## 2 System Model and Fading Statistics  
We consider an asymmetric dual-hop mixed RF/FSO AF relaying communication system where a source sensor (S) and a destination data center (D) are communicating through an intermediate relay node (R) (as in Figure 1). We assume that S-R link is RF and R-D link is FSO (FSO/RF link also can be similarly treated with the following analysis). This model is applicable to any sensor-network scenario that consists of low-power RF link among remote sensors (or terminals) and mid/long-distance FSO-relaying link between an intermediate cluster head (or gateway) and an information sink. In _Project Loon_, for an example, hot air balloons are connected using FSO mesh links to provide LTE-level internet services to infrastructure-poor areas [15]. Airborne weather sensors linked by RF in a balloon also can be connected to an information center on the ground using FSO link. Let \(\gamma_{RF}\) and \(\gamma_{FSO}\) denote the SNR of RF and FSO link, respectively. Since we consider a tightly power-constrained amplifying gain at the relay [8; 9], the end-to-end SNR is given by  
\[\gamma_{e}=\frac{\gamma_{RF}\gamma_{FSO}}{\gamma_{RF}+\gamma_{FSO}+1}. \tag{3}\]  
Figure 1: The system model. It is noted that the orders of RF-FSO and FSO-RF are equivalent in terms of the end-to-end SNR if the tightly power-constrained amplifying gain is used. If we assume that RF link suffers from Nakagami-\(m\) fading, \(\gamma_{RF}\) follows a Gamma distribution with fading parameter \(m\) and \(\Omega\) and an average SNR \(\tilde{\gamma}_{RF}\), the probability density function (PDF) of which is given by  
\[f_{\gamma_{RF}}(x)=\left(\frac{m}{\Omega\tilde{\gamma}_{RF}}\right)^{m}\frac{x^ {m-1}}{\Gamma(m)}e^{-\frac{m}{\Gamma\tilde{\gamma}_{RF}}x}, \tag{4}\]  
where \(\Gamma(\cdot)\) is a Gamma function such that \(\Gamma(x)=\int_{0}^{\infty}z^{x-1}e^{-z}dz\). For FSO link, let \(\eta\) and \(I\) denote the effective photo-electric conversion ratio and the channel coefficient of the FSO link, respectively. Then the SNR of FSO link is modeled by \(\gamma_{FSO}=(\eta I)^{w}/N_{0}\)[7], where \(N_{0}\) is the variance of zero-mean white Gaussian noise at the destination and \(w\in\{1,2\}\) represents a specific detection technique (i.e., \(w=1\) and 2 account for HD and IM/DD, respectively). It is further assumed that \(I=I_{l}I_{p}I_{f}\), where \(I_{l}\) is the path loss, \(I_{p}\) the pointing error, and \(I_{f}\) is the fading caused by atmospheric turbulence. We assume \(I_{l}=1\), since \(I_{l}\) is deterministic [16], whereas \(I_{p}\) and \(I_{f}\) are probabilistic with PDFs \(f_{I_{p}}(x)\) and \(f_{I_{f}}(x)\). According to [17], the PDF of \(I_{p}\) is given by  
\[f_{I_{p}}(x)=\frac{\delta^{2}}{\Phi^{\delta^{2}}}x^{\delta^{2}-1},\ \ 0\leq x\leq\Phi, \tag{5}\]  
where \(\delta\) is a ratio between the equivalent beam radius and the pointing error displacement standard deviation at the receiver (i.e., the larger the delta, the smaller the effect of the pointing error) and \(\Phi\) is a fraction of the collected power at radial displacement \(r=0\). The PDF of \(I_{f}\) is given in [3] such that  
\[f_{I_{f}}(x)=\frac{2(\alpha\beta)^{(\alpha+\beta)/2}}{\Gamma(\alpha)\Gamma( \beta)}x^{(\alpha+\beta)/2-1}K_{\alpha-\beta}(2\sqrt{\alpha\beta x}),\ x>0, \tag{6}\]  
where \(\alpha\) and \(\beta\) are atmospheric turbulence parameters and \(K_{a}(\cdot)\) is the modified Bessel function of the second kind of order \(a\).