Barrier Coverage: Foundations and Design

Abstract

The coverage of a wireless sensor network (WSN) characterizes the quality of surveillance that the WSN can provide. A deep understanding of the coverage is of great importance for the deployment, design, and planning of wireless sensor networks. Barrier coverage measures the capability of a wireless sensor network to detect intruders that attempt to cross the deployed region. The goal is to prevent intruders from sneaking through the network undetected. It is a critical issue for many military and homeland security applications. In this chapter we provide a comprehensive survey on the barrier coverage of wireless sensor networks. The main topics include the critical conditions and construction of barrier coverage in a 2-dimensional WSN, the barrier coverage under a line-based sensor deployment scheme, the effect of sensor mobility on barrier coverage, and the barrier coverage for a 3-dimensional underwater sensor network. For each topic we discuss the challenges, fundamental limits, and the solution for the construction of sensor barriers.

Appendix

1.1 Proof of Theorem 2

Proof

Consider the relative positions of nodes $s_i$ and $s_{i+1}$. Define $z^x = s^x_{i+1}-s^x_i$ and $z^y = s^y_{i+1}-s^y_i$. Then $z^x = \zeta + \delta ^x_{i+1}-\delta ^x_i$ and $z^y = \delta ^y_{i+1} - \delta ^y_i$. Since the $\delta ^s$ are Normally distributed with variance $\sigma ^2$, then $\delta ^x_{i+1}-\delta ^x_i$ and $\delta ^y_{i+1} - \delta ^y_i$ are both Normal random variable with variance $2 \sigma ^2$, i.e., $z^x \sim N(\zeta ,2\sigma ^2)$ and $z^y \sim N(0,2\sigma ^2)$.

The distance between $s_i$ and $s_{i+1}$ is equal to:

$$\begin{aligned} d_i \mathop {=}\limits ^{\Delta } |s_{i+1}-s_i| = \sqrt{(z^x)^2 + (z^y)^2} \end{aligned}$$

(A.1)

This implies that $d_i$ follows a Ricean distribution. Since the distribution is identical for all $i$, we drop the index and denote the distance between two consecutive nodes as $d$.

In particular, the probability that $d < \rho$ is given by (see for instance [57], Chap. 2):

$$\begin{aligned} P(d < \rho ) = 1 - Q_1(\frac{\zeta }{\sqrt{2}\sigma },\frac{\rho }{\sqrt{2}\sigma }) \end{aligned}$$

(A.2)

where $Q_1$ is the Marcum’s Q-function of the first order, defined by:

$$\begin{aligned} Q_1(\frac{\zeta }{\sqrt{2}\sigma },\frac{\rho }{\sqrt{2}\sigma }) = e^{-(\zeta ^2+\rho ^2)/4\sigma ^2} \sum _{k=0}^\infty \left( \frac{\zeta }{\rho }\right) ^k I_k\left( \frac{\zeta \rho }{2 \sigma ^2}\right) \end{aligned}$$

(A.3)

and $I_k$ is the $k$th order modified Bessel function of the first kind.

Thus, two sensors $s_i$ and $s_{i+1}$ provide barrier coverage with probability $P(d < \rho )$. If each pair of sensors $s_i$ and $s_{i+1}$ is within $\rho$ of each other and within $\rho$ of the boundary, then the sensor deployment provides barrier coverage over the all width of the area. For all $1 \le i \le n-1$, denote by $W_i$ the event that $d_i < \rho$. Denote by $W^b_0$ the event that $s_1$ is within distance $\rho$ of the boundary $x=0$, and $W^b_n$ that $s_n$ is within distance $\rho$ of the boundary $x=l$. Since this is not the only configuration that provides barrier coverage, we have

$$\begin{aligned} P(\text {Barrier Coverage}) \ge P\left( W^b_0 \bigcap W^b_n \bigcap (\bigcap _{i=1}^{n-1} W_i)\right) . \end{aligned}$$

(A.4)

Assumption (1) allows us to consider $W^b_0, W^b_n \text{ and } W_i$, $1 \le i \le n-1$, as independent events and approximate $P(W^b_0 \bigcap W^b_n \bigcap (\bigcap \nolimits _{i=1}^{n-1} W_i))$ with $P(W_0^b)P(W_n^b)$ $(P(d < \rho ))^{(n-1)}$. Indeed, if assumption (1) was violated, and $\rho$ was almost equal to $\zeta$, then a perturbation which brings node $s_i$ close to $s_{i-1}$ would also create a gap between $s_i$ and $s_{i+1}$. $W_i$ happening thus implies that $W_{i+1}$ would not happen, and that both events are conditioned on each other, not independent. However, by choosing the right parameter $\kappa$, the approximation by independent events is appropriate, as we confirm in the evaluation section.

We can also easily verify that $P(W_0^b) > P(d<\rho )$ and symmetrically, $P(W_n^b) > P(d<\rho )$, so that $P(\text{ Barrier } \text{ Coverage }) \ge P(d<\rho )^{n+1}$.

Assumption (2) ensures that the gap between $P(Barrier Coverage)$ and $P(\bigcap _{i=0}^n W_i)$ stays limited, and that the most likely configuration to provide coverage is indeed by having each $s_i$ and $s_{i+1}$ within $\rho$ of each other. Other configurations are possible, and a gap between $s_i$ and $s_{i+1}$ could be filled by having a third sensor out of position in the sequential ordering along the x-axis. However, assumption (2) ensures that such other configurations have a low likelihood. Simulation will show that, under assumption (2) the lower bound is actually tight.

Note that we do not put an explicit dependency of $\zeta$ on $n$, but as $n \rightarrow \infty$, the probability of barrier coverage goes to 1 as $\zeta \rightarrow 0$, all other parameters being constant. $\square$