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.

An arbitrary point on is covered if it is within the sensing range of at least one sensor. Here, we assume that the sensing area of sensor is a circle denoted by . The covered part of the path by each is its intersection with which is an arc, called . Thus, the total covered part of the path is

Notice that the length of 's depends on the location of the sensor within the ring-shaped network area and its sensing range. Considering an arbitrary point as the origin on and choosing the clockwise direction as the positive direction, each starts from and continues (clockwise) until , (Figure 2). In other words, determines the most left point of the arc and specifies the most right point of the arc. There are two noteworthy issues here. First, the size of 's and their positions are random because of the random placement of the sensors over the ring. Second, is not necessarily connected and there may exist several uncovered gaps on . The number of uncovered gaps on and their size can reflect the possible opportunities for intruders to pass without being detected by the sensors. If is fully covered.

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.

Assume that arcs are distributed independently with a uniform distribution over a circle of circumference 1. If shows the cdf of the arc length over , the distribution of the number of uncovered gaps on the circle, , is

where

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.

We equivalently show that the center points of the arcs are uniformly distributed over the circle. For this purpose, consider a small element with length on . Then, we build a sector of the ring based on this length element whose left and right sides pass the left and right ends of the length element (Figure 3). The center point of the arcs, resulted from the intersection of the sensing area of the nodes within the sector and falls within . Due to the independence of the sensors distribution from the polar angle, all elements with length on the circle have the same chance to include an arc center point. Therefore, the distribution of the arc centers, and consequently arc locations, is uniform on .

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.

The probability of the full path coverage, , is

Furthermore, one can show that the expected number of gaps on is [11]

where is the mean of 's.

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.

To guarantee a full path coverage with probability , the following relation holds

Proof.

Recall Markov's inequality for a positive random variable

where . If we let be the random variable of the number of gaps , and put , we have

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.

The probability of having no gaps larger than is

Proof.

Consider a realization of the random placement of arcs on the path. Now, one can consider a scenario where the length of each arc is increased by . If there exists a gap smaller than in the first scenario, this gap will be covered in the second scenario since the arcs are longer. On the other hand, a gap with any size in the second scenario will be a gap with length more than in the first scenario. Notice that the above discussion is valid for any realization of the network. Thus, instead of investigating the probability of having no gaps longer than in the first scenario, we look for the probability of the full coverage in the second scenario. Denoting the length of the arcs in the second scenario by , one can think of them as being drawn from the distribution or equivalently

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 .

We also like to investigate that is, the portion of which is covered by the nodes. To find , we first reorder the arcs based on their starting points, 's. Thus . Now, we divide to arcs , where is an arc starting from and ending at . Finally, starts from and ends at . Since we have random arcs intersecting with , there exist of such spacings on the circle. These spacings may or may not be covered by the network. Adding the covered parts of the path together, we have

where . Also, 's denotes the length of and

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.

The distribution of can be well approximated by a Gaussian distribution using central limit theorem (CLT) where the mean value of , , is . Here, denotes the mean value of . Also, where and represent the variance of and , respectively. In reality, one can safely simplify (12) to

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

### 3.2. Approximate Analysis

In the following, we present an approximate analysis simplifying our path coverage study. The idea of this approximate analysis is to consider a model where a set of fixed-length arcs are spread randomly over instead of using the actual random-sized arcs. The length of these fixed arcs is equal to the mean value of the random-sized arcs in the original case. We denote the mean value of these random arcs with . In this case, it can be shown that the number of uncovered gaps on is distributed as follows [12]:

where . The same technique as before is applicable to find the probability of having no gaps larger than on . For this purpose, we just need to use instead of in (14). In addition, the distribution of can be derived when the arc size is fixed [12]. In this case, we have

where is the cdf of .

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 .

An arbitrary point on remains uncovered when there is no covering it. This is equivalent to having none of 's within an arc with length whose right end point is . There are sensors in the network, hence, the probability of having uncovered, , is

Consequently,

### 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.