# Closed-Form Performance Analysis of the Inverse Power Lomax Fading Channel Model
## 2 IPL Channel Statistical Description  
### PDF and CDF of the Instantaneous SNR  
This research considers that the instantaneous SNR \(\gamma\) follows the inverse power Lomax distribution with the PDF \(w_{\gamma}(\gamma)\) and CDF \(F_{\gamma}(\gamma)\) defined as follows [28]:  
\[w_{\gamma}(\gamma)=\frac{\alpha\beta}{\lambda}\gamma^{-(\beta+1)}\bigg{(}1+ \frac{\gamma^{-\beta}}{\lambda}\bigg{)}^{-(\alpha+1)}, \tag{1}\]  
\[F_{\gamma}(\gamma)=\bigg{(}1+\frac{\gamma^{-\beta}}{\lambda}\bigg{)}^{-\alpha}, \tag{2}\]  
which are valid for arbitrary positive-distribution parameters (i.e., the scale parameter \(\lambda\) and the shape parameters \(\alpha\) and \(\beta\)). Although such a model can be used for further elaborations, it is conventional to perform a reparametrization in terms of the average SNR \(\tilde{\gamma}\) as given by the following Lemma:  
**Lemma 1**. : _The random variable \(\gamma\) is assumed to follow the two-parametric inverse power Lomax model (i.e., \(\gamma\sim\mathcal{IPL}(\alpha,\beta)\)) with the average value \(\tilde{\gamma}\) and channel parameters \(\alpha\), \(\beta\) if its PDF and CDF admit the following representation:_  
\[ \begin{cases}w_{\gamma}(\gamma)=\frac{\alpha\beta}{\Omega_{\alpha,\beta}}\frac{ \tilde{\gamma}^{\beta}}{\gamma^{\beta+1}}\bigg{(}1+\frac{1}{\Omega_{\alpha, \beta}}\bigg{(}\frac{\tilde{\gamma}}{\gamma}\bigg{)}^{\beta}\bigg{)}^{-( \alpha+1)},\\ F_{\gamma}(\gamma)=\bigg{(}1+\frac{1}{\Omega_{\alpha,\beta}}\bigg{(}\frac{ \tilde{\gamma}}{\gamma}\bigg{)}^{\beta}\bigg{)}^{-\alpha}.\end{cases}  \tag{3}\]  
Proof of Lemma 1.: To perform this step, one needs the expression for the moments of the instantaneous SNR, i.e., \(\mathbb{E}\{\gamma^{r}\}\) (where \(r\) is the order of the moment and \(\mathbb{E}\{\cdot\}\) is the expectation operator). Starting from the definition and performing the change in variable \(\gamma^{-\beta}=t\), after some straightforward manipulations, yields  
\[\mathbb{E}\{\gamma^{r}\} =\frac{\alpha\beta}{\lambda}\int_{0}^{\infty}\gamma^{r-(\beta+1 )}\bigg{(}1+\frac{\gamma^{-\beta}}{\lambda}\bigg{)}^{-(\alpha+1)}\text{d} \gamma= \tag{4a}\] \[=\alpha\lambda^{\alpha}\int_{0}^{\infty}\frac{t^{-\frac{r}{ \beta}}}{(t+\lambda)^{\alpha+1}}\text{d}t=\] (4b) \[=\lambda^{-\frac{r}{\beta}}\frac{\Gamma\Big{(}1-\frac{r}{\beta} \Big{)}\Gamma\Big{(}\alpha+\frac{r}{\beta}\Big{)}}{\Gamma(\alpha)}, \tag{4c}\]  
where \(\Gamma(\cdot)\) is the Euler gamma function [46]. It must be pointed out that (4c) is valid for any positive \(\alpha\) and \(\lambda\) but for \(\beta>r\), which implies a new restriction on \(\beta\), when evaluating the moments of \(\gamma\). Calculating the average SNR, i.e., \(\tilde{\gamma}=\mathbb{E}\{\gamma\}\), and evaluating \(\lambda\) leads to  
\[\lambda=\Omega_{\alpha,\beta}\frac{1}{\tilde{\gamma}^{\beta}}, \tag{5}\]  
where \(\Omega_{\alpha,\beta}\) is introduced for notation clarity in the following way:  
\[\Omega_{\alpha,\beta}=\left(\frac{\Gamma\Big{(}1-\frac{1}{\beta}\Big{)}\Gamma \Big{(}\alpha+\frac{1}{\beta}\Big{)}}{\Gamma(\alpha)}\right)^{\beta}. \tag{6}\]  
This allows one to rewrite (1) and (2) as follows:  
\[w_{\gamma}(\gamma)=\frac{\alpha\beta}{\Omega_{\alpha,\beta}}\frac{\tilde{ \gamma}^{\beta}}{\gamma^{\beta+1}}\bigg{(}1+\frac{1}{\Omega_{\alpha,\beta}} \bigg{(}\frac{\tilde{\gamma}}{\gamma}\bigg{)}^{\beta}\bigg{)}^{-(\alpha+1)}, \tag{7}\]  
\[F_{\gamma}(\gamma)=\bigg{(}1+\frac{1}{\Omega_{\alpha,\beta}}\bigg{(}\frac{ \tilde{\gamma}}{\gamma}\bigg{)}^{\beta}\bigg{)}^{-\alpha}, \tag{8}\]  
which finalizes the proof of Lemma 1 and will be used further as a starting point for a performance analysis. For further analysis, one resorts to \(\alpha\) as a \"power-transformation parameter\" and \(\beta\) as a \"shape parameter\" of the distribution. It must be pointed out that the form (3) is quite similar to the corresponding expression for the \(\alpha-\mathcal{F}\) model (see [42]); thus, some explanations are needed.