## 1 Introduction
After that, the same authors pointed out in [19] that the \(\kappa\)-\(\mu\)/IGA and \(\eta\)-\(\mu\)/IGA models not only provide a good fit for modeling the composite fading channels but also can be extended to the conventional and emerging wireless communication systems, such as cellular, wearable, and vehicular wireless applications. However, they only derived the statistical characteristics of the \(\eta\)-\(\mu\)/IGA and \(\kappa\)-\(\mu\)/IGA models, which include the PDF, the cumulative distribution function (CDF), the moment generating function (MGF), the higher-order moments, and the amount of fading. In [20], the effective capacity was analyzed over \(\eta\)-\(\mu\)/IGA and \(\kappa\)-\(\mu\)/IGA composite fading channels. In [21], the average symbol error probability (ASEP) and the channel capacity with adaptive schemes were obtained over \(\eta\)-\(\mu\)/IGA and \(\kappa\)-\(\mu\)/IGA composite fading channels. Moreover, some performance analysis was investigated over the inverse gamma pure shadowed fading channels in [22; 23]. As a counterpart of \(\eta\)-\(\mu\)/IGA and \(\kappa\)-\(\mu\)/IGA, the \(\alpha\)-\(\mu\)/IGA composite model has also been involved in [24; 25; 26], where [24] obtained the performance analysis of the single-branch system with the aid of Fox'H functions, [25] developed the energy detection of single-branch and maximal-ratio combining (MRC) systems over independent and identical distributed (i.i.d.) conditions, and [26] provided an approximated performance analysis of MRC and equal gain combining (EGC) by using a mixed IGA distribution under i.i.d. conditions. In recent work, the authors in [27] presented the \(\alpha\)-\(\eta\)-\(\mu\)/IGA and \(\alpha\)-\(\kappa\)-\(\mu\)/IGA channel models and evaluated the performance of multi-hop links. On the other hand, the Fisher-Snedecor \(\mathcal{F}\) fading channel model in [28] can be considered as an alternative version of the IGA-based composite model based upon the fact that the square root of an IGA random variable (RV) follows an inverse Nakagami-m distribution. So far, this model has obtained extensive interest and a great number of works have been done in [29; 30; 31] and references therein. By considering the nonlinearity of the composite envelope of fading signals, the \(\alpha\)-\(\mathcal{F}\), the \(\alpha\)-\(\eta\)-\(\mathcal{F}\), the \(\alpha\)-\(\kappa\)-\(\mathcal{F}\), and \(\alpha\)-\(\eta\)-\(\kappa\)-\(\mathcal{F}\) composite fading models were presented in [32; 33; 34], respectively. When \(\alpha=2\), these three models can reduce to the Fisher-Snedecor \(\mathcal{F}\), the \(\eta\)-\(\mu\)/IGA, and \(\kappa\)-\(\mu\)/IGA, respectively. Although the \(\alpha\)-\(\eta\)-\(\mu\)/IGA, \(\alpha\)-\(\kappa\)-\(\mu\)/IGA, the \(\alpha\)-\(\eta\)-\(\mathcal{F}\), the \(\alpha\)-\(\kappa\)-\(\mathcal{F}\), and \(\alpha\)-\(\eta\)-\(\kappa\)-\(\mathcal{F}\) are versatile, the mathematical forms of their PDFs are still intractable, which renders them rather difficult to analyze the performance of wireless communication systems exactly. Based on the above observation, most of the works involving the IGA shadowing model focused on the Fisher-Snedecor \(\mathcal{F}\), the \(\eta\)-\(\mu\)/IGA, and \(\kappa\)-\(\mu\)/IGA fading channels. Although the \(\alpha\)-\(\mu\)/IGA model has been considered in [24; 25; 26], to the best of the authors' knowledge, a comprehensive investigation of wireless communication systems over \(\alpha\)-\(\mu\)/IGA composite fading channels has not yet been involved in the independent and non-identical distributed (i.n.i.d.) scenarios and/or the correlated scenarios. To this effect, we will further extend our work in [24] to provide a thorough investigation and some significant insights into the \(\alpha\)-\(\mu\)/IGA and its application in wireless communication systems. In addition, this model can also be potentially applied to describe the distribution of population densities and urban activities investigated in [35].