## 3 Statistical Model of SNR  
### A. Statistical Model of SNR for Single Hop Links  
We adopt the IM/DD detection method with OOK modulation and assume that the transmitted signal \(x\) has an average power constraint \(E[x]=1\). Then, the SNR of the received signal for a single-hop link can be expressed as [35]:  
\[\gamma_{i}=\frac{P_{t}^{2}E^{2}[x]}{\sigma_{n}^{2}}|h_{i}|^{2}=\overline{ \gamma_{i}}|h_{i}|^{2} \tag{18}\]  
where \(P_{t}\) represents the emitted optical power. According to the principle of variable transformation, we obtain the PDF of \(\gamma_{i}\):  
\[f_{\gamma_{i}}(\gamma_{i})=\frac{1}{2\sqrt{\gamma_{i}\gamma_{i}}}f_{h_{i}} \bigg{(}\sqrt{\frac{\gamma_{i}}{\gamma_{i}}}\bigg{)} \tag{19}\]  
Putting Equation (17) into Equation (19), \(f_{\gamma_{i}}(\gamma_{i})\) is as follows:  
\[f_{\gamma_{i}}(\gamma_{i})=\frac{\alpha_{i}\beta_{i}\varepsilon_{\text{mod},i} ^{2}z_{i}^{k_{i}}}{2A_{\text{mod},i}\Gamma(\alpha_{i})\Gamma(\beta_{i})\sqrt{ \gamma_{i}}}L_{2,4}^{4,0}\left[ \begin{array}{c}\alpha_{i}\beta_{i}\\ A_{\text{mod},i}\end{array} \sqrt{\frac{\gamma_{i}}{\gamma_{i}}}\bigg{|} \begin{array} []{c}\left(\varepsilon_{\text{mod},i}^{2},1,1\right),(z_{i},1,k_{i})\\ \left(\varepsilon_{\text{mod},i}^{2}-1,1,1\right),(\alpha_{i}-1,1,1),(\beta_{ i}-1,1,1),(z_{i}-1,1,k_{i})\end{array} \right] \tag{20}\]  
Integrating (20) as \(F_{\gamma_{i}}(\gamma_{i})=\int_{0}^{\gamma}f_{\gamma_{i}}(\gamma_{i})d\gamma_ {i}\), we obtain:  
\[F_{\gamma_{i}}(\gamma_{i}) =\int_{0}^{\gamma}\frac{\alpha_{i}\beta_{i}\varepsilon_{\text{ mod},i}^{2}z_{i}^{k_{i}}}{2A_{\text{mod},i}\Gamma(\alpha_{i})\Gamma(\beta_{i})\sqrt{ \gamma_{i}}}L_{2,4}^{4,0}\Bigg{[} \begin{array}{c}\alpha_{i}\beta_{i}\\ A_{\text{mod},i}\end{array} \sqrt{\frac{\gamma_{i}}{\gamma_{i}}}\bigg{|} \begin{array}{c}\left(\varepsilon_{\text{mod},i}^{2},1,1\right),(z_{i},1,k_ {i})\\ \left(\varepsilon_{\text{mod},i}^{2}-1,1,1\right),(\alpha_{i}-1,1,1),(\beta_{ i}-1,1,1),(z_{i}-1,1,k_{i})\end{array} \Bigg{]}d\gamma_{i} \tag{21}\] \[=\frac{\alpha_{i}\beta_{i}\varepsilon_{\text{mod},i}^{2}z_{i}^{k_ {i}}}{2A_{\text{mod},i}\Gamma(\beta_{i})\sqrt{\gamma_{i}}}\times\int_{0}^{ \gamma}\gamma_{i}^{-\frac{1}{2}+\frac{\alpha}{2}d\gamma_{i}}\] \[\times\frac{1}{2\pi\gamma}\int_{1}\frac{\Gamma(-s+\varepsilon_{ \text{mod},i}^{2}-1)\Gamma(-s+\varepsilon_{i}-1)\Gamma(-s+\varepsilon_{i}-1) \Gamma^{k_{i}}(-s+z_{i}-1)}{\Gamma\left(-s+\varepsilon_{\text{mod},i}^{2} \right)\Gamma^{k_{i}}(-s+z_{i})}\Big{(}\frac{\alpha_{i}\beta_{i}}{A_{\text{ mod},i}\sqrt{\gamma_{i}}}\Big{)}^{s}ds\]  
where the internal integral about \(\gamma_{i}\) can be simplified as:  
\[\int_{0}^{\gamma}\gamma_{i}^{-\frac{1}{2}+\frac{\alpha}{2}d\gamma_{i}}=\frac{ \gamma_{i}^{\frac{1}{2}+\frac{\alpha}{2}\Gamma\left(\frac{s}{2}+\frac{1}{2} \right)}}{\Gamma\left(\frac{s}{2}+\frac{3}{2}\right)} \tag{22}\]  
Putting Equation (22) into Equation (21). After simplification, we can obtain:  
\[F_{\gamma_{i}}(\gamma_{i}) =\frac{\alpha_{i}\beta_{i}\varepsilon_{\text{mod},i}^{2}z_{i}^{k_ {i}}}{2A_{\text{mod},i}\Gamma(\alpha_{i})\Gamma(\beta_{i})}\sqrt{\frac{\gamma_{ i}}{\gamma_{i}}} \tag{23}\] \[\times\frac{1}{2\pi\gamma}\int_{1}\frac{\Gamma\left(-s+\varepsilon_ {\text{mod},i}^{2}-1\right)\Gamma(-s+\varepsilon_{i}-1)\Gamma(-s+\beta_{i}-1) \Gamma^{k_{i}}(-s+z_{i}-1)\Gamma\left(\frac{s}{2}+\frac{1}{2}\right)}{ \Gamma\left(-s+\varepsilon_{\text{mod},i}^{2}\right)\Gamma^{k_{i}}(-s+z_{i}) \Gamma\left(\frac{s}{2}+\frac{3}{2}\right)}\Big{(}\frac{\alpha_{i}\beta_{i}}{A_{ \text{mod},i}\sqrt{\frac{\gamma_{i}}{\gamma_{i}}}}\Big{)}^{s}ds\]  
The above formula can be expressed by \(I\)-function, and according to [36] (Equation 7.2), we can obtain:  
Substituting the above equation into Equation (28). After simplification, we can obtain:  
\[