Ratio of Products of Mixture Gamma Variates with Applications to Wireless Communications

TL;DR

This paper derives exact statistical expressions for the product and ratio of mixture Gamma variates and applies these to analyze various wireless communication performance metrics over complex fading channels.

Contribution

It provides novel closed-form expressions for the distribution of products and ratios of mixture Gamma variables and applies them to diverse wireless communication performance analyses.

Findings

Derived exact PDFs, CDFs, and MGFs using Meijer's G-function.

Analyzed outage probability, error rates, and secrecy metrics over new fading models.

Validated results with Monte Carlo simulations.

Abstract

In this paper, the statistical properties of the product of independent and non-identically distributed mixture Gamma (MG) random variables (RVs) are provided first. Specifically, simple exact closed-form expressions for the probability density function (PDF), cumulative distribution function (CDF), and moment generating function (MGF) are derived in terms of univariate Meijer's $G$-function. The statistical characterisations of the distribution of the ratio of products of MG variates are then derived. These statistics are used to analyse the outage probability (OP), the average error probability for different modulation schemes, the effective rate (ER) of communications systems and the average area under the receiver operating characteristics (AUC) curve of energy detection over cascaded fading channels. Additionally, the lower bound of secure outage probability (SOP$^L$) and probability of non-zero secrecy capacity (PNSC) of the physical layer and the OP of the multihop communications systems with decode-and-forward (DF) relaying protocol and co-channel interference (CCI) are studied by utilising the statistics of the ratio of the products. The derived performance metrics are applied for the Beaulieu-Xie and $\alpha-\lambda-\eta-\mu$ shadowed fading channels that have not been yet investigated in the literature. Accordingly, the equivalent parameters of a MG distribution for the aforementioned channels are given. A comparison between the numerical results and the Monte Carlo simulations is presented to verify the validation of our analysis.