Evaluation of the Probability of K-Hop Connection in Homogeneous Wireless Sensor Networks

Given a wireless sensor network (WSN) whose sensors are randomly and independently distributed in a bounded area following a homogeneous Poisson process with density rho and each sensor has a uniform transmission radius of r₀, we investigate the probability that two random sensors separated by a known distance x are fc-hop neighbors for some positive integer k in this paper. We give a closed-form equation for computing this probability for k = 2; and also give a recursive equation for evaluating this probability for k > 2 by using some approximations. The accuracy of the approximate analytical solution is validated by simulations. Furthermore, we present an empirical method to correct the discrepancies between the analytical results and the simulation results caused by the approximation. The result of this paper can be useful in a number of sensor network problems, e.g., estimating the transmission delay between two sensors and energy consumed in the transmission, and WSN routing problems.