Let us first discuss the case where the sensors have a fixed sensing range. Figure
13 shows the ring-shaped network containing
. 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
13 shows the distance of a sensor from the center of the ring. Since the sensors are uniformly distributed over the area, it can be easily shown that the cdf of
, is as follows:
We use
to derive
.In Figure
13, the intersection of the sensing area of an arbitrary sensor with
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
On the other hand, we have
Replacing
with
in (A.2) results in
Solving (A.4), we have
Equivalently,
Now having the cdf of
and using the relation between
and
in (A.5) and (A.6), we will derive
. To this end, one can state
Thus,
Replacing
in (A.8) using (A.1), we obtain
where
Moreover,
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:
In addition to the cdf of the arc length, we use the mean value of
for our approximate analysis. Recall that for an arbitrary random variable
distributed over
,
where
is the mean value of
and
is the cdf of
. Using (A.12),
can be found as follows:
Notice that when
.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
This can simply be justified using (A.6).To find
, 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
To evaluate two terms in the right side of the above equation, we use the joint distribution of
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
Notice 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
where
Having
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
In fact,
and
are the desired integral bounds. Thus,
Consequently,
The mean value of
, used in our approximate analysis, is also derived as follows: