# Characterizing the Path Coverage of Random Wireless Sensor Networks

- Moslem Noori
^{1}Email author, - Sahar Movaghati
^{1}and - Masoud Ardakani
^{1}

**2010**:716565

https://doi.org/10.1155/2010/716565

© Moslem Noori et al. 2010

**Received: **1 November 2009

**Accepted: **24 February 2010

**Published: **13 April 2010

## Abstract

Wireless sensor networks are widely used in security monitoring applications to sense and report specific activities in a field. In path coverage, for example, the network is in charge of monitoring a path and discovering any intruder trying to cross it. In this paper, we investigate the path coverage properties of a randomly deployed wireless sensor network when the number of sensors and also the length of the path are finite. As a consequence, Boolean model, which has been widely used previously, is not applicable. Using results from geometric probability, we determine the probability of full path coverage, distribution of the number of uncovered gaps over the path, and the probability of having no uncovered gaps larger than a specific size. We also find the cumulative distribution function (cdf) of the covered part of the path. Based on our results on the probability of full path coverage, we derive a tight upper bound for the number of nodes guaranteeing the full path coverage with a desired reliability. Through computer simulations, it is verified that for networks with nonasymptotic size, our analysis is accurate where the Boolean model can be inaccurate.

## Keywords

## 1. Introduction

Wireless sensor networks (WSNs) have many applications in security monitoring. In such applications, since it is essential to keep track of all activities within the field, network coverage is of great importance and must be considered in the network design stage.

Path coverage is one of the monitoring examples, where WSNs are deployed to sense a specific path and report possible efforts made by intruders to cross it. In a manual network deployment, the desired level of the path coverage can be achieved by proper placement of the sensors over the area. When it is not possible to deploy the network manually, random deployment, for example, dropping sensors from an aircraft, is used. Due to the randomness of the sensors location, network coverage expresses a stochastic behavior and the desired (full) path coverage is not guaranteed. Thus, a detailed analysis of the random network coverage can be ultimately useful in the network design stage to determine the node density for achieving the desired area/path coverage.

Path coverage by a random network (or barrier coverage which is a relaxed version of the path coverage) has been the focus of some previous work [1–6]. In [1], assuming that a random network is deployed over an infinite area with nodes following a Poisson distribution, authors investigate the path coverage of the network. They first study the path coverage over an infinite straight line when the sensor has a random sensing range. Then, they show that in the asymptotic situation, where the sensing range of the sensors tends to 0 and the node density approaches infinity, the results are extendible to finite linear and curvilinear paths. Further, a path coverage analysis is proposed for a high-density Poisson-distributed network in [2] where sensors have a fixed sensing range. The path coverage analysis of [1, 2] is based on the Boolean model of [7], where a Poisson point process is justified.

Kumar et al. study -barrier coverage provided by a random WSN in [3]. To this end, they develop a theoretical model revealing the behavior of the network coverage over a long narrow belt. It is assumed that the sensors are spread over the belt according to a Poisson distribution. The authors propose an algorithm determining whether an area is -barrier covered or not. Also, they introduce the concepts of weak and strong barrier coverage over the belt and derive the condition on the sensors density guaranteeing the weak barrier coverage.

The focus of [4] is on the strong barrier coverage. First, authors present a condition insuring the strong barrier coverage over a strip where the sensors locations follow a Poisson point process. Then, by considering asymptotic situation (on the network size and number of nodes) and using Percolation theory [8], they determine, with a probability approaching 1, whether the network has a strong barrier coverage or not. Then, they use their analysis to devise a distributed algorithm to build strong barrier coverage over the strip.

In this work, unlike most existing studies which focus on asymptotic setups, we study the path coverage of a finite random network (in terms of both network size and the number of nodes). As a result, the Boolean model is not accurate. Alternatively, the methodology of this work is based on some results from geometric probability. Our focus is on the path coverage for a circle, but extension to other path shapes is briefly discussed.

In the ideal case, all sensors are located exactly on the path. This, however, is not a practical assumption for randomly deployed networks. To consider the inaccuracy of the sensors locations, we assume that sensors are inside a ring containing the circular path. As a result, the portion of the path covered by any given sensor is not deterministic. Moreover, other factors may affect the sensing range of a sensor. Thus, our analysis is not based on a fixed sensing range. Indeed, we first develop a random model for the covered segment of the path by each sensor. Then, we study the distribution of the number of uncovered gaps on the path. The full path coverage is a special case where the number of gaps is zero. This is used to determine a tight bound on the number of active sensors assuring the full path coverage with a desired reliability. Also, we find the probability of having all possible gaps smaller than a given size. This probability reflects the reliability of detecting an intruding object with a known size.

In addition to studying the number of gaps, we present a simplified analysis for deriving the cumulative distribution function (cdf) of the covered part of the path. This simplified analysis is based on using the expected value of the covered part of the path by a sensor instead of considering the precise random model. We observe that the simplified analysis can provide a fairly accurate approximation of the path coverage.

Since our analysis studies the effect of the number of nodes on the path coverage of a finite size network, it can readily be used in the design of practical networks. In fact, using our results, one can determine the number of nodes in the network to satisfy a desired level of coverage. An example is provided.

The paper is organized as follows. Section 2 introduces the network model and defines the problem. Our coverage analysis is presented in Section 3. Section 4 includes computer simulations verifying our analysis. Finally, Section 5 concludes the paper.

## 2. Preliminary

### 2.1. Network Model

We also assume that sensors sensing range may vary from to . Obviously, for a fixed sensing range, model . Without loss of generality, it is assumed that the width of the ring is smaller than or equal to and the desired circular path is located at the middle of the ring. Since the sensors farther than to the path do not contribute to the path coverage, our assumption on the ring width does not hurt the generality of the analysis.

### 2.2. Motivation and Problem Definition

- (i)
*Distribution of the number of gaps.*Due to the randomness of the network implementation, sensors may not cover the whole path. In this case, one or more gaps appear. Assume that represents the number of gaps on . We are interested to find the probability of having gaps, shown by . - (ii)
*Full path coverage.*It is desired to provide a complete coverage of the path. Since the full path coverage is identical to having no gaps, one can equivalently find . This can simply be found from the derived distribution of . - (iii)
*Reliability of the network in detecting objects.*It is important to investigate whether the network is able to detect an object, while the path is not fully covered and there may exist some gaps. Basically, we need to consider the size of the gaps in addition to their number. If one knows the size of the intruders beforehand, it is not necessary to provide the full path coverage. Instead, it is possible to deploy a network such that while the path is not fully covered, the size of the possible gaps is smaller than the intruders. Clearly, implementing a network with possible small gaps requires fewer number of nodes and consequently is less expensive. To this end, we find the probability of having all gaps smaller than a given length , denoted by . - (iv)
*Distribution of the covered part of the path.*The covered part of the path, , has a stochastic nature and its distribution provides a general view of the entire path coverage. In fact, the covered part of the network reflects the combined effect of the number of gaps and their sizes. We derive the cdf of , .

## 3. Path Coverage Analysis

In this section, we present our analysis of the path coverage. For this purpose, we take advantage of existing results in geometric probability and extend them to our case. After the exact coverage analysis, a less complex approximate analysis is also presented.

The problem of covering a circle with random arcs has been studied in geometric probability [9–15]. In some cases, it is assumed that the arcs have a fixed length [9, 12, 13, 15] or the analysis is conducted in the asymptotic situation [10, 14]. Asymptotic analysis is suitable for the situation where the sizes of 's are significantly smaller than . In the following, we initially use the result of [11] on the coverage of a circle with random arcs of random sizes. This helps us to provide an exact explanation of the path coverage. Then, we use the mean value of 's to provide a simplified approximate analysis based on the fixed-length random arc placement over the circle [12].

### 3.1. Exact Analysis

We apply the following theorem from [11] to find the exact distribution of the number of gaps on .

Theorem 1.

Proof.

See [11].

To apply Theorem 1 for finding the number of uncovered gaps on , we first prove the uniformity of the arc distribution over in the following lemma.

Lemma 1.

For a symmetric distribution of the sensors over the path, the location of the intersection of the sensors sensing range and is uniformly distributed over .

Proof.

Following Lemma 1, in order to find the distribution of the number of gaps on we need or in our case the cdf of 's. Notice that 's are independent and identically distributed (i.i.d) random variables. We find in the appendix for arbitrary distributions of sensor location and sensing range.

As a result of Theorem 1 and Lemma 1, we have the following corollary.

Corollary 1.

can be used to find an upper bound on the number of nodes in the network guaranteeing the full path coverage with a given reliability. This is presented below.

Corollary 2.

Proof.

Combining (5) and (8) results in (6).

Using (6), it is straightforward to find an upper bound on guaranteeing a desired level of coverage, . Later, our simulations show that this bound is in fact very tight.

Another feature of the path coverage that we like to study is the quality of the coverage in terms of the size of the gaps on . Assume that we like to guarantee detecting any object bigger than a particular size, say . To assure detecting such objects, all of the gaps have to be smaller than . Hence, we like to find the probability of having no gaps larger than , where is the length of the largest gap on .

Corollary 3.

Proof.

This completes the proof.

Using the same approach taken for finding the upper bound on in (6), one can derive an upper bound on the number of nodes to guarantee having all gaps smaller than .

Notice that in (12) we assume rotational indices for 's. It means that if we replace the index with . In (12), is the length of the connected part of starting from and continuing clockwise. When , the whole spacing is covered and function should return . When , a portion of remains uncovered and there exists a gap at the right side of . Thus, function returns . It is noteworthy that because of the problem symmetry, 's are identically distributed random variables. Thus, we use a single random variable to refer to them.

This is because 's are i.i.d. and thus it is very unlikely that, for example, .

### 3.2. Approximate Analysis

One can also calculate the expected value of . To this aim, we first consider the uncovered part of the path, , and find its expected value, called . Then can be found using the fact .

### 3.3. Some Remarks

Remark 1.

Our path coverage analysis is applicable to any closed path, for example, ellipse, with finite length when the location of the path segment covered by an arbitrary sensor is uniformly distributed over the path. For this purpose, we just need to have the distribution of the intersection of sensors sensing range and the path. Also, the analysis is applicable to linear path coverage. In fact, the problem of covering a circle with random arcs can be transformed to the problem of covering a piece of line, say the interval , with random intervals. In this case, sensors are deployed randomly over a strip surrounding the linear path. It is notable that in the linear case, Torus convention [7] is applied. In Torus convention, it is assumed that if a part of the random interval goes out of the line segment, it comes in from the other side of the line piece. However, when the length of the random intervals is small compared to the line piece, one can remove the Torus convention and the analysis remains quite accurate.

Remark 2.

In many WSNs, the number of active sensors in the network changes with time. This can be due to, for example, sleep scheduling or death of some nodes. Since our analysis is provided for arbitrary it can accommodate such situations, simply by replacing with in relevant equations. Consequently, the coverage can be studied as a function of time.

## 4. Simulation

In this section, we demonstrate the accuracy of our analysis via computer simulations. We have inspected two scenarios for the sensors sensing range. In the first scenario, we assume a network with sensors all having a fixed sensing range equal to . The sensors are uniformly deployed inside a ring around the circular path, where has unit circumference. In the second scenario, the sensors sensing range is also uniformly distributed between and . A zero sensing range can represent a dead sensor.

We evaluate random properties such as the full coverage, number of uncovered gaps, tightness of the bound presented in (6), the intruder detectability, and the portion of the covered path using simulation, and compare the results with our theoretical analysis.

### 4.1. Uncovered Gaps

It is clear from Figure 4 that the results from the approximate analysis are fairly close to the exact analysis and the simulations. Due to the complexity of the evaluation of exact analysis, we compare the rest of our simulation results with the approximate analysis presented in Section 3.2 to characterize the coverage properties of the network.

In the case of fixed sensing range, as the width of the ring becomes smaller, the variance of decreases and the arc lengths become closer to , making the approximate analysis more accurate. To study the worst case, in our analysis, we assume the ring width is equal to . Notice that any node outside this ring does not contribute to the path coverage. For random sensing range, we choose . Notice that since , there will be nodes in the ring that will not contribute to the path coverage.

### 4.2. Detectability of the Object

### 4.3. Covered Part of the Path

## 5. Conclusion

In this paper, we studied the path coverage of a random WSN when neither the area size nor the number of network nodes were infinite. Hence, the widely used Boolean model was no longer valid. Moreover, due to the randomness of the sensors placement over the area, network coverage was nondeterministic. Thus, a probabilistic solution was taken for determining the network coverage features. Our analysis considered the number of gaps, probability of full path coverage, probability of having all uncovered gaps smaller than a specific size, and the cdf of the covered length of the path. All these characteristics were found as a function of the number of sensors We also proposed a tight upper bound on required for full coverage. Through computer simulations, we verified the accuracy of our approach. Since our study was performed for finite , using our results on various features of path coverage, one can find the necessary number of sensors for a certain quality of coverage.

## Appendix

In the following, we find the cdf of the intersection between the sensing area of the sensors and , called . First, we study the situation where sensors have a fixed sensing range and they are uniformly distributed over the ring. Then, we investigate the general case where sensors can have a random sensing range varying from to and have any symmetric distribution over the ring.

## Declarations

### Acknowledgment

The authors would like to thank Natural Sciences and Engineering Research Council of Canada (NSERC) and Informatics Circle of Research Excellence (*i* CORE) for supporting our research.

## Authors’ Affiliations

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## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.