1. Introduction
A wireless sensor network (WSN) has various applications in health-care, smart home, security, environmental exploration, and the military [
1,
2]. A sensor node (or simply node) is the basic component of a WSN. A WSN usually consists of numerous nodes deployed in a region of interest (ROI). Two nodes can communicate with each other if each is within the transmission range of the other, in which case we say that there is a link between them or that they are neighbors. Each node is able to collect data and process information and communicate with neighboring nodes.
Among various issues in WSNs, the coverage problem and the connectivity problem have been regarded as crucial foundations because many applications rely on them. Surveys for the coverage problem can be found in [
3,
4]. A good sensor deployment strategy should consider both coverage and connectivity. Sensor deployment not only determines the cost of constructing the network but also affects how well the given ROI will be monitored. This paper assumes that each node’s sensing region is of a disk shape (see
Figure 1a) and all nodes have the same sensing range
and communication range
.
Let
u be a location in the ROI,
be a node in the WSN, and
be the Euclidean distance between
u and
. Most of the past researches use the binary sensing model [
5,
6,
7], where nodes are assumed to be accurate in detecting targets within their sensing ranges. More precisely, under the binary sensing model (see
Figure 1b for an illustration), the detection probability
of
u by
is defined as
In physical scenarios, the distance decay effect cannot be ignored. That is, the detection probability of a node decays as the distance between the target and the node increases. Based on this, the coverage problem under a more realistic model, called the probabilistic sensing model, has been investigated; see [
7,
8,
9,
10,
11]. A survey for the coverage problem with uncertain properties can be found in [
12]. Do notice that unlike the binary sensing model, which has a unified definition of
, there is no unified definition of
under the probabilistic sensing model; see
Section 2 for details. We now give the definition of
used in [
7] and in this paper. Under the probabilistic sensing model (see
Figure 1c for an illustration), the detection probability
of
u by
is defined as
where
is a sensor-dependent parameter.
Many real-world applications impose the requirement of multilevel
coverage. For example,
for military or surveillance applications,
for positioning protocols using triangulation [
13], conducting data fusion [
14], and minimizing the impact of sensor failure [
15]. Sensor deployment with multilevel (
k) coverage have been discussed in [
7,
15,
16,
17].
Under the binary sensing model, an ROI is said to be
k-
covered if every location in the ROI can be detected by at least
k nodes (i.e., every location in the ROI is within
k nodes’ sensing regions). Unfortunately, to the best of our knowledge, there is no unified definition of
k-coverage under the probabilistic sensing model. We now are ready to elaborate this issue. Let
be the set of sensor nodes deployed in the ROI. In [
7], a location
u in the ROI is considered as
k-covered if the probability that there are at least
k nodes that can detect
u is not smaller than a predefined threshold
, where
. More precisely, in [
7], a location
u in the ROI is said to be
k-covered if
For clarity, we call this definition of
k-coverage the
k-threshold coverage. In [
18], a location
u in the ROI is said to be
k-covered if
For clarity, we call this definition of
k-coverage the
k-expectation coverage. In both [
7] and [
18], the ROI is said to be
k-covered if every location
u inside the ROI is
k-covered. The
used in [
18] is defined as (
4), which is a generalization of (
1) and is introduced in
Section 2. By (
3), the event of
u detected by
is independent of the event of
u detected by
for
. Reference [
18] proves that the ROI is
k-covered by
if, among all locations inside the ROI, the minimum expected number of nodes is
k (there is a typo and
k should be
at leastk k).
Three main considerations in WSNs’ coverage are: maximizing coverage quality, maximizing network lifetime, and minimizing the number of deployed nodes. The coverage quality and lifetime are two conflict factors with respect to energy consumption. Reference [
12] points out that the
k-coverage requirement makes the problem more sophisticated because more nodes are needed; deploying less nodes by decreasing the overlap region and prolonging the network lifetime become complicated. The
k-expectation coverage [
18] has the drawback that the user cannot specify his/her preference for the threshold
, meaning that the user cannot specify their desired coverage quality. Moreover, it is possible that the entire ROI is regarded as
k-covered but some locations inside it are detected by a lot of nodes each with a small detection probability. The
k-threshold coverage [
7] has the drawback that it tends to use too many sensor nodes; we give the calculation details in
Section 2. In short, Wang and Tseng [
7] first calculate
(notice that
is denoted as
in [
7]) and then replace the original
with
in the deployment.
The objective of this paper is to propose a sensor deployment scheme to use less number of nodes while ensuring the following two important coverage qualities: (i) the resultant WSN is connected and (ii) the detection probability satisfies a predefined threshold
, where
. Although
k-threshold coverage [
7] achieves the same objective, we find that the nature of
k-threshold coverage makes
tend to become
much smaller than the original
. For example, suppose
and m denotes meters. Then:
If and m, then for = 0.7, 0.8, 0.9 are 2.377 m, 1.487 m, and 0.702 m, respectively.
If and m, then for = 0.7, 0.8, 0.9 are 1.486 m, 0.929 m, and 0.439 m, respectively.
These very small ’s cause a large number of nodes to be deployed. Since the number of nodes is very large, the overall detection probability might be underestimated.
In this paper, we try to propose a “reasonable” solution to the k-coverage problem under the probabilistic sensing model. The main contributions of this paper are as follows.
We propose a “reasonable” definition of k-coverage under the probabilistic sensing model, called the k-layer coverage, and propose a scheme to achieve k-layer coverage.
We propose a novel “zone 1 and zone 1–2” strategy to fulfill k-layer coverage scheme and ensure good coverage quality. We propose an efficient algorithm to calculate the radius of zone 1, which takes at most 18 iterations when error tolerance .
Experimental results shows that our k-layer coverage scheme indeed uses less sensor nodes, thereby demonstrating the effectiveness of our scheme.
Our
k-layer coverage scheme partitions the nodes into
k subsets, each forming one layer of coverage, and ensures that for every location
u inside the ROI, the detection probability for
u by nodes in “each layer” is not smaller than
. In particular, our
k-layer coverage scheme ensures that every location
u inside the ROI is within zone 1 of at least
k nodes and within zone 1–2 of at least another
nodes (zone 1 and zone 1–2 are defined in
Section 4). When coverage quality is considered, our
k-layer coverage scheme provides a good solution. Different from the
k-threshold coverage [
7], which replaces the original
with
, our
k-layer coverage replaces the original
with
. We prove that as long as
, we have
, meaning that our
k-layer coverage will use less nodes. When the number of nodes is considered, our
k-layer coverage scheme also provides a good solution.
The rest of this paper is organized as follow.
Section 2 introduces the related works.
Section 3 gives preliminaries, assumptions, and objectives.
Section 4 gives the basics of the
k-layer coverage.
Section 5 gives our
k-layer coverage scheme.
Section 6 illustrates experimental results. Concluding remarks are given in the final section.
2. Related Works
Recall that a good sensor deployment should consider both coverage and connectivity. Assuming that the ROI is a convex set, Zhang and Hou [
19] investigate the relationship between coverage and connectivity and prove that
is both necessary and sufficient to ensure that coverage implies connectivity. With such a proof, one can then focus only on the coverage problem. As long as
, nodes can be deployed according to the regular triangular lattice pattern (triangular pattern for short; see
Figure 2) so that both coverage and connectivity can be ensured [
7,
20]. Using the triangular pattern, neighboring nodes will be regularly separated by a distance of
. A deployment using the triangular pattern is sometimes called an
optimal deployment since it is asymptotically optimal in terms of the number of nodes needed to achieve full coverage of the ROI.
In some applications,
or
may not hold. Wang and Tseng [
7] therefore consider an arbitrary relationship between
and
, thus relaxing the limitations of existing results. In [
7], for the the binary sensing model, two solutions to achieve
k-coverage are proposed: the naive duplicate placement scheme (duplicate scheme for short) and the interpolating placement scheme (interpolating scheme for short). The idea of the duplicate scheme is to use a good sensor placement method to ensure 1-coverage and connectivity and then duplicate
k nodes on each designated location. However, since the duplicate scheme may result in some regions in the ROI having a much higher coverage levels than
k, the interpolation scheme is therefore being proposed to “reuse” these regions to generate a multilevel coverage. When
or
or (
and
), it is found that the interpolation scheme will not save nodes compared to the duplicate scheme, thereby adopting the duplicate scheme. For clarity, we summarize the schemes used in [
7] in
Table 1. Notice that the interpolation schemes used in the (
)-case and the (
)-case are different.
Wang and Tseng [
7] adapt the schemes of the binary sensing model
to the probabilistic sensing model. Set (★) = (
and
) for convenience. We now use the (★)-case as an illustration to show how [
7] performs the adaption. According to
Table 1 (shown in
Section 2), under the binary sensing model, [
7] will use the duplicate scheme and triangular pattern in the (★)-case. Under the probabilistic sensing model, in the (★)-case, [
7] first calculates
(notice that
is denoted as
in [
7]). Then, [
7] replaces the original
with
in the deployment to ensure that every location inside the ROI is
k-covered under the probabilistic sensing model. Since the duplicate scheme places
k nodes on each designated location, [
7] calculates
by
and
where
u is a location in the ROI having the minimum
k-covered probability and
u is detected by a set
of
k nodes placed at a designated location with distance
to
u.
Let
u,
,
, and
be defined as in
Section 1. In [
11,
18,
21,
22], four parameters
are used to specify a probabilistic sensing model and
is defined as
where
and
are sensor-dependent parameters (see
Figure 3). That is, if
, then a target at
u will definitely be detected by
; if
, then the detection probability will be too small and will be totally ignored. If
, then the behavior of the detection probability obeys the function
. By taking
and
, (
4) coincides with (
1).
Notice that besides the binary sensing model and the probabilistic sensing model, some researchers consider the evidence-based sensor coverage model and use the theory of belief functions to solve the coverage problem; see [
23,
24,
25,
26]. Notice that [
23,
24,
25,
26] consider 1-coverage and their ROI is assumed as a two- or three-dimensional grid of points. In this paper, we consider
k-coverage and our ROI contains every location inside it. Some other references related to
k-coverage can also be found in [
27,
28,
29,
30,
31]. Before ending this section, for clarity, we summarize our most related works in
Table 2.