Ratio of Products of $\alpha$-$\mu$ Variates

Exact, closed-form expressions for the probability density function and the cumulative distribution function for the ratio of the products of an arbitrary number of independent and non-identically distributed α-μ variates are derived. The only restriction placed on the participating distributions is that there must be a rational relationship between their α parameters, which, in practical terms, represents no impairment. The expressions are given in terms of the Meijer G-function and, alternatively, in terms of a finite sum of hypergeometric functions. These results can be used to investigate the performance of wireless communication systems in a variety of realistic propagation environments in which the numerator and denominator products might be used to represent the signal and the interference. In addition, the results given comprise those for the ratio of the products of arbitrary combinations of other useful distributions such as one-sided Gaussian, negative exponential, Rayleigh, Weibull, Gamma, and Nakagami-m. Simulation is used to confirm the results. An application example is given in order to illustrate the use of the formulations.