## 1 Introduction  
Increasing demand for deploying wireless sensor nodes usually raises battery and spectrum issues. An idea of reducing battery consumption in sending the sensed data is to put a relay node, which is easily charged, between the sensors and the data sink. Minimizing the transmit power at the sensors with an intermediate relay has been investigated in [1,2]. The relay could be regarded as a gateway node that provides an interface between a group of sensor nodes and the data center network. Since wireless sensors normally exchange the data through RF channels, to mitigate a spectrum burden between the relay and the data sink, free-space optical (FSO) communication is considered in this paper. FSO used instead of RF also can enhance transmission security by letting eavesdroppers hard to access the communication link. FSO communication is already known as a cost-effective way of constructing a high-speed data tunnel between radio-frequency (RF) access network and optical fiber-based backbone network [3]. Mixed RF-FSOsystems have attracted a growing interest in a dual-hop configuration to support both RF-to-FSO and FSO-to-RF relaying. As a relaying technique for the mixed RF-FSO systems, both decoding-and-forward (DF) and amplifying-and-forward (AF) relaying have been widely considered under the assumption of either intensity-modulation/direct-detection (IM/DD) or heterodyne detection (HD) in the FSO reception side. HD is known to be more complicated than IM/DD while HD offers superior performance [4]. As RF link is affected by severe fading, FSO link experiences a similar fading phenomenon, so-called atmospheric turbulence that causes fluctuations in the refractive index [5]. Furthermore, FSO link also suffers from the pointing error that refers to the misalignment between the transmitter and the receiver due to dynamic wind loads or weak earthquakes [6]. In the evaluation of the mixed RF-FSO dual-hop systems in previous studies, RF link was represented by various models including Rayleigh, Rician, \(\kappa\)-\(\mu\), Nakagami-\(m\), generalized Nakagami-\(m\), generalized-\(K\) and Extended Generalized-\(K\) (EGK) fading. On the other hand, FSO link is assumed to suffer from Malaga-\(M\), exponentiated-Weibull, Gamma-Gamma with pointing errors, and Double Generalized Gamma (DGG) (see [7] and the references therein). In the dual-hop AF systems, the signal from the source received at the relaying node after suffering from a fading channel is amplified with gain \(G\). Since the maximum output power at the relay is usually limited, the amplifying gain should be bounded by  
\[G\leq\sqrt{\frac{P_{r}}{P_{s}|h_{1}|^{2}+\sigma}}\stackrel{{\text {def}}}{{=}}G^{U}, \tag{1}\]  
where \(P_{s}\) and \(P_{r}\) are transmitted power at the source and the maximum available power at the relay, respectively, \(|h_{1}|^{2}\) is the channel power of the first hop and \(\sigma\) is the noise power level [8; 9]. When the maximal gain \(G^{U}\) is used, the end-to-end signal-to-noise power ratio (SNR, \(\gamma_{e}\)) of the system is represented by  
\[\gamma_{e}=\frac{\gamma_{1}\gamma_{2}}{\gamma_{1}+\gamma_{2}+1}, \tag{2}\]  
where \(\gamma_{1}\) and \(\gamma_{2}\) are the SNR of the first and the second hop, respectively. For mixed RF/FSO systems, many works have been done to provide statistical analysis of \(\gamma_{e}\) that mainly determines the system performance including outage probability (OP), average bit-error rate (BER) and ergodic capacity, etc. ([7; 10] and the references therein). However, in the existing models, \(\gamma_{e}\) is not directly dealt with but an approximate version \(\gamma_{a}=\frac{\gamma_{1}\gamma_{2}}{\gamma_{1}+\gamma_{2}}\) is investigated instead of \(\gamma_{e}\)[7; 11]. \(\gamma_{a}\) is sometimes further approximated by \(\gamma_{m}=\min\{\gamma_{1},\gamma_{2}\}\)[10]. Obviously, \(\gamma_{e}<\gamma_{a}\leq\gamma_{m}\) and hence the existing models certainly overestimate the mixed RF/FSO AF relaying performances.