# Closed-Form Performance Analysis of the Inverse Power Lomax Fading Channel Model
## 3 IPL Fading Channel Model Application
### Hyper-Rayleigh Analysis  
Hyper-Rayleigh fading is identified in terms of the amount of fading, outage probability (OP), and ergodic capacity (EC) [54]. If one of these characteristics is worse than in the Rayleigh case, the channel is said to be in the weak hyper-Rayleigh regime (HRR); if two, in the strong HRR; and if all factors are below the corresponding levels for the Rayleigh case, in the full HRR. It is worth mentioning that these regimes are identified asymptotically. It is well known that for the classical Rayleigh channel, the assumed metrics are equal to  
\[ \begin{cases}\mathrm{AoF}^{Ray}=1,\\ \mathrm{P}_{\mathrm{out}}^{Ray}|_{\bar{\gamma}\to\infty}\backsimeq\frac{\gamma _{th}}{\bar{\gamma}},\\ \overline{\mathbb{C}}^{Ray}\left.\right|_{\bar{\gamma}\to\infty}\backsimeq\log_ {2}\bar{\gamma}+\frac{G_{e}}{\ln 2}.\end{cases}  \tag{45}\]  
Thus, one can state the following:  
**Lemma 2**. : _The IPL channel model exhibits hyper-Rayleigh behavior in the following ways:_  
* _In the OP sense if_ \(\alpha\beta<1\) _or_ \(\alpha\beta=1\) _and_ \(\alpha<\frac{1}{2}\)_;_
* _In the AoF sense if_ \(\alpha\beta=1\) _and_ \(\alpha<0.316\)_; otherwise, see Section_ 4.4_;_
* _In the EC sense if_ \(\alpha\beta=1\) _and_ \(\alpha<0.423\)_; otherwise, see Section_ 4.4_._  
_Thus, in the case_ \(\alpha\beta=1\)_, the IPL model is in the full HRR if_ \(\alpha<0.316\)_, in the strong HRR if_ \(0.316<\alpha<0.423\)_, in the weak HRR if_ \(0.423<\alpha<\frac{1}{2}\)_, and does not experience the HRR if_ \(\alpha>\frac{1}{2}\)_._  
Proof. : From the definition given in [54], it is clear that the IPL channel is said to be in the hyper-Rayleigh regime (in the AoF sense, OP sense, or EC sense) if the following corresponding conditions are satisfied:  
\[\mathrm{AoF}^{Ray}<\mathrm{AoF}^{IPL}, \tag{46a}\] \[\mathrm{P}_{\mathrm{out}}^{Ray}(\gamma_{th})|_{\bar{\gamma}\to \infty}<\mathrm{P}_{\mathrm{out}}^{IPL}(\gamma_{th})|_{\bar{\gamma}\to\infty},\] (46b) \[\overline{\mathbb{C}}^{Ray}\left.\right|_{\bar{\gamma}\to\infty}> \overline{\mathbb{C}}^{IPL}\left.\right|_{\bar{\gamma}\to\infty}. \tag{46c}\]  
This is equivalent to (recollecting the results from Theorems 2, 3, and 5):\[\Gamma(\alpha)\frac{\Gamma\left(1-\frac{2}{\beta}\right)\Gamma \left(\alpha+\frac{2}{\beta}\right)}{\left(\Gamma\left(1-\frac{1}{\beta}\right) \Gamma\left(\alpha+\frac{1}{\beta}\right)\right)^{2}}>2, \tag{47a}\] \[\left(\Omega_{\alpha,\beta}\right)^{\alpha}\left(\frac{\gamma_{ th}}{\gamma}\right)^{\alpha\beta-1}<1,\] (47b) \[\frac{1}{\beta\ln 2}\left(G_{e}(1-\beta)+\psi^{(0)}(\alpha)-\ln \Omega_{\alpha,\beta}\right)<0. \tag{47c}\]  
Starting from (47b) and noting that if \(\alpha\beta\geq 1\) for a sufficiently high SNR (i.e., the HRRs are defined asymptotically for \(\lim_{\gamma_{t}\to\infty}(\cdot)\)), the inequality holds true, i.e., the channel is in the HRR (in the OP sense). Thus, if \(\alpha\beta<1\), the IPL model experiences at least the weak HRR. If \(\alpha\beta=1\), the HRR requires \(\left(\Omega_{\alpha,\beta}\right)^{\alpha}\) (which is called \"power offset\" in [54]) to be less than one. Assuming the definition of \(\Omega_{\alpha,\beta}\) (see (6)) and substituting \(\beta=\frac{1}{\alpha}\) after some simplifications yields  
\[\left(\Omega_{\alpha,\frac{1}{\alpha}}\right)^{\alpha}=\frac{\Gamma(1-\alpha) \Gamma(2\alpha)}{\Gamma(\alpha)}<1. \tag{48}\]  
Noting that \(\beta>1\Rightarrow\alpha<1\) (since the definition does not require the second-order moment), the range of \(\alpha\) satisfying (48) is \(\alpha\in\left(0,\frac{1}{\alpha}\right)\).