## 2 System and Channel Models
### Outage Probability  
Outage probability is an important metric to evaluate the probability whether the receiver can successfully decode the message. Commonly, it is mathematically defined as the probability that the end-to-end output SNR is under a specific threshold \(\gamma_{th}\). Therefore, we have [38]  
\[P_{out}=\Pr(\gamma<\gamma_{th})=\int_{0}^{\gamma_{th}}f_{\gamma}(\gamma)d\gamma =F_{\gamma}(\gamma_{th}). \tag{20}\]  
By considering Equation (15), we have the outage probability \(P_{out}^{RA}\) of the system, i.e., \(P_{out}^{RA}\) as  
\[ \begin{array}{ll}P_{out}^{RA}=\left\{\begin{array}{ll}\frac{1}{\Gamma\left( m_{s,d}\right)}G_{1,2}^{1,1}\bigg{(}\frac{m_{s,d}\gamma_{th}}{\overline{ \gamma}_{s,d}^{RF}}\bigg{|}\begin{array}{c}1\\ m_{s,d},0\end{array}\bigg{)}\Bigg{\{}1-\exp\left[-\left(\frac{1}{\overline{ \eta}_{s,d}}\sqrt{\frac{\gamma_{th}}{\overline{\gamma}_{s,d}^{TS0}}}\right)^{ \beta_{s,d}}\right]\Bigg{\}}^{\alpha_{s,d}}\times\\ \Bigg{\{}1-\left\{1-\frac{1}{\Gamma\left(m_{i,j}\right)}G_{1,2}^{1,1}\bigg{(} \frac{m_{i,\gamma_{th}}}{\overline{\gamma}_{ij}^{RF}}\bigg{|}\begin{array}{c} 1\\ m_{i,j},0\end{array}\bigg{)}\right\}\Bigg{\{}1-\exp\left[-\left(\frac{1}{\overline{ \eta}_{ij}}\sqrt{\frac{\gamma_{th}}{\overline{\gamma}_{ij}^{TS0}}}\right)^{ \beta_{s,j}}\right]\Bigg{\}}^{\alpha_{s,i}}\Bigg{\}}^{M}\Bigg{\}}^{N}&\mbox{, without PE,}\\ \frac{B_{s,d}(\gamma_{th})}{\Gamma\left(m_{s,d}\right)}\sum\limits_{t=0}^{\infty}C_{ s,d}G_{1,2}^{1,1}\bigg{(}\frac{m_{s,d}\gamma_{th}}{\overline{\gamma}_{s,d}^{RF}} \bigg{|}\begin{array}{c}1\\ m_{s,d},0\end{array}\bigg{)}G_{2,3}^{2,1}\bigg{(}D_{s,d}(\gamma_{th})\bigg{|} \begin{array}{c}E_{s,d},1\\ 0,E_{s,d},E_{s,d}\end{array}\bigg{)}\times\\ \Bigg{[}1-\left[1-\frac{B_{i,j}(\gamma_{th})}{\Gamma\left(m_{s,d}\right)}\sum \limits_{t=0}^{\infty}C_{i,j}G_{1,2}^{1,1}\bigg{(}\frac{m_{i,\gamma_{th}}}{ \overline{\gamma}_{ij}^{RF}}\bigg{|}\begin{array}{c}1\\ m_{i,j},0\end{array}\bigg{)}\begin{array}{c}1\\ m_{i,j},0\end{array}\bigg{)}G_{2,3}^{2,1}\bigg{(}D_{ij}(\gamma_{th})\bigg{|} \begin{array}{c}E_{i,j},1\\ 0,E_{i,j},E_{i,j}\end{array}\bigg{)}\bigg{]}^{M}\Bigg{\}}^{N}&\mbox{, with PE.}\end{array}\right.\end{array}  \tag{21}\]  
By combing Equations (11) and (20), the outage probability of the hybrid direct link can be  
\[\begin{array}{ll}P_{out}^{SD}=\left\{\begin{array}{ll}\frac{1}{\Gamma\left(m_{s,d }\right)}G_{1,2}^{1,1}\bigg{(}\frac{m_{s,d}\gamma_{th}}{\overline{\gamma}_{s,d}^{ RF}}\bigg{|} \begin{array}{c}1\\ m_{s,d},0\end{array} \bigg{)}\Bigg{\{}1-\exp\left[-\left(\frac{1}{\overline{ \eta}_{s,d}}\sqrt{\frac{\gamma_{th}}{\overline{\gamma}_{s,d}^{TS0}}}\right)^{ \beta_{s,d}}\right]\Bigg{\}}^{\alpha_{s,d}}&\mbox{, without PE,}\\ \frac{B_{s,d}(\gamma_{th})}{\Gamma\left(m_{s,d}\right)}\sum\limits_{t=0}^{\infty}C_{ s,d}Gassisted hybrid system and atmospheric channel are shown in Table 3, some of which are also adopted in [30; 33; 34; 35]. When \(\tau\) and \(t\) are both selected as \(30t=30\), the approximate solution of the generalized Gaussian-Laguerre quadrature function tends to converge by simulations, and the average BER of the UAV-assisted hybrid system can be obtained according to Equations (18) and (19). Similarly, when \(t\) is chosen to be 30, the outage probability of a UAV-assisted hybrid system can be obtained according to Equation (21). For simplification, let the transmission power of FSO and RF links be the same, and each link has the same distance of \(1\)km\(1 km\) in simulations. The structure parameters \((N=1,M=3)\), \((N=5,M=3)\), \((N=3,M=5)\), \((N=3,M=2)\) have been selected to avoid entanglement. For simplicity, Figures 2-6 mainly discuss the impact of weather on the performance of the system, without considering pointing errors. Figure 2 illustrates the effects of different weather conditions (i.e., clear air, haze, light fog, and light rain) on the BER of the hybrid FSO/RF direct link under different modulation schemes.