## 3 Statistical Model of SNR
### A. Statistical Model of SNR for Single Hop Links
\begin{array}{rl}f_{\gamma_{\text{ub}}}(\gamma)&=\prod\limits_{i=1}^{N}\frac{ \varepsilon_{\text{mod},i}^{2}\varepsilon_{i}^{k_{i}}}{\Gamma(\alpha_{i}) \Gamma(\beta_{i})\gamma}\\ &\times\frac{1}{2\pi j}\int\limits_{i}\frac{\prod\limits_{i=1}^{N}\Gamma\left( \varepsilon_{\text{mod},i}^{2}+\frac{2l_{i}}{N}s\right)\Gamma\left(\alpha_{i} +\frac{2l_{i}}{N}s\right)\Gamma\left(\beta_{i}+\frac{2l_{i}}{N}s\right)\Gamma^ {k_{i}}\left(z_{i}+\frac{2l_{i}}{N}s\right)}{\prod\limits_{i=1}^{N}\Gamma\left( \varepsilon_{\text{mod},i}^{2}+1+\frac{2l_{i}}{N}s\right)\Gamma^{k_{i}}\left( z_{i}+1+\frac{2l_{i}}{N}s\right)}\\ &\left(N\gamma\prod\limits_{i=1}^{N}\psi_{i}\left(\frac{a_{i}\beta_{i}}{A_{ \text{mod},i}\sqrt{\gamma_{i}}}\right)^{\frac{2l_{i}}{N}}\right)^{-s}\!\!ds \end{array}  \tag{32}\]  
The above formula can be simplified by \(I\)-function:  
\[f_{\gamma_{\text{ub}}}(\gamma)=\prod\limits_{i=1}^{N}\frac{\varepsilon_{\text {mod},i}^{2}\varepsilon_{i}^{k_{i}}}{\Gamma(\alpha_{i})\Gamma(\beta_{i})\gamma }I_{2N,AN}^{4N,0}\left[N\gamma\prod\limits_{i=1}^{N}\psi_{i}\left(\frac{\alpha_ {i}\beta_{i}}{A_{\text{mod},i}\sqrt{\gamma_{i}}}\right)^{\frac{2l_{i}}{N}} \right|_{I_{1}}\right] \tag{33}\]  
where  
\[ \begin{array}{rl}I_{1}=\left(\left(\varepsilon_{\text{mod},1}^{2}+1,\frac{2 l_{1}}{N},1\right),\left(z_{1}+1,\frac{2l_{1}}{N},k_{1}\right),...,\left( \varepsilon_{\text{mod},N}^{2}+1,\frac{2l_{N}}{N},1\right),\left(z_{N}+1, \frac{2l_{N}}{N},k_{N}\right)\right),\\ \\ J_{2}=\left\{\begin{array}{l}\left(\varepsilon_{\text{mod},1}^{2},\frac{2 l_{1}}{N},1\right),\left(\alpha_{1},\frac{2l_{1}}{N},1\right),\left(\beta_{1}, \frac{2l_{1}}{N},1\right),\left(z_{1},\frac{2l_{1}}{N},k_{1}\right),...\\ \left(\varepsilon_{\text{mod},N}^{2},\frac{2l_{N}}{N},1\right),\left(\alpha_{N },\frac{2l_{N}}{N},1\right),\left(\beta_{N},\frac{2l_{N}}{N},1\right),\left(z_ {N},\frac{2l_{N}}{N},k_{N}\right)\end{array}\right\}.\end{array} \]  
Integrating (33) as \(F_{\gamma_{\text{ub}}}(\gamma)=\int_{0}^{\gamma}\gamma_{\text{ub}}(\gamma)d\gamma\), we obtain:  
\[ \begin{array}{rl}F_{\gamma_{\text{ub}}}(\gamma)&=\int_{0}^{\gamma}\prod \limits_{i=1}^{N}\frac{\varepsilon_{\text{mod},i}^{2}\varepsilon_{i}^{k_{i}}}{ \Gamma(a_{i})\Gamma(\beta_{i})}I_{2N,AN}^{4N,0}\left[N\gamma\prod\limits_{i=1}^ {N}\psi_{i}\left(\frac{a_{i}\beta_{i}}{A_{\text{mod},i}\sqrt{\gamma_{i}}} \right)^{\frac{2l_{i}}{N}}\right|\begin{array}{c}I_{1}\\ I_{2}\end{array}\right]d\gamma_{i}\\ &=\prod\limits_{i=1}^{N}\frac{\varepsilon_{\text{mod},i}^{2}\varepsilon_{i}^{k_ {i}}}{\Gamma(a_{i})\Gamma(\beta_{i})}I_{0}^{\gamma}\gamma^{-s-1}d\gamma\\ &\times\frac{1}{2\pi j}\int\limits_{i}\frac{\prod\limits_{i=1}^{N}\Gamma\left( \varepsilon_{\text{mod},i}^{2}+\frac{2l_{i}}{N}s\right)\Gamma\left(a_{i}+\frac{ 2l_{i}}{N}s\right)\Gamma\left(\beta_{i}+\frac{2l_{i}}{N}s\right)\Gamma^{k_{i}} \left(z_{i}+\frac{2l_{i}}{N}s\right)}{\prod\limits_{i=1}^{N}\Gamma\left( \varepsilon_{\text{mod},i+1}^{2}+\frac{2l_{i}}{N}s\right)\Gamma^{k_{i}} \left(z_{i}+1+\frac{2l_{i}}{N}s\right)}\left(N\prod\limits_{i=1}^{N}\psi_{i} \left(\frac{\alpha_{i}\beta_{i}}{A_{\text{mod},i}\sqrt{\gamma_{i}}}\right)^{ \frac{2l_{i}}{N}}\right)^{-s}\!\!ds\end{array}  \tag{34}\]  
where the internal integral with respect to \(\gamma\) is:  
\[\int_{0}^{\gamma}\gamma^{-s-1}d\gamma=\frac{\gamma^{-s}\Gamma(-s)}{\Gamma(-s+1)} \tag{35}\]  
Substituting the above equation into Equation (34). After simplification, we obtain:  
\[