## 2 Principles
\begin{array}{ll}P_{out}&=Pr\Big{\{}G_{FSO}\leq\frac{2^{[R-\Psi log_{2}(1+M_{RF }P_{RF})]}-1}{P_{FSO}}\Big{\}}\\ &=\int_{0}^{d}\frac{2}{\Gamma(\alpha)\Gamma(\beta)x}\Big{(}\frac{\alpha\beta} {\mu}x\Big{)}^{\frac{\alpha+\beta}{2}}K_{\alpha-\beta}\Big{(}2\sqrt{\frac{ \alpha\beta}{\mu}x}\Big{)}dx\end{array}  \tag{9}\]  
where \(d=\left[2^{[R-\Psi log_{2}(1+M_{RF}P_{RF})]}-1\right]/P_{FSO}\). Rewrite the modified Bessel function in terms of the Meijer-G function [23] (p. 665), and we can obtain  
\[P_{out}=\int_{0}^{d}\frac{2}{\Gamma(\alpha)\Gamma(\beta)x}\left(\frac{\alpha \beta}{\mu}x\right)^{\frac{\alpha+\beta}{2}}\frac{1}{2}C_{02}^{20}\bigg{(} \frac{\alpha\beta}{\mu}x|_{(\alpha-\beta)/2,-(\alpha-\beta)/2}\bigg{)}dx \tag{10}\]  
where \(G(\cdot)\) is the Meijer-G function. Using the integral formula of the Meijer-G function [23] (p. 46), a closed-form expression of the outage probability of cc-HARQ-aided FSO-RF hybrid communications can be expressed as follows:  
\[P_{out}=\frac{1}{\Gamma(\alpha)\Gamma(\beta)x}\bigg{(}\frac{\alpha\beta}{\mu}x \bigg{)}^{\frac{\alpha+\beta}{2}}\frac{1}{2}G_{13}^{21}\bigg{(}\frac{\alpha \beta}{\mu}x|_{-(\alpha-\beta)/2,-(\alpha-\beta)/2}^{1-(\alpha+\beta)/2,( \alpha-\beta)/2}\bigg{)}|_{x=d} \tag{11}\]  
When the messages on the FSO link are transmitted more than once, i.e., \(M_{FSO}>1\), according to the distribution of the sum of k independent identically distributed (i.i.d) random variables with a Gamma-Gamma distribution [22], it is easy to obtain the closed-form expression of outage probability when the FSO link is transmitted for \(M_{FSO}\) times and the RF link is transmitted for \(M_{RF}\) times  
\[ \begin{array}{ll}P_{out}=\frac{1}{\Gamma(\alpha_{M_{FSO}})\Gamma(\zeta_{M_{ FSO}})x}\bigg{(}\frac{\alpha_{M_{FSO}}\zeta_{M_{FSO}}}{\mu}x\bigg{)}^{\frac{ \alpha_{M_{FSO}}+\zeta_{M_{FSO}}}{2}}\\ \times\frac{1}{2}C_{13}^{21}\bigg{(}\frac{\alpha_{M_{FSO}}\zeta_{M_{FSO}}}{\mu }x|_{-(\alpha_{M_{FSO}}-\zeta_{M_{FSO}})/2-(\alpha_{M_{FSO}}+\zeta_{M_{FSO}})/ 2}^{1-(\alpha_{M_{FSO}}+\zeta_{M_{FSO}})/2}\bigg{)}|_{x=d}\end{array}  \tag{12}\]  
where \(d=\left[2^{[R-\Psi log_{2}(1+M_{RF}P_{RF})]}-1\right]/P_{FSO}\), \(\sigma_{k}\) and \(\zeta_{k}\) are respectively given by  
\[\sigma_{k}=k\upsilon, \tag{13}\]  
\[\zeta_{k}=k\tau, \tag{14}\]  
where \(\upsilon=\max\left\{\alpha,\,\beta\right\}\), \(\tau=\min\left\{\alpha,\,\beta\right\}\). \(\mu\) is the mathematical expectation that is equal to 1 if we assume that the FSO channel gain is normalized. \(\alpha\) and \(\beta\) are the fading/scintillation parameters related to the atmospheric turbulence conditions. These are the distribution parameters of a single variable that follows the Gamma-Gamma fading distribution. We can also analyze the throughput of the system under different RF over FSO rate ratios and retransmission numbers. The throughput is the average rate of data successfully decoded at the receiving end, and it is given by [21; 22]  
\[\eta=\frac{R(1-p(M))}{1+\sum_{m=1}^{M-1}p(m)} \tag{15}\]  
where \(R\) is the transmission rate of the first transmission round and \(p(m)\) denotes the probability of the receiver failing to correctly decode the message after m rounds. Under the assumption of random coding and typical set decoding with large packet lengths, \(p(m)\) is equivalent to the outage probability after m HARQ rounds. By substituting the above outage expression in Equation (15), we can obtain the closed-form approximation for the throughput of cc-HARQ.