder between successive coverage by mobile sensors can then be derived from the kinetic theory. Recall that, we use average uncovered distance (λ) to denote the average travel distance of an intruder between successive sensor coverage. Our goal is to obtain the probability of k-barrier coverage Pr(Λ ≥ k) in an MSN. It can be achieved by formulating the uncovered distance (λ) and sensor coverage rate (Θs ). We start to model the problem from an intruder point of view and we first assume sensors are stationary. We will relax the assumption in the later part of this section. Recall that the sensing range is R, when t = 0, a cross section of coverage can be modeled by using a circle with diameter 2R. The concept of cross section is used to express the likelihood of coverage between an intruder and sensors. After a period of time τ , the circle swept out an area (shown in Figure 4) and the number of sensor coverage can be estimated from the density of mobile sensors (nA = N (A)/|A|) inside the area. The average uncovered distance then can be taken as the travel distance of an intruder divided by the number of sensor coverage, or it equals to the intruder speed (vi ) divided by the coverage rate (Θs ). vi vi λ= = (2) Θs nA · S · vi The average uncovered distance in the static sensor case can be further written as: 1 λ= , (3) (nA ) · (2R + πR2 /vi τ ) where S = (2R + πR2 /vi τ ) is the cross section of coverage for static sensors. In order to calculate the average uncovered distance in an MSN, it is necessary to assess the average relative velocity of mobile sensors with respect to intruders. The relative velocity can be expressed in terms of intruders’ and sensors’ velocity vectors, which are shown in Figure 5b. Different sensor mobility models can result different relative velocity formulations. In this paper, for simplification, we use the random direction mobility model to describe the mobile sensor movement. We consider the case with homogeneous velocity of mobile sensors and calculate the coverage rate. The general formulation, however, is not restricted to specific mobility model and velocity assumption, as sensor mobility with different speed distributions PVm (vm ) can also be captured under the gas kinetic framework, for example, the Maxwell-Boltzmann speed distribution shown in [13]. By re-calculating the average relative speed (v rel ) un-
Figure 5: (a)The speed of the intruder relative to one of mobile sensors varies only with the angle between their respective directions of motion; (b)The relative speed vrel . der the distribution PVm (vm ) and different mobility models, we can still use the theoretical framework to compute the new uncovered distance and new coverage rate. Recall that the velocity vector of an intruder is denoted by vi and the speed of a mobility sensor is vm . The sensor coverage per unit time will be given by nA · S · v, and we need to replace the velocity v by the average relative speed of mobile sensors with respect to intruders, which is denote by v rel . Then the sensor coverage per unit time can be written as nA · S · v rel , where S is the cross section of coverage between the intruder and mobile sensors. Theorem 1. The coverage rate can be obtained by: Θv = nA · S · v rel
(4)
Proof. Consider an intruder i has certain probability to be covered by some mobile sensors j for j ∈ ∀N (A) with corresponding cross section Sj and sensor density nj . Then we have: X Θv = v rel · (nj · Sj ) j∈∀N (A)
=
nA · S · v rel
The inverse of coverage rate is the uncovered time duration between successive sensor coverage.
4.1
The k-Barrier Coverage in an MSN
With the formulation of the uncovered distance and the sensor coverage rate, we can now obtain the probability of k-barrier coverage Pr(Λ ≥ k) in an MSN. Theorem 2. Pr(Λ ≥ k) = 1 −
Γ(k, Θv · τ ) Γ(k)
(5)
where Γ(k, Θv · τ ) and Γ(k) are the incomplete Euler gamma function and the Euler gamma function respectively. Proof. With the sensor coverage rate (Θv ), we can calculate the probability of an intruder having exactly j number of sensor coverage along its traveling path with width W , which is Pr(Λ = j) =
e−Θv ·τ (Θv · τ )j j!
(6)
With this probability, we can further derive the probability of an intruder having at least k number of sensor coverage
4.2
along its traveling path with width W , which is Pr(Λ ≥ k)
=
∞ X
Pr(Λ = j)
j=k
= 1−
k−1 X „ −Θv ·τ
e
j=0
(Θv · τ )j j!
« .
The integration of Poisson process can be written as the Euler gamma function. R ∞ k−1 −t k−1 X t e dt Γ(k, Θv · τ ) Θv ·τ = Pr(Λ = j) = R ∞ k−1 −t Γ(k) t e dt 0 j=0
So far we have discussed the generalized formulation without specific sensor mobility model, we now proceed to calculate the average relative speed under the random direction mobility model. But as we illustrated previously, this can be applied to other mobility models. The speed of an intruder relative to mobile sensors varies only with the angle between their respective directions of motion, which is shown in Figure 5a. Since the mobile sensors move randomly with all possible directions (due to the random directional mobility model), a fraction dθ/2π of them move in directions that are with an angle θ of the intruder vi direction. Hence, for the average relative speed v rel , we have: Z 2π 1 v rel = vrel dθ (7) 2π 0 2 2 From Figure 5b, since vrel = vi2 + vm − 2vi vm cos θ, Z πq 1 2 − 2v v cos θ dθ v rel = vi2 + vm i m 2π 0
=
=
2(vi + vm ) · π « 12 Z π„ 2 4vi vm sin2 θ 1− 2 dθ 2 + 2v v vi + vm i m 0 2(vi + vm ) · E(u), π
where the incomplete elliptic integral E(u) is Z π/2 p E(u) = 1 − u sin2 θdθ,
(8)
(9)
(10)
We can further calculate the expected total number of sensor coverage E(Λ) as E(Λ) = =
Θv · τ
„ « πR2 (nA ) · v rel · 2R · τ + vi
dN
−N0 Θv e−Θv t dt Θv −Θv l/vi e dl −N0 vi
= =
(13)
Let ϕ(l) be the fraction of the original N0 intruders that are still going after traveled a distance l without being covered. And let Ψ(l) be the fraction of all the uncovered distances that have a length between l and l + dl. Then since ϕ = N/N0 and Ψ(l)dl = −dN/N0 , we have ϕ(l) =
e−Θv l/vi = e−l/λ 1 −l/λ e dl λ
(14) (15)
−(l + λ)e−l/λ |∞ 0 = λ
With the uncovered distance distribution, we can also calculate the probability (percentage) of uncovered distance (x), which exceeds the average uncovered distance (λ): Z ∞ Z 1 ∞ −l/λ Pr(x > λ) = Ψ(l)dl = e dl λ x x =
Theorem 3. With the coverage rate as Θv = nA · S · v rel , the average relative speed from Eq. (8) and the cross section of mobile sensor coverage as S = (2R + πR2 /vi τ ), we can obtain the average uncovered distance by: vi vi = , Θv nA · S · v rel
(12)
Let l be the uncovered distance for the length vi t, where an intruder has been covered at time t. The number of intruders that are covered between t and t + dt and terminate a path whose length lies between l and l + dl is
=
4vi vm 2 +2v v vi2 +vm i m
λ=
N = N0 e−Θv t = N0 e−Θv l/vi
where λ = Θviv . Both ϕ(l) and Ψ(l) are exponential functions, and λ equals to the intruder speed divided by the coverage rate, which is the average uncovered distance. We then recalculate the expected value of l by Z ∞ Z ∞ l −l/λ e E(l) = lΨ(l)dl = −∞ −∞ λ
0
and u =
In previous subsection, we only calculate the average value of the uncovered distance. We now compute the probability distribution function of the uncovered distance. We first consider a group of intruders initially are outside the region. Let the number originally in the group at time t = 0 be N0 , and at time t, let N of them still be going without covered by any mobile sensors. Then during the next time interval dt, the N Θv dt intruders will be covered and drop out of the group, where Θv denotes the coverage rate for an intruder with speed vi . The change of N is dN = −N Θv dt, which can be written as:
Ψ(l) =
Due to the symmetric of θ, we can rewrite the v rel as: v rel
Uncovered Distance Distribution
(11)
5.
e−x/λ
(16)
SENSITIVITY ANALYSIS
The formulation in the previous section mainly presents the dynamic aspects of the intrusion detection problem in an MSN. The coverage performance, however, is largely affected by many system parameters including the density of mobile sensors, sensing range, sensor and intruder mobility. In this section, we investigate the correlations and sensitivity on the coverage performance from a number of critical system parameters. Specifically, we study the relationships between k-barrier coverage probability, density of sensors and sensor mobility in this section.
coverage sensor mber of least k nu ty for at Probabili
Total number of sensor = 400 1.0
0.9
Probability of k-barrier coverage
1.0
0.8
0.6
0.4
0.2
ier cov era ge
0 10 20
0.0
30 800
40 600
num 400 ber of m obile sen sor
200
50
k=10 k=15
0.7
k=20 0.6
k=25
0.5
0.4
0.3
0.2
0.1
0.0
k-barr
1000
0.8
0
5
10
15
20
Proportion of the mobile sensor to static sensor population
0
Figure 7: Probability of k-barrier coverage against proportion of the mobile to static sensor population.
Random Direction Mobility Velocity ratio (Sensor:Intruder) = 1:1 Area = (50x100) Coverage Range = 1
Figure 6: The 3D plot for probability of k-barrier coverage, value of k and number of mobile sensors.
5.1
Density of Sensors and k-barrier Coverage Probability
We first study the correlation between the density of mobile sensors and the probability of k-barrier coverage. From Eqs. (5), (8) and (10), the density of sensors (nA ) is linearly related to the Poisson random variables (Θv · τ ), while the probability function of k-barrier coverage is the cumulative distribution function of the Poisson random variables. Recall that (Θv · τ ) is the expected total number of sensor coverage along the intruder traveling path from Eq. (11). In order to simplify the equations for analysis, here we consider a simple case, in which speeds of sensors and intruders are the same (vi = vm ). The relative speed v rel becomes: Z π/2 ` ´1/2 4(vi ) v rel = · 1 − sin2 θ dθ π 0 R π/2 p since 0 1 − sin2 θ dθ = 1, we have 4(vi ) . (17) π Then the uncovered distance and coverage rate become: v rel =
λ Θv
1 (nA ) · (8R/π + πR2 /vi τ ) „ « 8R 4R2 = (nA ) · · vi + π τ
=
(18) (19)
Figure 6 plots the relationships between the probability of k-barrier coverage, value of k and number of mobile sensors in an area with dimensions (50 × 100). The results are obtained by theoretical calculation from Eq. (5) by setting vi = vm and R = 1. The probability of k-barrier coverage (e.g. k = 20) increases with the Poisson distribution shape from low sensor density (nA = 1/50), and approaches to one when sensor density is ten times as large (nA = 1/5). This demonstrates that increase of mobile sensors can directly increase the probability of k-barrier coverage. With the above formulation, we can further consider the coverage problem under a hybrid sensor network, in which both static and mobile sensors are involved for intrusion
detection. We can calculate the probability of k-barrier coverage under the hybrid sensor network with a certain proportion (ρ) of the mobile to static sensor population. We can consider the hybrid sensor network as two independent sensor networks, one only consists with N (A)/(ρ + 1) static sensors; while another only consists with (N (A) · ρ)/(ρ + 1) mobile sensors. From the formulations in previous section, we calculate the coverage rates (Θs and Θv ) of the static sensor group and mobile sensor group respectively: Θv Θs
nA · ρ πR2 · (2R + ) ρ+1 vi τ nA πR2 = vi · · (2R + ) ρ+1 vi τ = v rel ·
The Poisson distribution parameters Θs · τ and Θv · τ are the intensity of the inhomogeneous Poisson processes in the static sensor group and mobile sensor group respectively. Due to the superposition property of the inhomogeneous Poisson process, the two original parameters (Θs · τ , Θv · τ ) can be combined as (Θs + Θv )τ , which is the intensity of the new inhomogeneous Poisson process in the hybrid sensor network. Pr(Λ ≥ k) = 1 −
Γ(k, (Θv + Θs ) · τ ) Γ(k)
(20)
Figure 7 illustrates the k-barrier coverage probability against the proportion of mobile to static sensor population by considering different values of k. The results are obtained under a hybrid sensor network with fixed number of sensors (N (A)=400). This clearly demonstrates a rapid improvement of k-barrier coverage probability when the proportion of mobile to static sensor population ρ increases initially from 0. The increment of k-barrier coverage probability slows down and saturates to a certain value as ρ tends to the infinity, e.g., the network consists of mobile sensors only. This result effectively demonstrates that sensor mobility can be exploited to compensate for the lack of sensors and significantly improve barrier coverage performance. Both Figures 6 and 7 illustrate that sensor mobility can improve the barrier coverage performance. It is worth pointing out that this is different from the work in [9], which proposed an deterministic algorithm to examine whether a stationary wireless sensor network can provide the k-barrier
1.3
40
30
1.1 20
mbe
Average
R
1.2
0.8
0.6
0.4
of
50
t k nu leas for at ge vera or co sens
1.4
Probability
elative Speed
0.2
0.0 2.0
R
Incomplete elliptic integral E(u)
R
60
elative Speed
Average
1.5
r
1.0
E(u)
to
y cit
4
6
de
We next analyze the correlation between sensor speed and probability of k-barrier coverage. From Eq. (4), we know that the average relative velocity (v rel ) is linearly related to the Poisson random variables (Θv · τ ). At the same time, Eqs. (8) and (9) describe that the average relative speed (v rel ) is affected by both speeds of sensors and intruders (vm and vi ). Let us consider the two extreme cases: (vm → 0) and (vi → 0). Under the both extreme cases, the parameter u of the incomplete elliptic integral E(u) from Eq. (9) becomes zero, and E(u) = π/2. Figure 8 illustrates the incomplete elliptic integral function E(u) against the ratio of sensor speed to intruder speed (vm : vi ). The result shows that the incomplete elliptic integral function E(u) is in the maximal value E(u) = π/2 at two ends (vm → 0 and vi → 0), and is in the minimal value (E(u) = 1) when vm = vi . Figure 8 also illustrates the average relative speed v rel against ratio of sensor speed to intruder speed (vm : vi ). The average relative speed increases slowly starting at vm → 0, and the slope of the line increases when the sensor speed is faster than the intruder speed. Figure 9 plots the relationships between the probability of k-barrier coverage, the value of k and the ratio of sensor speed to intruder speed. The results are obtained by theoretical calculations from Eqs. (5) and (8). It demonstrates that the increase of sensor mobility (speed of sensor) can directly increase the probability of k-barrier coverage. We can further examine the barrier coverage performance between a WSN and a MSN in Figure 10. The result shows the comparison of k-barrier coverage probability with different values of k. The horizontal lines present the k-barrier coverage probabilities for WSNs (as vm → 0), and all of the k-barrier coverage probabilities approach to one when the ratio of sensor speed to intruder speed is 3 : 1. By increasing the velocity of sensors from zero to three times as fast as intruders’, the probability that intruders covered by at least 30 mobile sensors can be increased from nearly zero to approximately one under the same network setting.
) :1
Sensor Speed and k-barrier Coverage Probability
50
k -b
ar
rie
er ov r c
ag
e
r so
5.2
x r (
coverage under the centralized manner. In a MSN, only the probabilistic barrier coverage can be obtained.
40
en S
Figure 8: Incomplete elliptic integral function and average relative speed v rel against ratio of sensor speed to intruder speed.
30
0.5
of
tru In
atio of Velocity of Sensor to Velocity of Intruder (x:1)
20
y cit
2
R
of
0
0 10
1.0
elo V
0.9
1.5
of
elo V
10
io at
1.0
Random Direction Mobility Number of Mobile Sensor = 400
Area = (50x100) Coverage Range = 1
Figure 9: The 3D plot for probability of k-barrier coverage, value of k and ratio of sensor speed to intruder speed.
6. SIMULATION RESULTS AND ANALYSIS In this section, we present numerical results verified by simulations. We develop a simulator that captures the essential aspects of the MSN described in the Section 3. The simulator also provides the flexibility in selectively changing the configuration with different parameter settings including: (i) the area size and dimension of A; (ii) the number of mobile sensors (N (A)); (iii) the coverage range of a sensor (R); (iv) the mobility of intruders; (v) the mobility of sensors. Unless otherwise specified, we use the following default settings: we deploy 400 mobile sensors randomly distributed in an area of size 50×100 with the ratio of sensor to intruder speed 1 : 1 and the coverage range of sensors R = 1. Figure 11 illustrates the probability density function for an intruder covered by at least k mobile sensors with different number of mobile sensors (N (A) = 100, 200 and 400). The simulation results for random direction mobility model are averaged by 100 simulations. The calculated distribution functions are almost in the same shape of the simulation results. We next focus on the k values that can result in the k-barrier coverage probability at least 90%, which we believe that might be reasonable in coverage requirement. The scenarios (N (A) = 100 and 200) can acquire fairly accurate results. For the N (A) = 400 case, it shows that there is more than 90% probability for intruders covered by at least 13 mobile sensors from simulations; while in theory, the value of k is 14 with difference less than 7%. We are also interested in knowing whether the theoretical result still hold if the velocity ratio between sensor and intruder is different from 1 : 1. Figure 12 shows the probability density function for intruders at least covered by k mobile sensors with three sets of velocity ratios between sensors and intruders (1 : 1, 2 : 1 and 1 : 3). In this simulation, we consider the dimension of area A to be 50 × 100 and sensor range as 1 with 400 mobile sensors. The calculated distribution functions are also almost in the same shape of the simulation results. By increasing the velocity ratio of sensors under the simulation scenarios from 1 : 1 to 2 : 1, the value of k that the k-barrier coverage probability are
Random Direction Mobility Number of mobile sensors = 400
Coverage Range = 1
Probability for k-barrier coverage
1.0
Area = (50x100)
k
WSN ( =10) 0.8
k
0.6
WSN ( =15)
k = 10 k = 15 k = 20 k = 30
0.4
0.2
k
WSN ( =20) 0.0
k
WSN ( =30)
0
1
2
3
4
Ratio of Velocity of Sensor to Velocity of Intruder (x:1)
Probability for at least k number of sensor coverage
Figure 10: The probability of k-barrier coverage against ratio of sensor speed to intruder speed.
Number of mobile sensors
N(A)=100 Simulation Results N(A)=100 Theoretical Calculation N(A)=200 Simulation Results N(A)=200 Theoretical Calculation N(A)=400 Simulation Results N(A)=400 Theoretical Calculation
1.0
0.8
0.6
Random Direction Mobility Velocity ratio (Sensor:Intruder) = 1:1
Area = (50x100)
0.4
Coverage Range = 1
0.2
0.0
0
5
10
15
20
25
30
35
40
k-barrier coverage
Figure 11: The probability density function of kbarrier coverage with different number of mobile sensors. higher than 90% increases from k = 9 to k = 24. Again, this result shows that sensor mobility can be exploited to compensate for the lack of sensors and improve barrier coverage performance. Finally, we discuss the impact from different mobility models. Figure 13 illustrates the comparative results of k-barrier coverage probabilities under different sensor mobility models including: the random walk mobility model, random direction mobility model and random way point mobility model. The results show that the random walk mobility model and the random way point mobility model cause higher k-barrier coverage probability than the random direction mobility model, when the value of k is larger than 10. This is consistent with the discussions in Section 3.2, with respect to the density waves generated by the random way point mobility model and the random walk mobility model.
7.
RELATED WORKS
The intrusion detection is a subset of the coverage problem, which was firstly introduced in sensor-based robotic systems [7] before it grew into a critical research problem in WSNs. Originally, coverage can be further divided into two categories: blanket coverage defined to achieve a static arrangement of sensing elements that maximizes the detection rate of targets appearing within the coverage area; barrier
coverage describes a static arrangement of sensing elements that minimizes the probability of undetected enemy penetration through the barrier. In WSNs, many existing works focus on the blanket coverage (or called full coverage) over a regular region. Hefeeda et al. proposed a probabilistic coverage protocol in [8] that considered probabilistic sensing models. [19] examined the issues associated with maintaining sensing coverage and connectivity by keeping the minimum number of sensor nodes in the active mode in WSNs. Recently, there has been an increased interest and in the area of barrier coverage in sensor networks. The work of k-barrier coverage [9] and the subsequent works, e.g., in [4], [5], [12], [17], have exclusively assumed sensors to be stationary nodes. There were several works that illustrated that node mobility could improve the blanket coverage performance of MSNs [11]. One of the applications was network surveillance presented in [6], in which the coverage problem ensured that targets of interest in the observed area are within the sensing range of one or more deployed sensors. It characterized the spatial distribution of a set of independent sensors in steady state. Chin et al. investigated the similar problem of detecting the presence/absence of a target using an MSN in [10]. It presented an analytic method to evaluate the detection latency based on a collaborative sensing approach with uncoordinated mobility. It analyzed the tradeoff between the number of nodes and detection latency in a MSN. Another direction of research in MSNs focuses on developing algorithms to reposition mobile sensors in order to achieve a static configuration with enlarged covered area [16], [18]. Specifically, the proposed algorithms attempted to spread sensors in the field for maximizing the covered area. Similarly, reposition algorithms for barrier coverage by mobile sensors were also examined. [15] studied the energy-efficient sensor relocation to utilize much fewer mobile sensors than stationary sensors to achieve barrier coverage with random deployment. Similar problem was addressed by Bhattacharya et al. in [2]. Our focus in this paper is completely different from all prior works. There are two distinctive features in our work: 1) we try to identify and characterize the dynamic aspects of the k-barrier coverage that depend on both sensor and intruder mobility; 2) we introduce a new barrier coverage performance metrics: k-barrier coverage probability. By leverage the kinetic theory from physics, we model the dynamic problem, and we examine its sensitivity under different network parameters and configurations. To the best our knowledge, we believe this is a completely new study on the k-barrier coverage in mobile sensor network.
8.
CONCLUSIONS
In this paper, we study the intrusion detection problem in mobile sensor networks (MSNs). Specifically, we introduce a performance metrics: k-barrier coverage probability and investigate the k-barrier coverage against moving intruders. By modeling the dynamic aspects of the k-barrier coverage that depend on both sensor and intruder mobility, we derive the inherent relationship between the k-barrier coverage performance and a set of crucial system parameters including sensor density, sensing range, sensor and intruder mobility. We obtain the k-barrier coverage probability by the (incomplete) Euler gamma functions of the system parameters. We further compare the coverage performance with mobile and
Probability for at least k number of sensor coverage
Velocity ratio (Sensor:Intruder) Simulation Results (1:3) Theoretical Calculation (1:3) Simulation Results (1:1)
1.0
Theoretical Calculation (1:1) Simulation Results (2:1) 0.8
[3]
Theoretical Calculation (2:1)
0.6
[4]
0.4
0.2
Random Direction Mobility
[5]
Area = (50x100) 0.0
Coverage Range = 1 Number of Sensor = 400 0
5
10
15
20
25
30
35
40
45
k-barrier coverage
Probability for at least
k number of sensor coverage
Figure 12: The probability density function of kbarrier coverage with different ratios of sensor speed to intruder speed.
[6]
[7]
[8]
1.0
0.8
[9]
0.6
0.4
[10] 0.2
Random Direction Mobility Model Random Walk Mobility Model
0.0
Random Waypoint Mobility Model
0
5
10
15
20
[11]
k-barrier coverage
Figure 13: The probability density function of kbarrier coverage with different sensor mobility models.
[12]
static sensors, which conclusively demonstrated that the mobility can be exploited to obtain better barrier coverage. There are several avenues for further research on this problem: (1) to consider the detection error of mobile sensors under varying sensor speeds. This can be formulated into an optimization problem for k-barrier coverage probability; (2) to study the optimal patrol route of controlled mobile sensors, which aims to maximize the k-barrier coverage probability among all possible sensor patrol routes.
[13]
9.
[16]
ACKNOWLEDGMENTS
The research was support in part by a grant from RGC under the contracts 615608, 623209, by a grant from NSFC/RGC under the contract N HKUST603/07, and by a grant from National Natural Science Foundation of China under the contract 60933012.
10.
REFERENCES
[1] American border patrol, operation b.e.e.f. border enforcement: Evaluation first, a plan for border security assessment. In http://www.americanpatrol.com/ABP/BEEF/PDF/ BEEFPLANDEC118.pdf, December 2006. [2] B. Bhattacharya, B. Burmester, Y. Hu, E. Kranakis, Q. Shi, and A. Wiese. Optimal movement of mobile
[14]
[15]
[17]
[18]
[19]
sensors for barrier coverage of a planar region. In International Conference on Combinatorial Optimization and Applications. Springer, August 2008. T. Camp, J. Boleng, and V. Davies. A survey of mobility models for ad hoc network research. Wireless Communications and Mobile Computing, 2(5):483–502, September 2002. A. Chen, S. Kumar, and T. Lai. Designing localized algorithms for barrier coverage. In in Proc. of Mobicom. ACM, September 2007. A. Chen, T. H. Lai, and D. Xuan. Measuring and guaranteeing quality of barrier-coverage in wireless sensor networks. In in Proc. of MobiHoc. ACM, May 2008. Y. Dong, W.-K. Hon, and D. K. Y. Yau. On area of interest coverage in surveillance mobile sensor networks. In in Proc. of IWQoS. IEEE, June 2007. D. W. Gage. Command control for many-robot systems. In The Nineteenth Annual AUVS Technical Symposium. AUVS, June 1992. M. Hefeeda and H. Ahmadi. Probabilistic coverage in wireless sensor networks. In Conference on Local Computer Networks (LCN). IEEE, November 2005. S. Kumar, T. H. Lai, and A. Arora. Barrier coverage with wireless sensors. In in Proc. of MobiCom. ACM, August 2005. T. lin Chin, P. Ramanathan, and K. K. Saluja. Analytic modeling of detection latency in mobile sensor networks. In in Proc. of IPSN. ACM, April 2006. B. Liu, P. Brass, O. Dousse, P. Nain, and D. Towsley. Mobility improves coverage of sensor networks. In in Proc. of MobiHoc. ACM, April 2005. B. Liu, O. Dousse, J. Wang, and A. Saipulla. Strong barrier coverage of wireless sensor networks. In in Proc. of MobiHoc. ACM, May 2008. R. Present. Kinetic Theory of Gases. McGraw-Hill Book, New York and London, 1958. E. Royer, P. Melliar-Smith, and L. Moser. An analysis of the optimum node density for ad hoc mobile networks. In International Conference on Communications (ICC). IEEE, June 2001. C. Shen, W. Cheng, X. Liao, and S. Peng. Barrier coverage with mobile sensors. In in Proc. of the International Symposium on Parallel Architectures, Algorithms, and Networks. IEEE, July 2008. G. Wang, G. Cao, and T. Porta. Movement-assisted sensor deployment. In in Proc. of Infocom. IEEE, March 2004. M. K. Watfa and S. Commuri. Power conservation approaches to the border coverage problem in wireless sensor networks. In International Conference on Wireless Networks (ICWN). IEEE, June 2006. C. W. Yu. Randomized coverage algorithms for mobile sensor networks. In in Proc. of the 25th Workshop on Combinatorial Mathematics and Computation Theory. Springer, April 2008. H. Zhang and J. C. Hou. Maintaining sensing coverage and connectivity in large sensor networks. Ad Hoc and Sensor Wireless Networks, 1(1-2):89–124, March 2005.
