Statistical Properties of Transmissions Subject to Rayleigh Fading and Ornstein-Uhlenbeck Mobility

In this paper, we derive closed-form expressions for significant statistical properties of the link signal-to-noise ratio (SNR) and the separation distance in mobile ad hoc networks subject to Ornstein-Uhlenbeck (OU) mobility and Rayleigh fading. In these systems, the SNR is a critical parameter as it directly influences link performance. In the absence of signal fading, the distribution of the link SNR depends exclusively on the squared distance between nodes, which is governed by the mobility model. In our analysis, nodes move randomly according to an Ornstein-Uhlenbeck process, using one tuning parameter to control the temporal dependency in the mobility pattern. We derive a complete statistical description of the squared distance and show that it forms a stationary Markov process. Then, we compute closed-form expressions for the probability density function (pdf), the cumulative distribution function (cdf), the bivariate pdf, and the bivariate cdf of the link SNR. Next, we introduce small-scale fading, modeled by a Rayleigh random variable, and evaluate the pdf of the link SNR for rational path loss exponents. The validity of our theoretical analysis is verified by extensive simulation studies. The results presented in this work can be used to quantify link uncertainty and evaluate stability in mobile ad hoc wireless systems.