## 2 System and Channel Models
### One-Hop FSO SublinkSubsystem under Various Weather Conditions  
Considering EW atmospheric turbulence only, according to Equation (8) in [29], the PDF of instantaneous signal to noise ratio (SNR) \(\gamma_{x,y}^{FSO}\) without PE in a one-hop FSO sublinklink  
Figure 1: Structure of a UAV-assisted multi-hop parallel hybrid FSO/RF communication system. (the subscripts \((x,y)\) denote the \(y\)-th hop hybrid FSO/RF link of the \(x\)-th path, especially \((x,y)=(s,d)\) represent the direct hybrid FSO/RF link and \((x,y)=(i,j)\) represent the \(j\)-th hop hybrid FSO/RF link of the \(i\)-th path) is:  
\[ \begin{split} f_{\gamma_{x,y}^{FSO}}\Big{(}\gamma_{x,y}^{FSO}\Big{)}& =\frac{\alpha_{x,y}\beta_{x,y}}{2\gamma_{x,y}^{FSO}\eta_{x,y}^{ \beta_{x,y}}}\Bigg{(}\sqrt{\frac{\gamma_{x,y}^{FSO}}{\gamma_{x,y}^{FSO}}}\Bigg{)} ^{\beta_{x,y}-2}\exp\left[-\left(\frac{1}{\eta_{x,y}}\sqrt{\frac{\gamma_{x,y}^ {FSO}}{\gamma_{x,y}^{FSO}}}\right)^{\beta_{x,y}}\right]\\ &\times\left\{1-\exp\left[-\left(\frac{1}{\eta_{x,y}}\sqrt{\frac {\gamma_{x,y}^{FSO}}{\gamma_{x,y}^{FSO}}}\right)^{\beta_{x,y}}\right]\right\} ^{\alpha_{x,y}-1},\end{split}  \tag{1}\]  
where the subscripts \((x,y)\) denote a one-hop hybrid FSO/RF link, and \((x,y)=(s,d)\) especially represent the direct hybrid FSO/RF link and \((x,y)=(i,j)\) represent the \(j\)-th hop hybrid FSO/RF link of the \(i\)-th path. \(\alpha_{x,y}\) and \(\beta_{x,y}\) are the shape parameters of EW turbulence, and \(\eta_{x,y}\) is the scale parameter. The values of these parameters are all greater than 0 and can be calculated according to [12]. \(\overline{\gamma}_{x,y}^{FSO}=\left(P_{x,y}^{FSO}\gamma_{x,y}^{FSO}\gamma_{x,y }^{FSO}/\sigma_{x,y}^{FSO}\right)^{2}\) is the average SNR of a one-hop FSO link [30], where \(P_{x,y}^{FSO}\) is the transmittedtransmission power of the FSO link, \(R_{x,y}^{FSO}\) is the sensitivity of the photodetector, \(\sigma_{x,y}^{FSO}\) is the standard deviation of Gaussian white noise, and \(g_{x,y}^{FSO}\) is the atmospheric loss of the FSO link. Considering EW atmospheric turbulence and pointing errors [31; 32], the PDF of instantaneous signal to noise ratio (SNR) \(\gamma_{x,y}^{FSO}\) with PE in a one-hop FSO sublink is:  
\[ \begin{split} f_{\gamma_{x,y}^{FSO}}\Big{(}\gamma_{x,y}^{FSO} \Big{)}&=\frac{\alpha_{x,y}\rho_{x,y}^{2}}{\left(\eta_{x,y}A_{x, y}^{0}\right)^{\beta_{x,y}}\gamma_{x,y}^{FSO}}\Bigg{(}\sqrt{\frac{\gamma_{x,y}^{ FSO}}{\overline{\gamma}_{x,y}^{FSO}}}\Bigg{)}^{\rho_{x,y}^{2}}\sum_{t=0}^{\infty} \frac{(-1)^{t}\Gamma\big{(}\alpha_{x,y}\big{)}}{t!\Gamma\big{(}\alpha_{x,y}-t \big{)}\big{(}1+t\big{)}^{-\frac{\rho_{x,y}^{2}}{\beta_{x,y}}}}\\ &\times G_{1,2}^{2,0}\left(\frac{1+t}{\left(\eta_{x,y}A_{x,y}^{0} \right)^{\beta_{x,y}}}\Bigg{(}\sqrt{\frac{\gamma_{x,y}^{FSO}}{\overline{ \gamma}_{x,y}^{FSO}}}\Bigg{)}^{\beta_{x,y}}\Bigg{|}\begin{array}{c}1\\ 0,1-\frac{\rho_{x,y}^{2}}{\beta_{x,y}}\end{array}\right),\end{split}  \tag{2}\]  
where \(\rho_{x,y}\) and \(A_{x,y}^{0}\) are parameters related to pointing errors, \(\rho_{x,y}=\omega_{x,y}^{eq}/2\sigma_{x,y}^{s}\) is the ratio of the equivalent beam radius to the standard deviation of the pointing error on the receiver plane, \(\left(\omega_{x,y}^{eq}\right)^{2}=\left(\omega_{x,y}^{L}\right)^{2}\sqrt{ \pi}erfc(v_{x,y})/\left[2v_{x,y}\exp(-v_{x,y}^{2})\right]\), \(\omega_{x,y}^{L}\) is the waist radius at the distance \(L_{x,y}\) from the light source, \(A_{x,y}^{0}=\left[erfc(v_{x,y})\right]^{2}\) is the fraction of optical power received without pointing errors, \(v_{x,y}=\left(\sqrt{\pi}d_{x,y}\right)/\left(\sqrt{2}\omega_{x,y}^{L}\right)\), \(d_{x,y}\) is the radius of the receiver aperture, and \(erfc(\cdot)\) is a complementary error function. The atmospheric loss of the FSO link \(g_{x,y}^{FSO}\) can be expressed by Beers Lamber Law [30], as follows:  
\[g_{x,y}^{FSO}=\left\{ \begin{array}{c}\frac{A_{x,y}}{\pi\big{(}\theta_{x,y}L _{x,y}/2\big{)}^{2}}e^{-\left(\omega_{x,y}^{FSO}L_{x,y}\right)}\quad\text{, without PE}\\ e^{-\left(\omega_{x,y}^{FSO}L_{x,y}\right)}\quad\text{, with PE}\end{array}