## 2 System and Channel Models
### One-Hop Hybrid FS0/RF System Based on a Selective Combination Scheme  
In a one-hop hybrid FSO/RF subsystem with a selective combination scheme, it detects the SNR of each sublink and selects the signal based on the maximum SNR. Therefore, the output SNR \(\gamma_{x,y}^{\rm{SC}}\) of the selection combiner on a one-hop link can be expressed as [37]:  
\[\gamma_{x,y}^{\rm{SC}}=\max(\gamma_{x,y}^{FSO},\gamma_{x,y}^{RF}). \tag{9}\]  
Therefore, the CDF of the SNR \(\gamma_{x,y}^{\rm{SC}}\) can be expressed as [37]:  
\[F_{\gamma_{x,y}^{\rm{SC}}}(\gamma) =\Pr(\max(\gamma_{x,y}^{FSO},\gamma_{x,y}^{RF})\leq\gamma) \tag{10}\] \[=\Pr(\gamma_{x,y}^{FSO}\leq\gamma.\gamma_{x,y}^{RF}\leq\gamma)= F_{\gamma_{x,y}^{FSO}}(\gamma)F_{\gamma_{x,y}^{RF}}(\gamma).\]  
By substituting Equations (4) and (8) into Equation (10), the CDF of the output SNR \(\gamma_{x,y}^{\rm{SC}}\gamma_{i,j}^{\rm{SC}}\) without PEat the j-th hop in i-th path can be obtained as:  
\[F_{\gamma_{x,y}^{\rm{SC}}}(\gamma)=\left\{ \begin{array}{cc}\frac{1}{\Gamma(m _{x,y})}G_{1,2}^{1,1}\bigg{(}\frac{m_{x,y}\gamma}{\gamma_{x,y}^{RF}}\bigg{|} \begin{array}{c}1\\ m_{x,y},0\end{array}\bigg{)}\left\{1-\exp\left[-\left(\frac{1}{\tilde{\eta}_{x,y}}\sqrt{\frac{\gamma}{\tilde{\eta}_{x,y}^{\rm{SC}}}}\right)^{\beta_{x,y}} \right]\right\}^{\alpha_{x,y}}\right.,\text{ without PE}\\ \frac{B_{x,y}(\gamma)}{\Gamma(m_{x,y})}\sum_{t=0}^{\infty}C_{x,y}G_{1,2}^{1,1 }\bigg{(}\frac{m_{x,y}\gamma}{\gamma_{x,y}^{RF}}\bigg{|}\begin{array}{c}1 \\ m_{x,y},0\end{array}\bigg{)}G_{2,3}^{2,1}\bigg{(}D_{x,y}(\gamma)\bigg{|} \begin{array}{c}E_{x,y},1\\ 0,E_{x,y},E_{x,y}\end{array}\bigg{)}\quad,\text{ with PE}\end{array} \right.. \tag{11}\]  
Similarly, the CDF of the output SNR \(\gamma_{s,d}^{\rm{SC}}\) of the combiner is rewritten as:  
**3. System Performance Analysis**  
According to the max-min criterion, the equivalent SNR \(\gamma_{\rm{eq}}^{\prime}\) of the optimal cooperative path from the source to the destination is denoted as [28]:  
\[\gamma_{eq}^{\prime}=\underset{i=1,\cdots,N}{\max}(\underset{j=1,\cdots,M}{ \min}(\gamma_{i,j})), \tag{12}\]  
Therefore, the CDF of the equivalent SNR \(\gamma_{eq}^{\prime}\) can be derived as:  
\[F_{\gamma_{eq}^{\prime}}(\gamma)=\bigg{[}1-\prod_{i=1}^{M}\bigg{[}1-F_{\gamma _{i,j}^{SC}}(\gamma)\bigg{]}\bigg{]}^{N}. \tag{13}\]  
For simplicity, consider that the channel of each hop follows an identical and independently distribution. Then, the above equation can be rewritten as  
\[F_{\gamma_{eq}^{\prime}}(\gamma) =\Big{[}F_{\gamma_{eq}}(\gamma)\Big{]}^{N}=\Bigg{[}1-\bigg{[}1-F_ {\gamma_{i,j}^{SC}}(\gamma)\bigg{]}^{M}\Bigg{]}^{N} \tag{14}\] \[=\left\{ \begin{array}{cc}\left\{1-\Bigg{\{}1-\frac{1}{\Gamma(m_{ i,j})}G_{1,2}^{1,1}\bigg{(}\frac{m_{i,j}\gamma}{\gamma_{i,j}^{RF}}\bigg{|} \begin{array}{c}1\\ m_{i,j},0\end{array}\bigg{)}G_{2,3}^{2,1}\bigg{(}D_{ij}(\gamma)\bigg{|} \begin{array}{c}E_{i,j},1\\ 0,E_{i,j},E_{i,j}\end{array}\bigg{)}\bigg{\}}^{M}\right\}^{N}\quad,\text{without PE}\\ \left[1-\bigg{[}1-\frac{B_{ij}(\gamma)}{\Gamma(m_{i,j})}\sum_{t=0}^{\infty}C_{ i,j}G_{1,2}^{1,1}\bigg{(}\frac{m_{i,j}\gamma}{\gamma_{t,j}^{RF}}\bigg{|} \begin{array}{c}1\\ m_{i,j},0\end{array}\bigg{)}G_{2,3}^{2,1}\bigg{(}D_{ij}(\gamma)\bigg{|} \begin{array}{c}E_{i,j},1\\ 0,E_{i,j},E_{i,j}\end{array}\bigg{)}\right]^{M}\right]^{N}\quad,\text{with PE} \end{array} \right..\]  
Furthermore, we can obtain the CDF of the system output SNR as [38]\[F_{\gamma}(\gamma) =F_{\gamma,s^{\text{SC}}_{d}}(\gamma)\times F_{\gamma^{\prime},q}( \gamma)=F_{\gamma,s^{\text{SC}}_{d}}(\gamma)\times\left[1-\left[1-F_{\gamma,s^{ \text{SC}}_{d}}(\gamma)\right]^{M}\right]^{N} \tag{15}\] \[=\left\{