Detection quality of border surveillance wireless sensor networks in the existence of trespassers’ favorite paths

Computer Communications 35 (2012) 1185–1199

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Detection quality of border surveillance wireless sensor networks in the existence of trespassers’ favorite paths Can Komar, Mehmet Yunus Donmez ⇑, Cem Ersoy Computer Networks Research Laboratory (NETLAB), Department of Computer Engineering, Bogazici University, Istanbul TR-34342, Turkey

a r t i c l e

i n f o

Article history: Received 17 May 2011 Received in revised form 29 February 2012 Accepted 1 March 2012 Available online 10 March 2012 Keywords: Surveillance wireless sensor networks Border monitoring Analytical model Detection quality metric Trespassers’ favorite path

a b s t r a c t The performance of a surveillance wireless sensor network is generally measured with its detection capability within a monitored zone. This capability is affected by various parameters such as the sensor count, the sensor range, the area width and the target mobility model. In real life examples it is observed that intruders prefer some favorite regions because of their geographical advantages within a monitored border area. These regions, which bounds the randomly chosen trespassing paths, are generally in close vicinity. In this paper, we introduce the notion of trespassers’ favorite paths (TFP) and provide a tool that can be used to forecast the detection probability of a surveillance network in such a network with TFPs. The detection probability is reduced to the geometric line intersection problem using bijection and the boundary conditions of intruder trajectories for the border area and the favorite region are determined. The line intersection problem is solved using tools from the integral geometry and geometric probability. The effect of the favorable region on the detection quality under different conditions is calculated using probabilistic models. The accuracy of the proposed quality metric is validated by both analytical methods and simulation results. Furthermore, the importance of the intrusion model on the network performance is presented using realistic scenarios. It is shown that the existence of favorite paths has significant impact on the detection quality of the network. The proposed detection quality metric provides analytical tools suitable for both real life and simulation scenarios to the network designers to forecast and optimize the expected detection performance, and is computationally inexpensive compared to both simulation based and distributed quality measurements. Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction The enhancement of the sensors with improved observation abilities such as ultrawide band radar sensing and video cameras have enabled wireless sensor networks [1] to be preferred for monitoring and tracking applications [2]. In this study, we focus on surveillance wireless sensor networks (SWSNs) which are used for watching border regions against security breaches. In surveillance applications, sensor nodes are generally deployed to the region via an airplane or artillery, thus, the distribution of the sensors is random. Following the deployment, sensors autonomously organize themselves and form a network and start sensing the area for intruders. When an intruder is detected, the detection information is sent to the sink node so that the network operator can take the necessary counter measures. The primary objective of a SWSN is to detect the border breaches successfully. In a border region, generally there are security measures such as watch towers, razor wires, and patrolling ⇑ Corresponding author. Tel.: +90 212 3597330. E-mail addresses: [email protected] (C. Komar), yunus.donmez@boun. edu.tr (M.Y. Donmez), [email protected] (C. Ersoy). 0140-3664/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.comcom.2012.03.002

guardians. In addition to these, the geographical conditions do not allow all of the border field easily accessible. Because of these reasons, an intruder prefers some paths over others. He/she tries to avoid detection and security measures using some advantageous paths which are also geographically feasible. The mobility of an intruder is shaped for achieving a goal, which is trespassing a region without being detected. An example of this behavior is shown in Fig. 1. In this figure, there are two watch towers observing some portions of the border field. The intruders try to avoid detection by the sentries by following paths away from the watch towers which are not in the observable areas. These paths are in the non-observable area, which is depicted as the shaded one and resides between the observable areas. In real life border scenarios, it is observed that these paths are generally in close vicinity and they can be bounded within a closed region. We call these paths trespassers’ favorite paths (TFP). An example of a TFP is given in Fig. 2, taken from the Turkey–Iran border. TFP notion is also being utilized in real life surveillance systems to increase detection rate. The border camera watch program that is installed along the USMexico border to detect drug smugglers takes the past trespassing pattern into account. As seen in Fig. 3, the locations of the surveillance cameras are selected according to the previous trespassing

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Fig. 1. A border area with a watch tower observing the border field.

patterns [4]. In this paper, we introduce the new concept of trespassers’ favorite paths (TFP), that has been proposed by us based on the real life observations and experiences. To the best of our knowledge, there is no notion of TFP used in the literature. There are some studies that have used paths in terms of detection probability or exposure but most of these studies take the minimum or maximum probability of detection as the system’s overall detection probability. SWSN applications rely on the overall detection capability of the network, which may be defined as the ability to detect any target passing through the deployment site. Note that, the instantaneous detection capability of the sensor network may altered by the network topology changes due to sensor losses or jammer attacks [5], which has been studied in [6] and can be easily integrated into our analysis. However, we aim to inspect the pure effect of TFP phenomenon on the detection capability, measured in terms of the detection quality metric (DetQM) proposed in this paper. Hence, the effect of sensor losses is beyond the scope of this paper. In this paper, we try to provide a tool to the network operator to derive the expected detection performance and risk analysis framework for a given sensor network with realistic border surveillance scenario parameters. For this purpose, we map the physical

problem definition of the trespasser preference and network parameters to the geometric domain and determine the probability of detection of a target following a linear trajectory by a single sensor in the geometric domain. We generalize this solution to derive the risk analysis metric proposed in this paper, which is the combined metric of sensing coverage and trespasser preference in the network. Based on the shortest path strategy, such a solution provides a conservative estimation for the trespasser detection performance. The mathematical solution is further simplified to provide a closed-form DetQM formulation. Providing a closed-form DetQM provides assistance to the WSN designers and supervisors in the following situations:

Fig. 2. A convoy of fuel smugglers in the Turkey–Iran border [3].

Fig. 3. An area frequently used by drug smugglers in US-Mexico border [4].

 A reliable and accurate detection quality measurement is needed for assessing the performance of the network and providing robustness during its operation.  In order to sustain the required detection quality, making decisions for critical activities such as sensor redeployment, relocation and waking up or putting into sleep some nodes at any time during the operation of the network.  Realtime monitoring of the detection performance is required to observe the current network status and measure the expected detection rate under such conditions.

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 The deployment parameters should be chosen carefully to get the optimum detection performance from the network using a limited deployment budget. These are some of the major problems that network designers and engineers need to cope with in a typical border SWSN setup. The network engineers can use the proposed metric to find a solution to these problems in the following ways:  SWSNs are real-time systems whose operation and response times are essential to the success of the detection mission. Due to the nature of their mission, the expected level of detection quality is very critical in terms of performance. The DetQM monitoring approaches based on simulations require long periods, whereas our proposed DetQM metric approach provides real-time results.  Most of the decisions about sensor networks are based on simulations. On the other hand, simulations require long processing times and needs accurate parameters in order to reflect the real life situations. Our proposed DetQM metric provides a low-cost calculation in terms of the processing time and can be easily adapted for different environmental and trespassing conditions.  The expected detection reliability of the network under different trespassing patterns and environmental conditions can be estimated by the calculation of the provided DetQM value.  The DetQM metric can be used as a performance benchmark to calculate the network parameters, such as the required sensor node count, and node sensing range, in order to monitor a region with a desired minimum detection probability under given environmental conditions and other parameters that are provided by the designer. In this paper, we address these problems and provide an analytical tool that can be adapted to different conditions easily. Moreover, the paper forms a solid mathematical background of our DetQM model and presents a simple base for further more complex studies. In Section 2, we summarize the related work and compare their contributions and differences with our work. In Section 3, we present the network model used in our analysis and simulations. In Section 5, we give a formal definition of the border SWSN intruder detection problem with TFP region. In Section 5, we provide the analytical representation of the DetQM metric for a border SWSN scenario with a TFP region. In Section 6, we present our analytical results and their interpretations. In Section 7, we conclude our work and present possible future work.

2. Related work Monitoring an area against movements is an important topic in surveillance systems. The capability of a surveillance wireless sensor network is generally inspected from two aspects: coverage and detection. In this section, we will provide some related work about coverage and detection from the literature. Covering an area using various surveillance methods have been formulated in different domains in the past. One of the famous problems is the Art Gallery Problem in which the objective is to place guards in an art gallery such that every point of rooms is covered by at least one observer [7]. The coverage problem in the wireless sensor networks are differentiated according to the requirements and properties of the network. In [8], the authors make a survey about the coverage problem in sensor networks and define three types of network coverage which are area coverage, point coverage and barrier coverage. In the area coverage, the objective of the network is to monitor a given region of interest with respect to different performance

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criteria such as coverage ratio, minimum number of sensors providing desired minimum coverage level during the maximum lifetime of the network. Generally, node sleep schedule algorithms are used for maximizing the network lifetime. Since the sleep scheduling problem is NP-hard, there is no certain optimal algorithm existing and several heuristics are proposed [8–11]. In the point coverage, the object of the network is to monitor given targets during its lifetime. In [12], the authors model the problem as a maximum cover tree problem and show that it is an NP-complete problem. They propose heuristic approximation algorithms to increase the lifetime of the network. In [13], the authors analyze the sparse surveillance networks against moving targets. For a moving target moving with a given speed a pill shaped region called target pill is formed which includes all possible locations of sensors that can detect the target. Deterministic deployment is another method in point coverage. In that case, the locations of the sensors are determined according to target positions and required coverage level. In [14], authors proposed an approximation to the sensor deployment plan with constraints of minimum sensor cost and maximum target coverage which are spread across a geographical region. The barrier coverage aims forming a line made up from sensor coverage areas and tries to detect every object that breaches this line. One of the advantages of the barrier coverage over full coverage is its requirement for less number of sensors. In addition to this, they try to provide certain detection against breaches. One of the major limitations of the barrier coverage against the full coverage is that, it is not suitable for developing localized algorithms since the central nodes must be aware of other nodes’ states to decide if the barrier is intact. Most of the algorithms used in the barrier sensor networks are centralized. In addition to this, the loss of a sensor node in a critical point can form a breach hole within the barrier. The barrier is as strong as the weakest node in the network. That makes the barrier networks vulnerable to jamming and physical attacks. Another weak point of the barrier coverage networks is their inability about determining the path of the intruders. The detection is performed within the coverage area of the barrier nodes which means that the observed section of the intrusion path is at most in the length of the coverage diameter. The path of the intruder before entering and after exiting the sensor coverage is not known. Most of the studies in the barrier coverage try to improve the detection quality and lifetime of the barrier coverage using different sleep schedule and communication algorithms. In [15], to localize the barrier coverage problem, the authors divide a long border region into shorter rectangular segments. Within each segment, a horizontal barrier is constructed and along the connection strips of these segments, a vertical barrier is constructed. Compared to a centralized method, they use more sensor nodes but gain in terms of communication overhead and computation cost in the central node in addition to the reduced delay of forming a centralized barrier. In [16], the authors assume that most of the intrusions are performed within a narrow slice of the longer border region. They propose a localized sleep-wakeup protocol called Localized Barrier Coverage Protocol (LBCP) for maximizing the network lifetime. LBCP does not guarantee global barrier coverage like a centralized method does, but performs well under given assumptions. In [17], the authors claim that sensor deployment may be linear instead of uniform random in some deployments such as dropped from an aircraft. They study the properties of barrier coverage of the network under such conditions. The detection capability of a sensor network is discussed in [18]. In this study, authors define the path exposure phenomenon to quantify the detection capability of a sensor network. They define the exposure of a target as the probability of detecting the target traversing along a minimum detection path. Their sensor model is based on the power of the signal emitted by the target

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at a distance. The detection probability is calculated according to the distance between the sensor node and the target. A sensor deployment strategy is proposed to reach the desired minimum exposure. In a similar study, authors propose a formulation to determine the target detection probability using minimum exposure path [19]. They investigate the effects of the shape of the field and the sensing range on the detection quality of the network. They assume that the target position is determined randomly and does not take the target mobility into account. In [20], authors analyze the k-sensed ratio of a path by a sensor network, which is the part of the path sensed by at least k sensors. Other performance metrics such as distance until first sensed, sensing holes, breach and support paths are also investigated using their model. They also provide a generalized version of their model using m-dimensional straight line paths sensed by an n-dimensional sensor network. In [21], authors formulate the target detection problem as a line-set intersection problem and use integral geometry to analyze it. They analytically evaluate the target detection probability and the mean free path until a target is detected, and show that detection probability is independent of the shape of the monitored zone. Their work assumes that the targets traverse all the monitored zone randomly and do not have a direction of travel over the zone. Similarly, most of the studies that investigate the sensor network detection ignore the behavior of the intruder when analyzing the detection probability or utilize mobility models inherited from previous wireless network and ad hoc network experiments. Most of these models employ random mobility either in single entity or a group of entities that is related to each other according to some properties [22]. More detailed survey about mobility models can be found in [23–25]. Though these mobility models may be acceptable for representing the movement of an individual intruder, the intrusion preferences over the whole border region are generally not random. In a recent study, the authors propose an environment-aware mobility model to be used in wireless ad hoc networks [26]. They use obstacles and doorways to model different pathways. They present that the existence of regions of interest has significant impact on node mobility. In our study, we show that the environmental conditions affect the mobility of the intruders dramatically. In contrary to most of the models, we take the aim of the mobility into account. In fact, the mobility of an intruder is shaped for achieving a goal, which is trespassing a region without being detected through a feasible terrain. In a border region, generally there are physical counter measures to prevent the intruders. In addition to these, the terrain and other geographical parameters do not allow all of the border field easily accessible. Because of these reasons, an intruder prefers some regions over others and tries to avoid detection using some advantageous regions which are also geographically feasible. An example of this behavior is shown in Fig. 1. In this figure, there are two watch towers observing some portions of the field. The intruders try to avoid detection from the sentries by following paths away from the watch towers in the middle of the border area which are not seen from them.

3. Network model In this section, we will provide the definitions and parameters used in our wireless sensor network model. In the analysis, we assume a convex 2-D border area, S that can have any shape as long as it is convex. In case of concavity, the convex hull of these geometric shapes can be used for approximation without loss of generality as in [21]. In our simulations, we simplify the convex border area constraint as rectangle border area that has a width of w and a height of h for demonstrative purposes. There are Ns sensors deployed randomly within the area. Each sensor si is assumed to have

convex sensing coverage of Si with radius ri, uniformly and independently distributed within S. The border area separates the secure side from the insecure side. The direction of the trespassing is from insecure side to the secure side, thus along the y-axis of the region. The trespassers prefer the paths that remain in a smaller area compared to the whole border area. This paths are known as TFPs and the area that consists these paths is known as the TFP region. In order to form a mathematical basis and to evaluate the effect of the existence of TFP region independent of TFP count on the detection, the number of TFP regions is restricted to 1. After the construction of the mathematical basis, the formulation can be generalized for higher number of TFP regions. The location of the TFP region is not known by the network operators but known by the trespassers. The probability of selecting the TFP region for trespassing by an intruder is pt. It means that if the total number of trespassing in the border area is nt, the number of trespassing through the TFP region is pt xnt . The width of the TFP region is wt which is generally much more smaller than the whole border width w. The list of the parameters used in the study is summarized in Table 1. We assume that, the intruders have limited or no knowledge about the deployment of the sensor network similar to the assumptions in [27]. On the other hand, the intruders have knowledge about other security measures such as locations of watch towers, mine fields and highly patrolled areas. They try to avoid Table 1 List of symbols. Symbol

Explanation

ATð!/ Þ

Area of S covered by lines in ! with parameter /. Minimum value for the parameter / of lines in !. Maximum value for the parameter / of lines in !. Angle limit of a line in the TFP region. A bounded set. A bounded subset. The event that C \ !/ – ;. The event that C \ !/ ¼ ;. Expected area of S covered by lines in !. Expected area of TFP region covered by lines in !t . Expected thickness of S wrt !. Expected thickness of set Si wrt !. Expected thickness of TFP region wrt !t . Intruder path line. Angle limit of a line in a rectangular deployment site. The height of the rectangular border. Thickness of the lines with slope a in S. Thickness of the lines with slope a in TFP region.

a1 a2 b C Ci d d EðAT Þ EðAT t Þ EðTÞ EðT i Þ EðT t Þ G

c h 0 h 0 ht Ns / p PD P Djt P Djt PD P si P si pt pt r S si Si t Tð!/ Þ

! !/ !t !t;/ !t w wt

Number of deployed sensors. Angle of line perpendicular to intruder line wrt. x axis. Distance of the intruder line to the origin. Probability that the network detects an intruder. Probability of detection in a TFP path. Probability of detection in a non-TFP path. Probability that the network misses an intruder. Probability that a sensor si detects an intruder. Probability that a sensor si misses an intruder. The probability of selecting the TFP region by an intruder. The probability of selecting the non-TFP region by an intruder. Radius of sensing coverage of a sensor Convex deployment area. Index of a sensor. Sensing coverage area of a sensor si. TFP region. Thickness of line set crossing S with parameter /. Set of lines crossing S. Set of lines crossing S with parameter /. Set of lines in the TFP region. Set of lines in the TFP region with parameter /. Set of lines that are not in the TFP region. !t ¼ ! n !t The width of the rectangular border. The width of the TFP region.

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these non-sensor detection measures with selecting preferable paths which they possibly have learnt from previous experiences. Thus, we can say that the location of the TFP paths are indirectly determined by the conventional intrusion surveillance methods. The coordinates of the TFP region that consists of the TFP paths is not known by the network operators. In a recent study, the authors propose an environment-aware mobility model to be used in wireless ad hoc networks [26]. They use obstacles and doorways to model different pathways. They present that the existence of regions of interest has significant impact on node mobility. In our work, we call these regions of interest as TFP regions. Most of the trespassers use TFP regions when crossing across the border. The entry and exit points and the path they follow are contained within the TFP region. The sensor deployment is done with an airplane or artillery because of the accessibility difficulties to the border region. The sensor nodes are distributed in a random manner over the area. We assume that sensors are deployed uniformly in the area as in [6,21]. The intruder has no knowledge about the coordinates of the sensor nodes in the monitored zone. If the intruder had the deployment map of the sensor network, he could avoid detection by following longer but safer paths around the sensor nodes. However, the lack of such information prevents the intruder to take such evasive actions and enforces him to choose a path which will reduce his exposure probability. In this case, increasing the path length in the network causes an increase in the area in which the resident sensors are able to detect the intruder. Since the probability of detection for this intruder by a single sensor is equal to the ratio of this area to the whole deployment area, this increase will also increase the probability of detection. The path which minimizes the detection probability under these conditions is the shortest path from the entry point to the exit point. Such linear paths is the least detection path with respect to the intruder. Although this situation limits the path diversity and excludes the curved and segmented paths, it is the optimal solution for the benefit of the intruder. One advantage of using linear trajectories approach is that, the worst detection performance of the sensor network can be calculated which is an important threshold for the network operator. As seen in Fig. 1, segmented paths are generally longer than the straight path, and the detection probability of the intruder following a segmented or curved path increases significantly. Other advantage of the straight line path approach is the parametrization of linear trajectories it provides which can be easily applied both in the analytical studies and physical representation of the network model. The path selection of the intruders are assumed to be random. In other words, the intruder passes through a random point in the region with a random slope parameter, given that the entry

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Fig. 4. Theoretical and real sensing areas of a sensor.

and the exit points of the path resides in the insecure and the secure sides of the border respectively. In this study, our aim is primarily to demonstrate the effects of trespassers’ behavior, geographical preferences and the properties of terrain on the detection performance of a border SWSN. In order to reduce the effects of the remaining factors, the sensors are assumed to operate using a binary detection model as in [6,21]. In this model, if the target is within the sensing range of the sensor node, it is certainly detected. If the target is out of the sensing range, it avoids detection. Binary detection model’s main advantage is its simplicity, increasing its understandability by the reader while building a theoretical sensing model, which may be replaced by more complex models once a solid theoretical base is constructed. In the real world, the sensing range of a sensor is affected from various factors such as the battery power, the terrain, temperature and humidity. Hence in most of the sensing models, for a sensor si, the effective sensing area can be divided into two as shown in Fig. 4, a sensing boundary ðS0i Þ wherein the detection of the tar  get is certain and another boundary further away S00i where the detection happens with some probability. Elfes detection model [28] is devised to represent such probabilistic detection, plotted in Fig. 5. In this model, the region between re and r denotes the uncertain detection range, where the detection probability is determined by an exponential function. In our study, we further simplify the sensing area as the largest circle Si around the sensor si with radius r that can be fitted into the certain detection area S0i (the circle with radius re in the Elfes model). This circular area represents a theoretical lower bound for the detection as compared to actual sensing capability of the sensor.

Fig. 5. Elfes detection model.

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DetQM using this sensing model gives us the lower bound for detection probability, enabling us to calculate the expected minimum performance of the wireless network which is a desirable metric in high security systems such as border surveillance. Though it is a simple model, binary sensing model can be safely used for micropower impulse radar (MIR) sensors. The MIR sensors are radar-based low-power sensors that use ultra-wideband (UWB) pulses [29,30]. Their range are relatively larger compared to other types of sensors which make them more preferable in border surveillance sensor networks. There are commercially available MIR sensors such as SEGA-Node TIMS Radar [31]. In the MIR sensors, a transmitting antenna radiates a pulse. Reflections from the target are listened by a receiving antenna, and using the time difference, the distance of the target is calculated [32]. 4. Problem definition In this section, we will provide a formal definition of the border SWSN intruder detection problem with TFP region. Moreover, we will provide two additional problems which will be mapped to each other to derive a solution for the geometric domain utilizing metrics and tools from Integral Geometry and Geometric Probability. Problem 1 (Border surveillance intruder detection with TFP region problem). We have a convex deployment area, S, sensed by Ns sensors where each sensor si has a sensing coverage Si with a radius of ri, uniformly and independently distributed within S. The deployment area separates the insecure region from the secure one. The trespassers enter into the deployment area from the insecure side and exit from the secure side, following a linear trajectory. The trespassers prefer the TFP region with probability pt and make their passage through this region. They choose any other path within the border area with probability of pt . What is the probability PD that an intruder X randomly crossing S is detected by at least one sensor? A graphical representation of Problem 1 is given in Fig. 1. The probability PD is defined as the detection quality metric (DetQM) of such a sensor network. The solution of this problem requires the calculation of the probability of detection for a set of sensors in the network. In order to provide a solution for this problem, we introduce two additional problems. The first one, Problem 2, is the simplified version of Problem 1 which does not consider any TFP regions. The second one, Problem 3, corresponds to the geometric interpretation of such a sensor network. We will provide a mapping between these two problems to show that they are equivalent, i.e., both can be reduced to each other. Problem 2 (Border surveillance intruder detection problem). We have a convex deployment area, S, sensed by Ns sensors where each sensor si has a sensing coverage Si with a radius of ri, uniformly and independently distributed deployed within S. The deployment area separates the insecure side from the secure one. The trespassers enter into the deployment area from the insecure side and exit from the secure side, following a linear trajectory. What is the probability PD that an intruder X randomly crossing S is detected by at least one sensor?

Our analytical study relies on the fact that these two problems can be reduced to each other with a bijection between physical and geometric domains. The following lemma presents this bijection. Lemma 1. Border surveillance intruder detection problem and the constrained line-set intersection problem are equivalent, i.e., can be reduced to each other with a bijective mapping. Proof. We provide our proof extending the method used in [21]. The deployment area S is represented in the map domain as a collection of points, C, each representing a coordinate in S. Clearly, the 2-D geometric representation, C, is of identical shape with S. Hence we map the deployment area, S, to the bounded and convex set C. By the same rationale, the sensing area, Si of sensor si is mapped to the bounded set Ci of identical shape with Si, uniformly and independently distributed in C. The geometric representation of a linear trajectory of an intruder X is a straight line Gðp; /Þ in the plane, defined by p as the shortest distance of G to the origin O of a coordinate system, and / as the angle of the line perpendicular to G with respect to the x axis. Clearly, the intersection constraints of the line G with set C and Ci are the same as the trespassing constraints in the geographic domain. This provides a mapping between the mobile target detection problem for a stochastic sensor network and the line-set intersection problem. Hence, we conclude that both problems are equivalent. h The mapping stated in Lemma 1 between Problems 2 and 3, provides a bijection between the physical sensor network domain and the geometric domain. In the next section, we will provide a solution for the probability of detection of a target by a single sensor in the geometric domain since we have shown that Problems 2 and 3 can be reduced to each other. The provided solution will be extended to derive the detection for a given set of sensors. Next, we will integrate the effect of the TFP region and TFP probability, in this formulation. Finally, we will further simplify our result to provide a closed-form solution for the probability of detection in Problem 1. 5. Detection quality of a border SWSN with TFP region In this section, we will find an analytical representation of the DetQM metric for a border SWSN with TFP region. Since we have shown that the probability of target detection PD is equivalent to the probability of lines to intersect geometric shapes, the problem can be simplified as a frequency count of lines intersecting geometric shapes. In our problem, we consider only the set ! of all possible linear trajectories crossing the deployment area from the insecure side to secure side with a given constraint induced by the deployment area and application requirements. Therefore, PD is equal to the quotient of the number of lines that intersect any of the sensing areas, over the number of lines that intersect the deployment area given that these lines are in !. However, the set of lines in ! intersecting a given convex set is uncountable. Hence we will introduce the line measure tool from the integral geometry to derive an analytical solution to the Line-Set intersection problem. For any straight line Gðp; /Þ, the density of the line is formulated as

dG ¼ dp ^ d/ Problem 3 (Constrained Line-Set intersection problem). We have a bounded convex set C of identical shape with S and Ns sets Ci of identical shape with Si in Problem 2, uniformly and independently distributed inside C. What is the probability PD that a random line G intersecting C with the same constraints as in Problem 2, also intersects at least one of the sets C i ; i ¼ 1 . . . N s ?

ð1Þ

The measure m(G) of a set of lines Gðp; /Þ is the integral of the density of the line, which is in the differential form, over the set [33]. Hence,

mðGÞ ¼

Z

dp ^ d/

ð2Þ

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where EðT i Þ is the average thickness of the sensor coverage area, EðTÞ is the average thickness of the deployment area for the given line-set intersection constraint which determines !, EðAT Þ is the average area of the deployment site covered by lines in ! and A denotes the area of the deployment site. Proof. The probability that an intruder randomly passing through S with a trajectory in ! is detected by a single sensor, si, is derived as follows. By the mapping of the intruder detection problem to line-set intersection problem provided in Lemma 1, this probability is equivalent to the probability that a line G intersecting C in line set !, also intersects Ci. In terms of the measures, this probability is equal to the ratio of the measure of the set of lines in ! that intersect both C and Ci to the measure of the set of lines that intersect C.

Psi ¼ Pr½G \ C i \ C – ;jG \ C – ;; G 2 ! ¼ Pr½G \ C i \ C – ;jG \ C – ;; G 2 !; dPr½d     þ Pr G \ C i \ C – ;jG \ C – ;; G 2 !; d Pr d

Fig. 6. The representation of thickness TðhÞ [33].

Our problem requires these lines to cross insecure and secure sides of the deployment area with possibly more constraints. Such lines constitute the subset ! of the lines measured in Eq. (2). Hence we need a geometric interpretation of the measure of a given set of lines. For this purpose, we adapt the definition of the thickness of a bounded set for lines determined by angle / in set !; Tð!/ Þ, where !/ ¼ fGðp; /Þ : G 2 !g. In our analysis, we use the thickness definition as given in [33]. According to this definition, the thickness TðhÞ of the set K in the direction h is equal to the length of the projection of K onto a line with direction h which is shown in Fig. 6. Hence, the thickness of the set K can be used as a measure for the set of lines along the direction perpendicular to h, that intersect K. Let U be the intersection of set C with !/ , the area of the region traversed by the lines determined by angle / in the line set !. Tð!/ Þ is defined as the length of the projection of U to a line with direction parameter / [33]. The measure of a set of lines intersecting with C at a fixed direction / and in line set ! is equal to the thickness Tð!/ Þ of the set in that direction. Let d be the event C \ !/ – ; and d be the complement of d, i.e. C \ !/ ¼ ;. Given that the trajectories are constrained to be in the set, !, the measure of the set of lines in ! that pass over a bounded convex set, C, defined by the support function p ¼ pð/Þ of the convex set, is given as:

mðG : G \ C – ;jG 2 !; dÞ ¼ ¼

Z Z

dp ^ d/ Tð!/ Þd/ ¼ ða2  a1 ÞEðTÞ

where a1 and a2 are the minimum and maximum values respecRa tively for parameter / of lines in ! and EðTÞ ¼ a12 a2 1 a1 Tð!/ Þd/. Since the border area is convex and the trespasser trajectories are linear from the insecure to the secure side, the target trajectories in ! are allowed to have parameters in range ½a1 ; a2 . By Eq. (3), the average thickness of the set of lines for a given constraint on line-set intersection, ! can be used to calculate the intersection of a random line with a convex set Ci within C, which will be used for the proof of the following lemma. Lemma 2. Let S be the deployment area of a sensor network and let si be any of the sensors deployed in the area. The probability that an intruder randomly passing through S with a trajectory from the line set, !, is detected by the single sensor, si is equal to

EðT i Þ EðAT Þ ; EðTÞ A

mðG : G \ C i \ C – ;jG 2 !; dÞ   Pr d mðG : G \ C – ;jG 2 !; dÞ

ð5Þ

The measure of the set of lines is 0 if the convex set does not intersect with the area ATð!/ Þ of the region traversed by the lines determined by angle / in the line set !. In addition, Ci is known to be inside the convex set C. Hence, Eq. (5) can be reduced to

P si ¼

mðG : G \ C i – ;jG 2 !; dÞ Pr½d mðG : G \ C – ;jG 2 !; dÞ

ð6Þ

The expected area of the regions determined by / measured in ! is given as:

EðAT Þ ¼

Z

a2

a1

1

a2  a1

ATð!/ Þ d/

ð7Þ

The probability that a sensor si to reside in this expected area determined by ! is found as:

Pr½d ¼

EðAT Þ A

ð8Þ

Using Eqs. (3) and (8), Eq. (6) becomes

ða2  a1 ÞEðT i Þ EðAT Þ EðT i Þ EðAT Þ ¼ ða2  a1 ÞEðTÞ A EðTÞ A



ð9Þ

ð3Þ

a1

P si ¼

mðG : G \ C i \ C – ;jG 2 !; dÞ Pr½d mðG : G \ C – ;jG 2 !; dÞ þ

P si ¼

G\C–;jG2!;d

a2

¼

ð4Þ

Thus, using the mapping provided in Lemma 1, the probability that an intruder randomly passing through S is detected by a single sensor, si, is found in terms of EðTÞ, the expected thickness of deployment area, and EðT i Þ, the expected thickness of the sensing coverage of a sensor. Now, we need to derive the detection quality of a sensor network with a trespasser trajectory in line set ! with no favorite paths, i.e. the selection of all trajectories are equally likely. Theorem 1. The DetQM of any sensor network with Ns sensors deployed in the area with a trespasser trajectory pattern determined by the line set ! with no favorite paths is

PD ¼ 1 

  EðT i Þ EðAT Þ 1 EðTÞ A i¼1 Ns Q

ð10Þ

where EðT i Þ is the average thickness of the sensor coverage area, EðTÞ is the average thickness of the deployment area for the given

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C. Komar et al. / Computer Communications 35 (2012) 1185–1199

line-set intersection constraint which determines !, EðAT Þ is the average area of the deployment site covered by lines in ! and A denotes the area of the deployment site. Proof. By Lemma 2, the probability of all sensors to miss the target, PD and the overall target detection probability, PD are calculated as:

  EðT i Þ EðAT Þ 1 EðTÞ A i¼1 i¼1   Ns Q EðT i Þ EðAT Þ PD ¼ 1  PD ¼ 1  : 1 EðTÞ A i¼1

PD ¼

Ns Q

ð1  Psi Þ ¼

Ns Q

ð11Þ 

ð12Þ

Next, we will generalize Theorem 1 for the cases where trespassers follow favorite paths in the deployment site. Theorem 2. The DetQM of any sensor network with Ns sensors deployed in the area with a trespasser trajectory pattern determined by the line set ! with favorite paths in the TFP region is

  EðT i Þ EðAT t Þ PD ¼ 1  pt  1 EðT t Þ A i¼1   N s Q EðT i Þ EðAT  AT t Þ  ð1  pt Þ  1 EðT  T t Þ A i¼1

Eq. (10). In Theorem 2, we elaborate Theorem 1 and include the favorite path probabilities into the formulation with given TFP region size and preference probability. We analyze the DetQM of such sensor network and derive Eq. (13). In our derivations, we only require all sets to be convex and bounded. However for simplicity, we assume circular sensing coverage, and rectangular deployment area. The following corollary is a demonstrative example for our analysis and simulations are based on this corollary. Corollary 1. If the sensors have identical circular sensing coverage, then the DetQM of a sensor network with a rectangular deployment area is equal to:

 N 2r EðAT t Þ s PD ¼ 1  pt  1   EðT t Þ h  w  Ns 2r EðAT t Þ  ð1  pt Þ  1   EðT t Þ h  w and closed form equations are given as

b ¼ arctanðh=wt Þ

Ns Q

c ¼ arctanðh=wÞ ð13Þ Tð!p2þa Þ ¼ w sinðaÞ  h cosðaÞ

where pt is the probability of the trespassers to follow favorite paths, EðT t Þ and EðT  T t Þ are the average thickness of TFP and non-TFP regions respectively, whereas EðAT t Þ and EðAT  AT t Þ are the average area of the TFP and non-TFP regions respectively covered by lines in ! which satisfies the region crossing constraints.

Tð!t;p2þa Þ ¼ wt sinðaÞ  h cosðaÞ EðT t Þ ¼

Z pb Tð!t;pþa Þ 2

p  2b

b

Proof. Since, TFP region is a subset of the deployment area, the target detection probability in the TFP region is calculated using the subset of lines satisfying the region crossing constraints, !t :

  EðT i Þ EðAT t Þ 1 EðT t Þ A i¼1 Ns Q

PDjt ¼ 1 

ð14Þ

Similarly, for the non-TFP region which is the complement of the TFP region, we calculate the target detection probabilities using the measure of the set !t ¼ ! n !t :

PDjt

  Ns Q EðT i Þ EðAT t Þ ¼1 1 EðT t Þ A i¼1

EðT t Þ ¼ ¼

a2

a1 a2

Z

a1 a2

EðAT t Þ ¼ ¼

Z

a1 Z a2 a1

1 Tð!t;/ Þd/ a2  a1   1 Tð!/ Þ  Tð!t;/ Þ d/ ¼ EðT  T t Þ a2  a1 1

a2  a1

p=2 w cosðaÞ  h sinðaÞ  p  2b b

Z pc Tð!p2þa Þ  Tð!t;p2þa Þ

p  2c

c

da

p=2 p=2 w cosðaÞ  h sinðaÞ wt cosðaÞ  h sinðaÞ ¼2  2    p  2c p  2c c

EðAT t Þ ¼

Z pb Tð! p Þ h t; 2 þa sinðaÞ b

EðAT t Þ ¼

p  2b

p=2 2 wha  h sinðaÞ da ¼ 2    p  2b b



Z pc Tð!pþa Þ  Tð!t;pþa Þ h sinðaÞ 2 2 c

b

p  2c

da

p=2 p=2  2 2 wha  h sinðaÞ wt ha  h sinðaÞ ¼2  2   p  2c p  2c 

ð20Þ

b

ð16Þ

where r is the radius of the sensing coverage of a sensor, wt is the width of the TFP region, w is the width of the deployment are and h is the height of the deployment area.

ð17Þ

Proof. A sample rectangular deployment site is illustrated in Fig. 7 to be used throughout this proof. The whole border region is represented with the rectangle ABCD. The width of the border region is w and the height of the border region is h. The shaded region within the rectangle KLMN shows the TFP region which includes the TFPs in the border. The width of the TFP region is wt whereas the height is equal to h. The sensors deployed within the border area have circular sensing with a radius of r. On the figure, only one of the deployed sensors is shown as a sample. The rest of the figure is explained within the analysis. By the assumption of identical circular sensing coverage for the sensors, the average thickness of each sensor is equal to the diameter of the circle which is 2r.

By the definition of the conditional probability, the detection probability of the trespasser is found as:

PD ¼ PDjt  pt þ P Djt  pt

da ¼ 2 

c

ATð!t;/ Þ d/

  1 ATð!/ Þ  ATð!t;/ Þ d/ ¼ EðAT  AT t Þ a2  a1

EðT t Þ ¼

ð15Þ

Since !t and !t are disjoint, by the definitions of EðT t Þ and EðAT t Þ,

Z

ð19Þ

ð18Þ

which yields Eq. (13). h With Theorems 1 and 2, we show that the DetQM of any sensor network with given number of sensors deployed can be calculated according to the field properties and the trespasser trajectory patterns. In Theorem 1, we take the case where there is no favorite path within the field and analyze its DetQM accordingly and derive

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C. Komar et al. / Computer Communications 35 (2012) 1185–1199

  Similar to the derivation of E T t , the area of the non-TFP region covered by !t;/ is derived by subtracting the area of the TFP region covered by !t;/ from the area of the deployment area covered by !/ . Hence the average area EðAT t Þ is equal to:

0

Z pb E AT t ¼ b

¼

Fig. 7. The representation of a rectangular deployment site.

The trajectories in ! are restricted to pass through the deployment site from the insecure side (bottom side) to the secure side (top side). Hence the lower bound of these linear trajectories is the angle c, which is the angle between the diagonal of the rectangle and the x axis. Similarly, the upper bound is the angle of the opposite diagonal which is p  c. In addition, since the height of the site is h and the width of the site is w, we have:

c ¼ arctanðh=wÞ In the same way, trajectories in the subset !t pass through the TFP region of width wt and are bounded by the angles, b and p  b, where

b ¼ arctanðh=wt Þ The thickness of the lines with slope a is determined by measuring the set of lines in !/ and !t;/ where / ¼ p2 þ a since these lines are parameterized by the slope of the perpendicular line passing from 0 the origin. The thickness of !/ and !t;/ are equal to the height h 0 and ht of the parallelograms respectively formed by the lines with slope a. Clearly 0

h ¼ Tð!p2þa Þ ¼ w sinðaÞ  h cosðaÞ 0

ht ¼ Tð!t;p2þa Þ ¼ wt sinðaÞ  h cosðaÞ

ð21Þ

The average thickness of the TFP region is equal to the average of 0 the thicknesses, ht , of the parallelograms formed by lines in all directions between b and p  b.



EðT t Þ ¼

Z pb T !t;pþa 2



p  2b

b

da

ð22Þ

Since the TFP trajectories does not belong to !t;/ , the thickness of the non-TFP region can be found by subtracting the measure of lines in !t;/ from the measure of !/ . This is equivalent to subtracting the 0 0 thickness ht from thickness of the deployment area, h . Hence, the average thickness of the non-TFP region is equal to the average of 0 0 the thicknesses, h  ht , of the parallelograms formed by lines in all 0 directions between c and p  c. Note that, ht ¼ 0 if a R ½b; p  b, since !t;p2þa ¼ ;.

EðT t Þ ¼

Z pc Tð!p2þa Þ  Tð!t;p2þa Þ

p  2c

c

da

ð23Þ

The average area of the TFP region covered by !t is equal to the average of the areas, ATð!t;/ Þ , of the parallelograms formed by lines 0 in all directions between b and p  b. The thickness ht of the region formed by trajectories with slope a is the height of the parallelogram. The length of these trajectories in TFP region is equal to h . Hence the average area EðAT t Þ is equal to: sinðaÞ

Z pb h  h sinðaÞ 0

EðAT t Þ ¼

b

p  2b

da ¼

Z pb Tð! p Þ  h t; 2 þa sinðaÞ b

p  2b

da

ð24Þ

A

T !t;pþa

2

da



p  2c



A

2

p  2c

Z pb h0  h  h0  h t sinðaÞ sinðaÞ

da

0

Z pb h0  h  h t sinðaÞ b

¼

T !pþa



b

¼

1

@A

Z pc

da p  2c

Tð!p2þa Þ  Tð!t;p2þa Þ sinðh aÞ

p  2c

c

da

ð25Þ

We provide the derivations for the list of equations in Eq. (20). By substituting these equations in Eq. (13), we provide Eq. (19). h In this section, we derived a formulation for the probability of intruder detection in a border surveillance sensor network using the line measure and thickness tools from the geometric probability. The initial formulation covers the heterogenous sensing coverage and convex bounded condition. The initial formulation is further simplified by the assumption of homogenous and circular sensing coverage and a rectangular deployment site. Note that in this paper we do not concentrate on the node or connectivity losses since we aim to determine the direct effect of TFP on the DetQM without considering any degradation due to sensor losses. However, the node or connectivity losses may be identified using the periodic ALIVE messages transmitted by the sensor nodes, which may be used to determine the size of the coverage holes in the deployment area. After determining the count and dimensions of the coverage holes, we can integrate their effect on DetQM formulation using the probability that a sensor does not reside in any of the coverage holes in the deployment area and in the TFP region (P a and Pat ) into our formulas as introduced in [6]. In this case, Eq. (19) can be rewritten as:

 N s 2r EðAT t Þ  P at PD ¼ 1  pt  1   EðT t Þ h  w  Ns 2r EðAT t Þ  Pa  ð1  pt Þ  1   EðT t Þ h  w

ð26Þ

In the next section, we will present our simulation scenarios and provide a comparison between the analytical and simulation results. 6. Analytical results In this section, we provide a comparison between the analytical and the simulation results to present the validity of our analytical estimation. In order to observe the effects of the individual parameters in the model, the following assumptions are made to minimize the effects of external parameters on our results.  A binary sensing model is assumed, where for a given sensor s located at zs, probability of detection P D ðsÞ of a point x is:

PD ðsÞ ¼



1; if dðx; zs Þ < Rs 0; otherwise

ð27Þ

 A circular sensing range is assumed, on a 2-D flat world deployment region.

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C. Komar et al. / Computer Communications 35 (2012) 1185–1199

Table 2 Tested parameters for different TFP scenarios. Parameter

Tested Values

Border Width

2000, 4000, 6000, 8000, 10000, 12000, 14000, 16000, 18000, 20000a (m) 2000 m 200a, 400, 600, 800, 1000, 1200, 1400, 1600, 1800, 2000 (m) 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9⁄, 1.0

Border Height TFP Region Width TFP Region Preference Sensor Count (low) Sensor Count (high) Sensing Range a

50, 100, 150, 200a, 250, 300, 350, 400, 450 500, 600, 700, 800a, 900, 1000 25 m

These are the default values used in the simulations.

 Intruders prefer the TFP region more than the other for trespassing.  TFP region consists a small ratio of the deployment area.  Intruders follow a straight line while they are trespassing the deployment area.  Sensing equipments of sensors are awake and able to detect the intruders at all times. For testing the validity of analytical results, simulations are run and tested against different parameter sets in the MATLAB platform. The simulations are repeated for different parameter sets. For each set, 1,000 random sensor network deployments and TFP region placements are performed. For each random sensor network deployment, 10,000 lines, each representing one trespassing, are generated. Thus, for each parameter, we have a result set in the size of 107 detection logs and the probability of detection is calculated using these logs. The results represented in the graphs are the mean values of these simulation runs. In Table 2, the values indicated with ⁄ are used as the default values for the simulations. The region dimensions extend from 2,000 m  2,000 m that represents a square border region to 2,000 m  20,000 m that represents a long narrow border region.

1

1

0.9

0.9

0.8

pt = 0.0 p = 0.5

0.8

pt = 1.0

0.7

t

0.7 0.6

DetQM

DetQM

By executing simulations using different dimensions, various border region shapes are analyzed. For the sensor count, two different parameter sets (low and high) are defined to simulate low and high density sensor deployment networks. Since the aim of the study is to analyze the effect of the TFP region and its properties, sensing range is selected as constant and fixed at 25 m which is a plausible value for MIR type sensor networks. However, we also designed an experiment to investigate the effect of different sensing ranges on our DetQM metric. In our simulations, we analyzed the effects of system parameters on the detection performance to introduce the characteristics of the TFP and compared the simulation results and analytical predictions to show the accuracy of the analytical model. The values used in these simulations are chosen for demonstrative purposes. In our figures, we presented the results only for 3 different levels of TFP preference for the sake of clarity. The DetQM metric is used as an estimator for the network quality and is proposed to estimate adjustable network parameters such as sensing range and sensor count to sustain a desired level of detection quality. Since it is not possible to adjust the uncontrollable parameters such as TFP region width, which is formed based on the trespassers preferences, it is feasible to determine the factors such as the sensor count for a given area size, or the area size for a given sensor count. To emphasize the effects of these parameters on the DetQM values, some reference DetQM values are provided in the feasible operation range of a sample SWSN, which requires the DetQM value to be above 90%. Note that since DetQM is proposed as a metric for the rate of detection, it is measured in terms of percentage (%). Our proposed model can be applied to different scenarios very easily. The given DetQM metric can be easily utilized in developing network optimization tools. The expected DetQM of a sensor network can be easily calculated for different possible network configuration parameters and the corresponding DetQM values can be stored in a lookup table. While the deployment is being planned, the required sensor number and the deployment area dimensions can be found out from this table by a simple search algorithm. For example, if the network budget is limited, the maximum area that can be covered with a minimum required detection probability using the available sensor count can be easily calculated using

0.5

0.6 0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

Simulated DQM values Analytical DQM Values

0

p = 0.0 t p = 0.5 t pt = 1.0

0.1

Simulated DQM values Analytical DQM Values

0 50

100

150

200

250

300

350

400

450

500

600

700

800

Sensor Count

Sensor count

(a)

(b)

900

1000

Fig. 8. Effect of sensor count and TFP preference probability on analytical and simulated DetQM values. Lines are ordered from the lowest TFP preference probability at the top (0.0, no trespassing through TFP region) to the largest TFP preference probability at the bottom (1.0, all trespassing through TFP region) for the respective scenarios. (a) Low density sensor deployment (50–450). (b) High density sensor deployment (500–1000).

1195

1

1

0.9

0.9

0.8

0.8

0.7

0.7

pt = 0.0 pt = 0.5 pt = 1.0

0.6

0.6 0.5

pt = 0.0

0.4

pt = 1.0

p = 0.5 t

DetQM

DetQM

C. Komar et al. / Computer Communications 35 (2012) 1185–1199

0.5 0.4

0.3

0.3

0.2

0.2

0.1

0.1

Simulated DQM values Analytical DQM Values

Simulated DQM values Analytical DQM Values

0 2000 4000 6000 8000 100001200014000160001800020000

0 2000 4000 6000 8000 100001200014000160001800020000

Region width (m)

Region width (m)

(a)

(b)

Fig. 9. Effect of border width with fixed TFP region ratio and TFP probability on DetQM values. Lines are ordered from lowest TFP preference (0.0, no passage through the TFP region) at the top to the smallest TFP preference (1.0, all passages through the TFP region) at the bottom for the respective scenarios. (a) Low density sensor deployment (200 sensors). (b) High density sensor deployment (800 sensors).

1

1

0.9

0.9

pt = 0.0 p = 0.5 t

0.8

0.8

0.7

0.7

0.6 pt = 0.0

0.5

pt = 0.5 pt = 1.0

0.4

DetQM

DetQM

p = 1.0

0.6 0.5 0.4

0.3

0.3

0.2

0.2

0.1

Simulated DQM values Analytical DQM Values

0 2000 4000 6000 8000 10000 1200014000160001800020000

Region width (m)

(a)

t

0.1

Simulated DQM values Analytical DQM Values

0 2000 4000 6000 8000 10000 1200014000160001800020000

Region width (m)

(b)

Fig. 10. Effect of border width with fixed TFP region width and TFP preference ratio on detection probability. Lines are ordered from lowest TFP preference (0.0, no passage through the TFP region) at the top to the smallest TFP preference (1.0, all passages through the TFP region) at the bottom for the respective scenarios. (a) Low density sensor deployment (200 sensors). (b) High density sensor deployment (800 sensors).

the lookup table. The effect of the other network parameters also can be easily observed with the help of such a table. 6.1. The combined effect of sensor count and TFP probability on DetQM values In these simulations, the sensor count deployed is increased step by step according to the values given in the parameter sets for different TFP preference probabilities to see its effect on the DetQM values. Separate test cases are designed for high and low sensor counts. In all simulations, the width of the border region is 20,000 m and the width of the TFP region is 200 m and the height of the border region is 2,000 m. The results are presented in Fig. 8. The narrow gap between the simulation results and DetQM shows the accuracy of the theoretical model and shows that DetQM can

be used as a close estimate of the actual detection quality of the network. The gap is at most 2.8% for the low density deployment scenarios and 1.4% for the high density deployment scenarios. When the sensor network is very sparse and trajectory count is restricted, the assumption that the sensors are distributed uniformly along the border is not always satisfied for given sample size. The uniformity property of the distribution is hard to maintain using the Monte Carlo simulation approach with small number of sensors and trajectories distributed over a large area, e.g. only 50 sensors in a 2,000 m  20,000 m border region. This situation causes a bigger gap between the theoretical values and simulation results for sparse sensor network cases. On the other hand, in the dense network scenarios, as the number of sensors increases, the problem of clustering becomes more apparent. In this case, the area covered by the sensors is generally less than the expected total

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C. Komar et al. / Computer Communications 35 (2012) 1185–1199

sensor network are much more significant than the results presented in this study.

0.6

0.55

6.2. Combined effect of network density and TFP on DetQM

DetQM

0.5

0.45

0.4

0.35 Simulated DQM values Analytical DQM Values 0.3 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

TFP preference (pt ) Fig. 11. DetQM vs TFP preference (pt).

coverage of the sensors, causing at most 1.4% absolute difference between the simulation and analytical results. The sensor count parameter is observed to be proportional to the probability of detection, as indicated in the DetQM formulation. Both the analysis and simulation results show that, when the TFP preference probability is increased, the detection ratio falls. This behavior is more clear in the low density deployment scenarios compared to high density deployment scenarios. For example, when 50 sensors are deployed to the border region, the DetQM value falls from 15.8% for the pt ¼ 0 to 11.8% for the pt ¼ 1. This represents an absolute 4% and a relative 25.3% fall (relative to the DetQM value at pt ¼ 0) in the detection performance of the system. When the sensor count is increased to 450, the corresponding DetQM values are 78.7% and 67.6% for the respective TFP probabilities. This difference means an absolute 11.1% and a relative 14% decrease in the detection performance of the system. Hence, the rate of relative performance degradation due to increasing TFP preference decreases with the increase in the sensor count. A similar behavior is also observed in the high density deployment scenarios, where the sensor count is increased from 500 to 1000 in steps of 100 (Fig. 8b). As seen from Eq. (13), when the TFP region is selected, the DetQM is calculated using the combination of partial probabilities of the TFP region and non-TFP region. Because of the angular calculations in E(T) and EðAT Þ, the detection probability of the trespassers preferring the TFP region is smaller than the ones that do not prefer the TFP region. The smaller angle range of the possible paths for TFP trespassers has a restricting effect on the length of average length of the shortest paths. The reduced path length results in reduced detection probability. Under the given assumptions and default parameter values, the feasible range of operation, where the DetQM value is above 90%, requires the sensor count to be above 900 for pt ¼ 0:5. Similarly, for a sensor count of 1000, the area can be monitored for a feasible DetQM even if the trespassers prefer using only the TFPs. In our simulations, we assumed uniform distribution of the sensors on the field and there are no sensor failures or deaths. In the real life, the sensors deployed in the TFP region make more detection compared to the rest of the network, and communicate more frequently. This higher load causes these sensors die earlier and soon, a detection hole appears in the TFP region. In other words, the effects of the TFP region on the performance and lifetime of a

In the previous simulations, we changed the network density directly by changing the sensor count deployed on the area. These simulations gave us an insight about the relationship between the sensor count and the TFP preference on the network DetQM. In order to see the effect of the network density and the TFP region on the DetQM, we performed additional simulations. In the first set of the simulations, we fixed the ratio of the TFP region width to the border area width at 0.01. We increased the border width from 2,000 m to 20,000 m while keeping the TFP region width ratio fixed. In the second set of the simulations, we fixed the absolute width of the TFP region at 200 m. By increasing the border width, we basically decreased the sensor density in the border area. The simulations are repeated for low and high number of sensor deployment scenarios, 200 and 800 sensors, respectively. The results are shown in Figs. 9 and 10. The narrow gap between analytical and simulation results indicate that the provided DetQM metric is a close estimate of the simulated detection probability in the given cases. The gap is at most 2.5% for the low density deployment scenarios and 1.4% for the high density deployment scenarios. The area width parameter is observed to be inversely proportional to the probability of detection, as indicated in the DetQM formulation.The figures show that there is a significant detection performance decrease if there is a TFP region in the border area. This decrease is especially observed in the sparse sensor deployment cases. When we increase the region width to 20,000 m, i.e. decrease the sensor density 10-fold by increasing the deployment area 10-fold in the first set of simulations, the DetQM value decreases from 56.3% for pt ¼ 0 to 48.5% for pt ¼ 1. This corresponds to an absolute 7.8% and relative 13.9% decrease in the detection performance when all of the trespassers prefer the TFP region instead of the non-TFP region. When the same value is calculated for the second set of the simulations, we get a DetQM value decrease from 57.6% to 46.5% for pt ¼ 0 and pt ¼ 1 values, respectively. This corresponds to an absolute 9.1% and relative 15.28% decrease in the detection performance. We also see that, our analysis matches with the simulation results. It can also be concluded that decreasing the sensor density increases the importance of TFP width on detection quality. When the fixed ratio TFP region and fixed width TFP region scenarios are compared to each other, we do not observe a significant change between the scenarios for the same border width and sensor count scenarios. This comparison can be interpreted as that, the existence of a TFP region in the border area has a more important effect on the results, than the width of the region. This phenomenon can be explained by Eq. (13). As seen from the partial probability in the equation, the coefficients of the major components are the TFP preference probabilities. The remaining factors are embedded in the calculation of E(T) and EðAT Þ, determining expected path length and expected number of sensors in the TFP region respectively. The DetQM of the system is significantly affected from the TFP preference probability, pt. Under the given assumptions and default parameter values, the feasible range of operation, where the DetQM value is above 90%, requires the width of the deployment area to be below 400 m for all TFP preference values when the network consists of 200 sensors in both scenarios. The width of the deployment area should be below 18,000 m and 16,000 m for all TFP preference values when the network consists of 800 sensors in the first and second scenarios respectively. Similarly, for an area width of 18,000 m, the area

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1

1

0.9

0.9

0.8

0.8

0.7

0.7 0.6 pt = 0.0

0.5

p = 0.5 t

DetQM

DetQM

0.6

pt = 0.0 p = 0.5 t pt = 1.0

pt = 1.0

0.4

0.5 0.4

0.3

0.3

0.2

0.2

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0.1

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Fig. 12. Effect of sensing range on DetQM. Lines are ordered from lowest TFP preference (0.0, no passage through the TFP region) at the top to the smallest TFP preference (1.0, all passages through the TFP region) at the bottom for the respective scenarios. (a) Low density sensor deployment (200 sensors). (b) High density sensor deployment (800 sensors).

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Fig. 13. Effect of meandering path on DetQM. (a) Detection probability using straight and segmented trajectories. (b) Detection probability at different number of segmented trajectory.

can be monitored for a feasible DQM when the trespassers prefer using the TFPs at most half of the time for fixed TFP width scenario. 6.3. DetQM vs TFP preference In previous simulations, we observed that the effect of TFP preference is dominant over the effect of TFP region width. In this set of simulations, we will analyze the effect of TFP preference on DetQM more in detail. In order to perceive this effect, we measure the DetQM of the system with all system parameters set constant except the pt. The border region width is 20,000 m and the TFP width is 200 m. The pt which denotes the TFP preference is increased from 0 to 1 with steps of 0.1. The result of the simulation is seen in

Fig. 11. The pt parameter is observed to be proportional to the probability of detection, as indicated in the DetQM formulation. The gap between the simulation results and DetQM value is at most 2.2%. The DetQM value decreases from 49.7% for pt ¼ 0 to 39.4% for pt ¼ 1. This fall means about an absolute 10.3% and relative 20.7% degradation in the DetQM performance between the values pt ¼ 0 and pt ¼ 1 which indicates the importance of the TFP preference on the detection quality. Although, no parameter is changed within the network including the deployed sensor count and sensor node configurations, the network performance deteriorates with increasing pt. Another consequence of the TFP region is the inequality among the nodes in terms of duty and usefulness. Most of the detections are being achieved by a small subset of all

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sensors which cover TFP paths. This means a heavy load on a small part of the network whereas the rest of the sensor network remains idle most of the time and rarely used. This setting results in early exhaustion of active sensors and potentially creating detection holes in those areas. Thus, it can be concluded that network operators should carefully inspect the network deployment area, uncover TFP paths within the border region and deploy sensors accordingly for an efficient surveillance network. 6.4. Effect of sensing range on DetQM In our previous simulations, we fixed the sensing range at 25 m which is a plausible range for most of the sensor types used in surveillance sensor networks. On the other hand, there are various sensor types which have different sensing ranges and generally shorter than our fixed value. In order to observe the effect of sensing range on the DetQM of a sensor network and to show the adaptability of our model to different sensing ranges, we performed simulations using different sensing ranges between 5 m and 25 m and compared the simulation results with the analytical model results. The simulations are repeated for low density (200 sensors) and high density (800 sensors) deployments. In all simulations, the width of the border region is 20,000 m and the width of the TFP region is 200 m and the height of the border region is 2,000 m. The results of the simulations and analytical results are shown in Fig. 12. The part (a) shows the DetQM values for the low density deployment and the part (b) shows high density deployment. The sensing range is observed to be proportional to the probability of detection, as indicated in the DetQM formulation. The gap between the simulation results and DetQM is acceptable and shows the accuracy of the theoretical model and shows that DetQM can be used as a close estimate of the actual detection quality of the network. The gap is at most 2.8% for the low density deployment scenarios and 2.5% for the high density deployment scenarios. The wide gaps are caused by the sparseness and coverage problems discussed in Section 6.1. As the sensing range increases, the total coverage of the sensors increases and the gap between the DetQM values and simulation results narrows. For very low sensing ranges, the total coverage decreases and the gap between the lines in high density deployment case is observed to be similar to the performance gap in low density deployment case. The first set of simulations are performed for the low density deployment scenario. Both the analysis results and simulations show that in all different sensing ranges, the detection ratio decreases when the TFP probability increases. This behavior is more significant when the expected detection ratio of the system is on the average very low. For example when the sensing range is 5 m in the low density deployment, the DetQM value is 12.8% for the non-TFP scenario and 9.5% for TFP scenario. The detection difference between the non-TFP scenario and TFP scenario is only 3.3% in this case. On the other hand, when the sensing range is increased to 25 m, the DetQM values are 49.7% for the non-TFP case and 39.4% for the TFP case, and the difference increases to 10.3% which is more than 3 times of the 5 m sensing range scenario. The same set of simulations are also run for the high density deployment scenario. Unlike the low density deployment scenario, the detection ratio differences between the non-TFP scenario and the TFP scenario for different sensing ranges are very similar. When the sensing range of the sensors is set at 5 m, the detection ratios for the non-TFP scenario and TFP scenario are 42.2% and 33% respectively, having a difference of 9.2%. When it is increased to 10 m, the corresponding values are 66.6% and 55.1%, having a difference of 11.5%. The DetQM difference is calculated as 10.7%, 9% and 7.1% for the rest of the scenarios in increasing sensing range order. Although, we did not run simulations for larger sensing

ranges, it can be guessed that the gap between the non-TFP and TFP scenarios get smaller with the increased network detection ratio. The maximum difference is observed when the sensing range is at 10 m. Under the given assumptions and default parameter values, the feasible range of operation, where the DetQM value is above 90%, requires the sensing range of the sensor to be above 25 m when trespassers prefer TFP all of the time. Similarly if the trespassers are known to utilize the TFP region half of the time, then the appropriate sensing range should be 25 m or more.

6.5. Segmented trajectory vs. straight trajectory We assume that the intruder follows a straight path within the monitored zone. This assumption is made to observe the worst case performance of the network. If the intruder follows a segmented path instead of a straight path, the detection probability increases and so the performance of the network. The performance difference between two cases are shown using simulations. Straight and segmented trajectory simulations are run within a homogeneous preferred border area with uniform random deployment using the same dimension and deployment parameters. Two sample trajectories are depicted in Fig. 1. In the straight trajectory case, the intruder X follows the straight path L between the entry point P and exit point R. In the segmented trajectory case, the intruder follows a path L0 with w waypoints between the entry point P0 and exit point R0 such that:

L0 ¼ fðP0 ðx0 ; 0Þ; w1 ðx1 ; y1 Þ; . . . ; wi ðxi ; yi Þ; R0 ðxiþ1 ; hÞg where

1 6 i 6 wmax 0 6 y1 6    6 yi 6 h

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The simulations are run in a single region border scenario with height h = 500 m and width w changing between 2000 m and 10,000 m. There are 100 sensors deployed to the region. The result is shown in Fig. 13.a. The maximum waypoint value is wmax ¼ 5. This means that an intruder moving along a segmented trajectory will change direction at most 5 times following a path consisting of 6 segments until reaching the other side. The number of waypoints is selected to be uniform random between 1 and 5 for each trajectory. The direction change at each waypoint is limited to forward direction, i.e. there is no turning back in the followed path. Another set of simulations are designed in order to see the effect of the segment count on the detection probability. In this scenario, the height and width of the region are kept constant, h = 2000 m and w = 10,000 m, respectively. There are 100 sensors deployed to the region. The number of segments within the segmented path is increased from 1 to 8, and kept constant at each simulation set. The detection probability is measured and plotted for different segment counts. The result is shown in Fig. 13.b. The initial point of the figure in which there is only one segment within the trajectory also corresponds to the straight trajectory case. The detection probability increases with each segment count and reaches to the certain detection probability of PD=1 at 6 segments. The comparison of trajectory simulations clearly shows that the detection probability in the segmented path scenario is always bigger than the straight path scenario. It can be concluded that, in terms of detection performance, our straight path analysis gives the worst performance of the sensor network. For the network operators, knowing the worst case performance is generally more important than the average case, especially in SWSN applications.

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7. Conclusion In this paper, we introduce a new concept, trespassers’ favorite paths (TFP), based on the real life observations and experiences. We develop an analytical and closed form formulation for the detection quality which takes the pattern of TFPs into consideration and show the effect of the TFPs on the network detection performance. Described analytical formula can be highly useful both for the deployment of real life SWSNs and their simulation by the research community. Such a formulation can guide the network operators to forecast the performance of a SWSN before deployment and allow them to improve the detection quality in the design phase. In addition, it can be used easily for different TFP border scenarios by simply modifying the input parameters to reflect the deployment environment. To test the validity of the analytical work, we studied the detection quality and ran corresponding simulations under different intruder TFP preferences with different parameter sets. The simulation results match with the analytical results with a high accuracy, on the average about 5% gap, thus proving the validity and applicability of the proposed analytical work in the real life scenarios. The impact of the TFP region on the detection quality is shown to be significant when the density of the nodes is about medium and the TFP preference rate is high. In that case, the detection probability in a TFP scenario is almost 20% less than the nonTFP scenario, meaning a 20% performance degradation in case of a deployment made without taking the TFP region into account. The results clearly show that the existence of TFP is an important factor that should not be disregarded during the design and deployment phases. In this work, we assumed a binary sensing model, which is used to build the theoretical base of the analysis. Currently, we aim to replace this model with more complex sensing models such as a probabilistic sensing model in our calculations. Moreover, the deployment sites are assumed to be a flat world. For relatively flat borders, a two-dimensional formulation is sufficient and very useful for the network engineers to determine the physical properties of the border surveillance sensor network. However, the formulation can be improved to support three-dimensional deployment models. Acknowledgement This work is supported by the Turkish State Planning Organization (DPT) under the TAM Project, number 2007K120610, the Scientific and Technological Council of Turkey (TUBITAK) under the Grant No. 108E207 (COST Action IC0906, WINEMO) and Bogazici University Research Fund (BAP) under the Grant No. 5146. References [1] I.F. Akyildiz, W. Su, Y. Sankarasubramaniam, E. Cayirci, Wireless sensor networks: a survey, Computer Networks 38 (4) (2002) 393–422. [2] I.F. Akyildiz, T. Melodia, K.R. Chowdhury, A survey on wireless multimedia sensor networks, Computer Networks 51 (4) (2007) 921–960. [3] Millet haber, http://www.millethaber.com/, 2010. [4] B.V. BorderWatch, Blueservo virtual borderwatch, URL http://www.blueservo. net/about.php, 2010. [5] J. Yick, B. Mukherjee, D. Ghosal, Wireless sensor network survey, Computer Networks 52 (12) (2008) 2292–2330.

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