## 2 System and Channel Models
### Channel Model  
For the FSO links, the joined impacts of atmospheric turbulence-induced fading (modeled using the gamma-gamma distribution [24]), pointing errors (modeled using the Rayleigh distribution [25]), and path loss (based on the Beers-Lampert law [26]) have been taken into account. In that case, it can be proved that the cumulative distribution function (CDF) of the received SNR is given by ([17] [Equation (18)])  
\[F_{\gamma_{1}}(\gamma)=\mathcal{A}G_{b+1,3b+1}\Bigg{(}\frac{D^{b}\gamma}{b^{2b }\bar{\gamma}_{1}^{f}}\Big{|}_{\Lambda_{b,b^{2}}\Lambda_{b,\sigma}\Lambda_{b, \rho}\Lambda_{b,\rho}}\Bigg{)}, \tag{5}\]  
where \(D=\alpha\beta\kappa\), \(\mathcal{A}=\frac{\xi^{2}b^{\alpha+\beta-2}}{(2\pi)^{b-1}\Gamma(a)\Gamma(\beta )}\), \(\Lambda_{x,y}=\frac{y}{x}\), \(\frac{y+1}{x}\), \(\frac{y+x-1}{x}\), while \(\bar{\gamma}_{1}^{f}\) denotes the average received SNR defined as  
\[\bar{\gamma}_{1}=\frac{\eta P_{f}G_{Tf}G_{R}\kappa l_{P}^{f}A_{0}}{F_{f}^{b} \sigma_{f}^{2}}, \tag{6}\]  
with \(\kappa=\frac{\xi^{2}}{\xi^{2}+1}\). Moreover, \(l_{P}^{f}\) denotes the path loss attenuation, \(A_{0}\) is the fraction of total power collected at the receiver aperture, and \(F_{f}\) is the free space loss defined as \(F_{f}=\frac{4\pi d_{k}}{\lambda_{f}}\). Moreover, \(\zeta\) denotes the pointing error parameter coefficient, while \(\alpha\) and \(\beta\) are large- and small-scale turbulence parameters, respectively, related to the scattering environment, whose expressions are analytically provided below. Finally, \(G_{p,d}^{nn}[\cdot]\) denotes the Meijer's G-function ([27] [Equation (9.301)]), and \(\Gamma(\cdot)\) the gamma function ([27] [Equation (8.310/1)]). The corresponding PDF expression is given by ([17] [Equation (17)])  
\[f_{\gamma_{1}}(\gamma)=\frac{\mathcal{B}}{\gamma}G_{1,3}^{3,0}\Bigg{(}D\Bigg{(} \frac{\gamma}{\gamma_{1}^{f}}\Bigg{)}^{1/b}\Big{|}_{\zeta^{2},\alpha,\beta}^{ \zeta^{2}+1}\Bigg{)}, \tag{7}\]  
where \(\mathcal{B}=\frac{\zeta^{2}}{b\Gamma(a)\Gamma(\beta)}\). As far as the large- and small-scale turbulence parameters are concerned, they are, respectively, defined as ([24] [Equations (7a) and (7b)])  
\[ \begin{split}\alpha=\Bigg{\{}5.95(h_{k}-h_{\ell})^{2}& \sec(\theta)^{2}\bigg{(}\frac{2W_{0}}{r}\bigg{)}^{5/3}\bigg{(}\frac{ \Delta_{pe}}{W}\bigg{)}^{2}\\ &+\Bigg{[}\exp\bigg{(}\frac{0.49\sigma^{2}}{\big{(}1+0.56\sigma^ {12/5}\big{)}^{7/6}}\bigg{)}-1\bigg{]}\Bigg{\}}^{-1},\end{split}  \tag{8}\]  
\[\beta=\Bigg{[}\exp\Bigg{(}\frac{0.51\sigma^{2}}{\big{(}1+0.69\sigma^{12/5} \big{)}^{5/6}}\bigg{)}-1\Bigg{]}^{-1}, \tag{9}\]  
where the pair \(k,\ell\) takes values \(s,h\) when S-H link is considered and \(h,p\), when H-G link is considered. Next, the various parameters included in (8) and (9) will be analytically discussed. More specifically, in (8), \(\sigma^{2}\) denotes the Rytov variance and is given by  
\[ \begin{split}\sigma^{2}&=2.25\xi_{1}^{7/6}(h_{k}-h _{\ell})^{5/6}\sec(\theta)^{11/6}\\ &\quad\times\int_{h_{\ell}}^{h_{k}}C_{n}^{2}(h)\bigg{(}1-\frac{h- h_{\ell}}{h_{k}-h_{\ell}}\bigg{)}^{5/6}\bigg{(}\frac{h-h_{\ell}}{h_{k}-h_{\ell}} \bigg{)}^{5/6}dh.\end{split}  \tag{10}\]Moreover, \(C_{n}^{2}(h)\) denotes the refractive index structure parameter, which is defined as [28]  
\[ \begin{split} C_{n}^{2}(h)&=0.00594\Big{(}\frac{w}{27} \Big{)}^{2}\left(10^{-5}h\right)^{10}\exp\left(-\frac{h}{1000}\right)\\ &+2.7\cdot 10^{-16}\exp\left(-\frac{h}{1500}\right)+C_{n}^{2}(0) \exp\left(-\frac{h}{100}\right),\end{split}  \tag{11}\]  
where \(C_{n}^{2}(0)=1.7\times 10^{-14}m^{-2/3}\) and \(w\) denotes the wind velocity. Moreover, in (8), \(W_{0}\) denotes the beam size at the transmitter, while the corresponding parameter at the receiver is given by \(W=W_{0}\sqrt{\Theta^{2}+\Lambda^{2}}\), where \(\Theta=1-\frac{d_{k}}{R_{0}}\) and \(\Lambda=\frac{2d_{k}}{k_{1}W_{0}^{2}}\). Furthermore, \(F_{0}\) denotes the phase front radius of the curvature of the beam at the transmitter and \(d_{k}=\frac{h_{k}}{\cos(\theta)}\).