  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.
