On the Statistics of the Ratio of Non-Constrained Arbitrary {\alpha}-{\mu} Random Variables: a General Framework and Applications

TL;DR

This paper derives comprehensive, exact statistical expressions for the ratio of squared alpha-mu random variables, applicable to various wireless network scenarios, using advanced mathematical functions and series expansions.

Contribution

It introduces a general framework for the ratio of alpha-mu variables with arbitrary parameters, including closed-form expressions and computational tools, extending previous limited models.

Findings

Validated expressions through Monte Carlo simulations.

Applicable to interference, spectrum sharing, and security in wireless networks.

Provides special case distributions as byproducts.

Abstract

In this paper, we derive closed-form exact expressions for the main statistics of the ratio of squared alpha-mu 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 non constrained arbitrary values of the parameters of the alpha-mu distribution. Thus, the probability density function, cumulative distribution function, moment generating function, and higher order moments are given in terms of both (i) theFox 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 alpha-mu distribution, are provided as byproducts. The validity of the proposed expressions is confirmed via Monte Carlo simulations.