## 3 Performance Analysis
### Secrecy Outage Probability  
SOP is defined as the probability that the instantaneous secrecy capacity, \(\mathcal{C}_{s}\), falls below a predetermined secrecy rate, \(\mathcal{R}_{s}\geq 0\). The secrecy capacity can be written as [32] [Equation (41)]  
\[\mathcal{C}_{s}(\gamma_{\ell 2e},\gamma_{E})=\max\{\log_{2}(1+\lambda\gamma_{ \ell 2e})-\log_{2}(1+\lambda\gamma_{E}),0\}, \tag{27}\]  
where \(\log_{2}(1+\lambda\gamma_{\ell 2e})\) and \(\log_{2}(1+\lambda\gamma_{E})\) denote the capacities of the legitimate receiver at the destination and the eavesdropper at the relay, respectively. SOP can be mathematically described as  
\[\text{SOP}=\text{Pr}[\mathcal{C}_{s}(\gamma_{\ell 2e},\gamma_{E})\leq R_{s}]= \int_{0}^{\infty}F_{\gamma_{\ell 2e}}\Big{(}\theta\gamma+\theta\lambda^{-1}- \lambda^{-1}\Big{)}f_{\gamma_{E}}(\gamma)d\gamma, \tag{28}\]  
where \(\theta=\)2\({}^{\mathcal{R}_{s}}\), \(f_{\gamma_{E}}(\gamma)\) is the PDF of eavesdropper's instantaneous SNR which is given in (11), and \(F_{\gamma_{\ell 2e}}(z)\) is the CDF of \(\gamma_{\ell 2e}\) which is given in (15). Finding the closed-form solution of the integral in (28) is challenging [42] [Equation (7)]. However, the lower bound of SOP (the case when \(\gamma\rightarrow\infty\)) [30] [Equation (18)] can be calculated as  
\[\text{SOP}_{L}=\mathcal{P}\{\gamma_{\ell 2e}\leq\theta\gamma_{E}\}=\int_{0}^{ \infty}F_{\gamma_{\ell 2e}}(\theta\gamma)f_{\gamma_{E}}(\gamma)d\gamma. \tag{29}\]  
Now, using (15) and (11) in (29), we can write  
\[\text{SOP}_{L} = \sum_{k=1}^{\beta_{R}}\sum_{l=1}^{\beta_{D}}\sum_{m=1}^{\beta_{E }}\frac{\Phi_{1}\phi_{2R}\phi_{1E}}{\mu_{0}^{\alpha_{1}}}\mathcal{I}_{9}+\sum _{k=1}^{\beta_{R}}\sum_{l=1}^{\beta_{D}}\sum_{m=1}^{\beta_{E}}\sum_{n=0}^{N} \Phi_{2}\frac{\phi_{2R}\phi_{1E}}{\mu_{0}^{\alpha_{2}}}\mathcal{I}_{10} \tag{30}\] \[+ \sum_{k=1}^{\beta_{R}}\sum_{l=1}^{\beta_{D}}\sum_{m=1}^{\beta_{E }}\sum_{n=0}^{N}\frac{\Phi_{3}\phi_{2R}\phi_{1E}}{\mu_{0}^{\alpha_{3}}} \mathcal{I}_{11}+\sum_{k=1}^{\beta_{R}}\sum_{m=1}^{\beta_{E}}\Big{(}1-F_{ \gamma_{D}^{\prime}}(1)\Big{)}\phi_{2R}\phi_{1E}\mathcal{I}_{12},\]where  
\[\mathcal{I}_{9}=\int_{0}^{\infty}\gamma^{-1}G_{4,8}^{7,1}\Bigg{(} \frac{\theta\psi_{1R}^{2}\gamma}{2^{4}\mu_{R}}\Bigg{|} \begin{matrix}1,\kappa_{1R },1+\varrho_{1}\\ \varrho_{1},\kappa_{2R},0\end{matrix} \Bigg{)}G_{1,3}^{3,0}\Bigg{(}\frac{\psi_ {1E}\gamma^{\frac{1}{2}}}{\mu_{E}^{2}}\Bigg{|} \begin{matrix}\xi_{E}^{2}+1\\ \xi_{E}^{2},\alpha_{E},m\end{matrix} \Bigg{)}d\gamma;\] \[\mathcal{I}_{10}=\int_{0}^{\infty}\gamma^{-1}G_{4,8}^{7,1}\Bigg{(} \frac{\theta\psi_{1R}^{2}\gamma}{2^{4}\mu_{R}}\Bigg{|} \begin{matrix}1,\kappa_{1 R},1+\varrho_{2}\\ \varrho_{2},\kappa_{2R},0\end{matrix} \Bigg{)}G_{1,3}^{3,0}\Bigg{(}\frac{\psi_ {1E}\gamma^{\frac{1}{2}}}{\mu_{E}^{2}}\Bigg{|} \begin{matrix}\xi_{E}^{2}+1\\ \xi_{E}^{2},\alpha_{E},m\end{matrix} \Bigg{)}d\gamma,\] \[\mathcal{I}_{11}=\int_{0}^{\infty}\gamma^{-1}G_{4,8}^{7,1}\Bigg{(} \frac{\theta\psi_{1R}^{2}\gamma}{2^{4}\mu_{R}}\Bigg{|} \begin{matrix}1,\kappa_{1 R},1+\varsigma_{3}\\ \varsigma_{3},\kappa_{2R},0\end{matrix} \Bigg{)}G_{1,3}^{3,0}\Bigg{(}\frac{\psi_ {1E}\gamma^{\frac{1}{2}}}{\mu_{E}^{2}}\Bigg{|} \begin{matrix}\xi_{E}^{2}+1\\ \xi_{E}^{2},\alpha_{E},m\end{matrix} \Bigg{)}d\gamma;\] \[\mathcal{I}_{12}=\int_{0}^{\infty}\gamma^{-1}G_{3,7}^{6,1}\Bigg{(} \frac{\theta\psi_{1R}^{2}\gamma}{2^{4}\mu_{R}}\Bigg{|} \begin{matrix}1,\kappa_{1 R}\\ \kappa_{2R},0\end{matrix} \Bigg{)}G_{1,3}^{3,0}\Bigg{(}\frac{\psi_{1E}\gamma^{\frac{1}{2}}}{\mu_{E}^{ 2}}\Bigg{|} \begin{matrix}\xi_{E}^{2}+1\\ \xi_{E}^{2},\alpha_{E},m\end{matrix} \Bigg{)}d\gamma. \tag{31}\]  
The integrals in (31) are solved by employing [40] [Equation (07.34.21.0013.01)] as  
\[\mathcal{I}_{9}=\frac{2^{\alpha_{E}+m-2}}{\pi}G_{10,10}^{7,7}\Bigg{(}\frac{ \psi_{1E}^{2}\mu_{R}}{\psi_{1R}^{2}\mu_{E}}\Bigg{|} \begin{matrix}1-\varrho_{1},1-\kappa_{2R},1,\kappa_{1E}\\ \kappa_{2E},0,1-\kappa_{1R},-\varrho_{1}\end{matrix}