## Appendix A Derivation of the e2e SNR's CDF Expression  
To derive the CDF expression for the e2e SNR, we rewrite \(\gamma_{D}\) in (14) as \(\gamma_{D}=\gamma_{D}^{\prime}\gamma_{R}\), where \(\gamma_{D}^{\prime}=\mu_{0}h_{D}^{2}\), and \(\mu_{0}\) is given as  
\[\mu_{0}=\frac{0.75V_{t}(\eta_{D}\mathcal{R}_{D}A_{D}\zeta_{D})^{2}\mathbb{E}^{ 2}[h_{D}]}{\xi_{D}\sigma_{D}^{2}}\frac{\rho_{2}}{\rho_{1}}\Bigg{(}\frac{2\tau \mathcal{B}_{\max}^{2}}{(1-\tau)}+\bigg{(}\frac{\mathcal{B}_{\max}+\mathcal{B} _{\min}}{2}\bigg{)}^{2}\Bigg{)}, \tag{11}\]  
where \(\rho_{1}=\mathcal{P}_{S}/\sigma_{R}^{2}\) and \(\rho_{2}=\mathcal{P}_{S}/\sigma_{D}^{2}\). Now, we can rewrite (14) as  
\[F_{\gamma_{e2}}(z)=\Pr\big{[}\min\big{(}\gamma_{R},\gamma_{D}^{\prime}\gamma_{ R}\big{)}\leq z\big{]}. \tag{12}\]  
It can be noted from \(\gamma_{e2e}=\min(\gamma_{R},\gamma_{D}^{\prime}\gamma_{R})\) that \(\gamma_{e2e}=\gamma_{D}^{\prime}\gamma_{R}\) when \(0\leq\gamma_{D}^{\prime}<1\), and \(\gamma_{e2e}=\gamma_{R}\) when \(\gamma_{D}^{\prime}\geq 1\). Therefore, the CDF in (12) can be calculated as  
\[F_{\gamma_{e2}}(z)=\underbrace{\int_{0}^{1}F_{\gamma_{R}}\Big{(}\frac{z}{c} \Big{)}f_{\gamma_{D}^{\prime}}(c)dc}_{\mathcal{I}_{A1}}+\underbrace{\int_{1}^ {\infty}F_{\gamma_{R}}(z)f_{\gamma_{D}^{\prime}}(c)dc}_{\mathcal{I}_{A2}}. \tag{13}\]  
The RV \(\gamma_{D}^{\prime}\) follows the Malaga distribution with pointing error effect, and its PDF and CDF expressions are given as [2]  
\[f_{\gamma_{D}^{\prime}}(x)=\sum_{k=1}^{\beta_{D}}\frac{\phi_{3}}{x}G_{1,3}^{3,0}\Bigg{(}\frac{\psi_{2}x^{\frac{1}{2}}}{\mu_{0}^{\frac{1}{2}}}\Big{|}_{ \mathcal{S}_{D}^{\prime}}^{2}+1}{\mu_{0}^{\frac{1}{2}}}\Big{|}_{\mathcal{S}_{ D}^{\prime}}^{2},\alpha_{D},k\Bigg{)}, \tag{14}\]  
\[F_{\gamma_{D}^{\prime}}(x)=\sum_{k=1}^{\beta_{D}}\phi_{4}G_{3,7}^{6,1}\Bigg{(} \frac{\psi_{2}^{2}x}{2^{4}\mu_{0}}\Big{|}_{\mathcal{K}_{4}}\Bigg{)}, \tag{15}\]  
where \(\kappa_{3}=\Delta(2\colon 1+\xi_{D}^{2})\), \(\kappa_{4}=\Delta(2\colon 5_{D}^{2})\), \(\Delta(2\colon\alpha_{D})\), \(\Delta(2\colon k),\phi_{3}=\frac{\xi_{D}^{2}A_{D}b_{k}}{4},\phi_{4}=\frac{\xi_ {D}^{2}A_{D}b_{2}x^{D}+k}{4\pi}\), and \(\psi_{2}=\frac{\xi_{D}^{2}\alpha_{D}\beta_{D}\left(\beta_{D}+\Omega_{D}^{ \prime}\right)}{\left(\beta_{D}\beta_{D}+\Omega_{D}^{\prime}\right)\left(1+ \xi_{D}^{2}\right)}\). \(A_{i}\) and \(b_{k}\) are given in (6). To solve the integral in (13), we write the PDF of \(\gamma_{D}^{\prime}\) in (14) in terms of power series, by employing [44] [Equation (9.303)] and [40] [Equation (07.31.02.0001.01)], as  
\[f_{\gamma_{D}^{\prime}}(x)=\sum_{k=1}^{\beta_{D}}\frac{\Phi_{1}x^{\frac{\xi_{D }^{2}}{2}-1}}{\mu_{0}^{\frac{\xi_{D}^{2}}{2}}}+\sum_{k=1}^{\beta_{D}}\sum_{n= 0}^{\infty}\frac{\[PERSON]_{2}x^{\frac{n+\alpha_{D}}{2}-1}}{\mu_{0}^{\frac{n+ \alpha_{D}}{2}}}+\sum_{k=1}^{\beta_{D}}\sum_{n=0}^{\infty}\frac{\[PERSON]_{3}x^{ \frac{n+\alpha_{D}}{2}-1}}{\mu_{0}^{\frac{n+\alpha_{D}}{2}}}, \tag{16}\]  
where  
\[\Phi_{1} =\phi_{3}\Gamma[\alpha_{D}-\xi_{D}^{2}]\Gamma[k-\xi_{D}^{2}] \psi_{2}^{\xi_{D}^{2}};\] \[\[PERSON]_{2} =\frac{\phi_{3}\Gamma[\xi_{D}^{2}-\alpha_{D}]\Gamma[k-\alpha_{D}] \big{(}\alpha_{D}-\xi_{D}^{2}\big{)}_{n}\psi_{2}^{n+\alpha_{D}}}{n!\Gamma[1+ \xi_{D}^{2}-\alpha_{D}]\big{(}1+\alpha_{D}-\xi_{D}^{2}\big{)}_{n}(1+\alpha_{D}- k)_{n}};\] \[\[PERSON]_{3} =\frac{\phi_{3}\Gamma[\xi_{D}^{2}-k]\Gamma[\alpha_{D}-k]\big{(}k- \xi_{D}^{2}\big{)}_{n}\psi_{2}^{n+k}}{n!\Gamma[1+\xi_{D}^{2}-k]\big{(}1+k-\xi_ {D}^{2}\big{)}_{n}(1+k-\xi_{D})_{n}}. \tag{17}\]  
In (1), \(\big{(}a\big{)}_{n}\) represents the Pochhammer symbol [40] [Equation (06.10.02.0001.01)]. Now, the integral \(\mathcal{I}_{A1}\) in (13) can be rewritten by using (12) and (16) in \(\mathcal{I}_{A1}\), and with the help of [40] [Equation (07.34.17.0012.01)], as\[\mathcal{I}_{A1} =\sum_{k=1}^{\beta_{R}}\sum_{l=1}^{\beta_{D}}\frac{\Phi_{1}\phi_{2R} }{\mu_{0}^{c_{1}}}\int_{0}^{1}c^{\varrho_{1}-1}G_{7,3}^{1,6}\left(\frac{2^{4}\mu _{R}c}{\psi_{1R}^{2}z} \begin{vmatrix}1-\kappa_{2R,1}\\ 0,1-\kappa_{1R}\end{vmatrix}