# \(\alpha\)-Fluctuating Nakagami-\(m\) Fading Model for Wireless Communications
## 5 Simulation and Results' Analysis
### \(\alpha\)-Fluctuating Nakagami-m Model Performance  
The current section presents the results of the numerical analysis of system-level performance metrics (i.e., outage probability, average error rate, and ergodic capacity) derived in closed-form in Section 4. One starts the system performance assessment in terms of the Amount of Fading plotted in Figure 5 across various channel parameters. As was mentioned, AoF serves as a key metric to quantify the severity of fading in wireless channels. The figure combines two plots. The right-hand vertical axis encompasses \(\mathrm{AoF}(m_{s})\) (the results are plotted with dashed lines) and the left-hand \(\mathrm{AoF}(m)\) (the results are plotted with solid lines). Qualitatively, the left plot shows that as \(m\) increases, indicating reduced multipath fading, the AoF systematically decreases for all values of \(\alpha\). This trend confirms that higher \(m\) contributes to more stable wireless channels. Quantitatively, when \(\alpha=3\), the AoF decreases from approximately 1 at \(m=0.5\) to less than 0.2 at \(m=10\), highlighting an almost order of  
Figure 4: MGF comparison of numerical evaluation and numerical simulation across various channel parameters. magnitude reduction in fading severity. A similar pattern is observed for other values of \(\alpha\). In particular, the decline in AoF (as \(m\) increases) is roughly the same for different values of \(\alpha\), indicating that although the nonlinearity of the model helps suppress the effects of fading (in absolute terms), the rate of improvements (in relative terms) depends weakly on \(\alpha\). The right-hand plot analyzes AoF as a function of \(m_{s}\), where increasing \(m_{s}\) corresponds to weaker shadowing. It shows that AoF significantly drops as \(m_{s}\) increases from roughly 2 to 10, especially for smaller \(\alpha\). For example, when \(\alpha=3\), AoF reduces from about 100 at \(m_{s}=2\) to roughly 1 at \(m_{s}=10\), demonstrating a substantial improvement in channel stability. Cross-comparison between the plots reveals that both \(m\) and \(m_{s}\) contribute to reducing AoF, but their impact is determined by the value of \(\alpha\). The effect of \(\alpha\) is more pronounced at lower \(m\) and \(m_{s}\), making it particularly useful in harsh fading conditions. Figure 6 illustrates the outage probability as a function of signal-to-noise ratio (SNR) for various values of the transformation parameter \(\alpha\), with fixed parameters \(m=1\), \(m_{s}=2.5\), and a threshold SNR of \(\gamma_{th}=0\) dB. The outage probability is a critical metric that quantifies the likelihood of the instantaneous SNR falling below a predefined threshold, directly impacting link reliability in wireless communication systems. The plot demonstrates that increasing \(\alpha\) leads to a significant reduction in outage probability, highlighting the role of \(\alpha\) in mitigating extreme fading effects. At \(\bar{\gamma}=25\) dB, the outage probability for \(\alpha=1\) is approximately \(10^{-1}\), whereas for \(\alpha=4\), it decreases to nearly \(10^{-4}\). This implies that higher \(\alpha\) values lead to an order-of-magnitude improvement in link reliability at moderate SNRs. The numerical integration results (circle markers) and Monte Carlo simulations (cross markers) closely match the analytically derived outage probability (solid lines), validating the theoretical formulation. The high-SNR asymptotic expressions (dashed lines) provide excellent approximations for small \(\alpha\) (i.e., \(\alpha<2\)) beyond \(\bar{\gamma}=20\) dB, and for \(\alpha>2\), beyond 10 dB, simplifying performance evaluation at practical operational SNR levels. Notably, for low \(\alpha\), the outage probability saturates at high SNRs, indicating that deep fading events remain dominant despite increasing transmission power. This effect is particularly pronounced in the hyper-Rayleigh regime, reinforcing the necessity of advanced signal processing techniques for mitigating fading-induced performance degradation.