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.
-barrier coverage provided by a random WSN in [
-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.
sensors monitoring a circular path with unit circumference, called 
. In an ideal case, the sensors are precisely located on the circular path, but this is not usually true for a randomly deployed network. In order to take this fact into account, we assume that sensors are randomly spread over a ring containing 
(See Figure 
Thus, distributions with larger values close to 
are preferred. When no effort is made to put the sensor as close as possible to the path (
sensors are spread totally randomly), the uniform distribution is obtained. Hence, in the sense of placement efforts, uniform distribution reflects the worst case. We consider uniform distribution to verify our analysis by computer simulation in Section 4. Our analysis, however, is presented for any given symmetric distribution. Also, notice that since the number of sensors is finite and known, Poisson distribution, which has been the focus of existing asymptotic analysis, is not applicable.
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.
represents the number of gaps on 
. We are interested to find the probability of having 
gaps, shown by 
.
. This can simply be found from the derived distribution of 
.
, denoted by 
.
, 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 
, 
.
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
'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 
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.
.
's are significantly smaller than 
. In the following, we initially use the result of [
's to provide a simplified approximate analysis based on the fixed-length random arc placement over the circle [
.
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
, we first prove the uniformity of the arc distribution over 
in the following lemma.
is uniformly distributed over 
.
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 
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 
.
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.
, is
is [
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.
, the following relation holds
. If we let 
be the random variable of the number of gaps 
, and put 
, we have
guaranteeing a desired level of coverage, 
. Later, our simulations show that this bound is in fact very tight.
. 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 
.
is
. 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
in (6), one can derive an upper bound on the number of nodes to guarantee having all gaps smaller than 
.
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
. Also, 
's denotes the length of 
and
'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.
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
's are i.i.d. and thus it is very unlikely that, for example, 
.
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 [
. 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 [
is the cdf 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 
.
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
, 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 [
it can accommodate such situations, simply by replacing 
with 
in relevant equations. Consequently, the coverage can be studied as a function of time.
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.
, was derived in Section 3.1. Figure 
. Here, we have assumed that the sensing range of all sensor nodes is fixed and is equal to 
. The theoretical results using (2) have also been sketched for comparison. It can be concluded that the formulation derived in Section 3.1 quite accurately describes the pmf of the number of uncovered gaps on the path. The third curve in Figure 
in (14) is set to be the expected value of random variable 
, derived in the appendix.
and 
.
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.
is estimated through simulation for different values of sensing range, 
, and is compared with the theoretical results using (14). As seen from the curves, our theoretical formulation can effectively predict probability of full path coverage. Figure 
and 
. We have run the simulation based on three uniform distributions, 
, and compared with the theoretical results. For theoretical calculations, we have computed the average arc length for the case of random sensing range using (A.22) in the appendix, and then substituted the resulting 
into the approximate formula (14). From Figure 
, versus the number of sensor nodes for fixed sensing range.
, versus the number of sensor nodes for random sensing range.
when sensing range is fixed and is equal to 
. The solid line is the result of Poisson assumption in [
when 
.
, after deploying 
sensors in the ring is given by (5). In Figure 
has been calculated versus 
for three values of fixed sensing range, 
, using simulation as well as the analysis. The expected number of gaps for variable sensing ranges is shown in Figure 
found via search by 
. Table 
calculated for probability of full coverage being equal to 
. The results found by inequality (6) and simulation are shown for comparison. For probability of full coverage closer to one (the region of interest in practice), 
gets even closer to the value of 
satisfying the desired reliability found via simulation. For example, for probability of full coverage equal to 
, we found 
for various values of 
It can be inferred that (6) provides a tight upper bound on the number of nodes needed for full path coverage.
.
. The probability of this kind of coverage, 
, was given by (9). We use simulation to find 
for values of 
equal to 
, when 
. Again, comparing simulation results with theoretical ones in Figure 
.
when 
.
, is another important metric for path coverage in a WSN. Indeed, 
is a random variable whose cdf is approximated in Section 3.2. Figure 
, for 
and 
. As it can be seen, our path coverage analysis is more accurate for larger values of 
.
.
, is derived in Section 3.2. Figure 
versus 
, when 
.
.
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.
, 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.
. As mentioned previously, the circumference of 
is 1, hence, the radius of 
is 
. It is also assumed that the ring width is 
and 
, where 
is the sensing radius of the sensors. Notice that 
in Figure 
, is as follows:
to derive 
.In Figure 
is denoted by 
. By forming a triangle whose vertices are the center of 
, sensor location, and one of the points where the sensing circle of the sensor meets 
, one can write
with 
in (A.2) results in
and using the relation between 
and 
in (A.5) and (A.6), we will derive 
. To this end, one can state
in (A.8) using (A.1), we obtain
when 
and 
when 
. We use (A.9) for our exact analysis in order to characterize the path coverage features of the network. Notice that when 
is small, 
can be approximated as follows:
for our approximate analysis. Recall that for an arbitrary random variable 
distributed over 
,
is the mean value of 
and 
is the cdf of 
. Using (A.12), 
can be found as follows:
.Now assume that both sensing range and sensor location are random and we like to find 
. Sensing range of the sensors, 
, varies over 
with probability density function (pdf) 
. Also, 
such that 
, because sensors located farther than 
from the path do not contribute in the path coverage. It is noteworthy that 
where
, we partition the problem to two separate cases. In the first case, sensing area of the sensor does not intersect with 
, that is, 
. This happens when 
or 
. If 
, this never happens and sensing area of the sensor always intersects with 
and consequently 
. If 
, we have
and 
. Notice that in the case where sensors sensing range is independent from their location, 
, where 
is the pdf of 
over 
. To evaluate 
and 
, we simply have to integrate from 
over the area where 
or 
. It can be shown that
states that the pdf of 
, has a Dirac delta function at 
.When sensing area of a sensor intersects with path, 
. To find 
in this case, we first find 
. For this purpose, we apply Jacobian transformation to derive 
, the joint distribution of 
and 
, from 
. Using (A.6) and Jacobian transformation, one can show that
is found by integration over 
. To integrate over 
, the region of integration has to be determined carefully. For any arbitrary value of 
, there exist an infinite number of pairs 
satisfying (A.6); however, to guarantee an intersection between the sensing range of the sensor and 
, 
should fall within 
where
and 
are the desired integral bounds. Thus,
, used in our approximate analysis, is also derived as follows: