# Closed-Form Performance Analysis of the Inverse Power Lomax Fading Channel Model
## 4 Simulation and Results' Analysis
### Model Verification and Positioning  
First, to validate the correctness of the model, the simulated instantaneous SNR samples generated for specific channel parameters were compared with the theoretical PDF for the same parameter values. The results are presented in Figure 1 for three sets of \(\alpha,\beta\): \(\alpha=0.1,\beta=1.1\); \(\alpha=1.5\), \(\beta=2.1\); and \(\alpha=5.5,\beta=7\).  
The plots show how the PDF varies with different channel conditions and demonstrate that a decrease in the channel parameters leads to a shift in the distribution mode toward the origin. This means that the smaller the values of \((\alpha,\beta)\), the heavier the fading (i.e., the more probable extremely small SNR values are). It must be highlighted that the Monte Carlo simulation data correspond closely with the theoretical PDF.  
Figure 8: Experimental results from [60] fitted with the IPL and fdRLoS [59] models.  
Figure 7: Experimental results from [58] fitted with the IPL and fdRLoS [59] models.  
To better understand the impact of channel parameters, the CDF was plotted for various sets of \((\alpha,\beta)\) (see Figure 2), including the following:  
* The increase in the shape parameter \(\beta=(2.1;3;5;15)\) for a fixed and small power-transformation parameter (i.e., \(\alpha=0.1\));
* The increase in the power-transformation parameter \(\alpha=(0.1;0.2;0.5;1;5)\) for a fixed-shape parameter (i.e., \(\beta=2.1\)). To provide a baseline for comparison, Figure 2 includes the Rayleigh CDF. It is clear that the proposed model can handle fading that is heavier (even much heavier) than Rayleigh (i.e., a small \(\alpha\) and up to a moderate \(\beta\)), lighter than Rayleigh (see the curves with larger values of \(\alpha\)), as well as cases very close to Rayleigh (see the curve for \(\alpha=0.5\) and \(\beta=2.1\)).  
As pointed out earlier, there are numerous classical models used in wireless communications to describe fading channels. It is interesting to see how the IPL model is positioned among them.  
Figure 3 illustrates the results (in terms of the CDF) for common fading channel models, such as Rayleigh, Rice (with \(k=3\) dB), Nakagami-\(m\) (with \(m=0.5\)), Lomax (with \(\alpha=1.1\)), Hoyt (with \(q=0.05\)), and \(\alpha-\mu\) (with \(\alpha=0.1\), \(\mu=5\)). The parameters for Nakagami, Hoyt, Lomax, and \(\alpha-\mu\) were chosen to correspond to heavy fading conditions. It can be seen that the presented IPL model with small parameter values demonstrates extremely deep fading (much heavier than the classical models). Combining the results depicted in Figures 2 and 3, one can state that by choosing a specific \(\alpha\) and \(\beta\), the IPL model can cover the fading regimes achieved with the classical models, demonstrating its high flexibility.  
Figure 4 presents a comparison between the MGF obtained through numerical evaluation and the derived closed-form expression for different channel parameters. The parameters include \(\alpha=1.1\), \(\beta=3.5\); \(\alpha=2,\beta=2\); and \(\alpha=1.1\), \(\beta=1.1\). The high-SNR asymptotic expressions are also depicted. The comparison validates the accuracy of the closed-form expression in capturing the behavior of the MGF across various channel conditions.  
It must be pointed out that it is quite cumbersome to convert Monte Carlo simulation samples to the sampled MGF. Therefore, a different approach for validating the analytics was chosen: it was tested against the direct numerical integration in (10a).