On the statistics of the ratio of nonconstrained arbitrary α-μ random variables: A general framework and applications

In this paper, we derive closed-form exact expressions for the main statistics of the ratio of two squared α-μ random variables, which are of interest in many scenarios for future wireless networks where generalized distributions are more suitable to fit with field data. Importantly, different from previous proposals, our expressions are general in the sense that are valid for nonconstrained arbitrary values of the parameters of the α-μ distribution. Thus, the probability density function, cumulative distribution function, moment generating function, and higher-order moments are given in terms of both (i) the Fox H-function for which we provide a portable and efficient Wolfram Mathematica code and (ii) easily computable series expansions. Our expressions can be used straightforwardly in the performance analysis of a number of wireless communication systems, including either interference-limited scenarios, spectrum sharing, full-duplex, or physical-layer security networks, for which we present the application of the proposed framework. Moreover, closed-form expressions for some classical distributions, derived as special cases from the α-μ distribution, are provided as byproducts. The validity of the proposed expressions is confirmed via Monte Carlo simulations.