# Stable Throughput Analysis of Heterogeneous Hybrid FSO/RF Networks with Cognitive Radio Capability
## 2 System Models
### Physical Layer Model  
As shown in Figure 1, let \(h_{i}^{j}\) denote the channel between communicating pair \(j\in\{\texttt{RO},\texttt{HY}\}\) over the frequency channel \(i\in\{f_{0},f_{1},f_{2}\}\). When a pair \(j\) is communicating on an RF channel, but the other pair \(k\) with CR falsely senses it to be idle, \(k\) may send a signal using the channel and cause interference at the receiver of \(j\). Interfering channel from the transmitter of \(k\) to the receiver of the other pair \(j\) over channel \(l\) is denoted by \(g_{l}^{j}\). The FSO channel is assumed not to be borrowed. And, thus \(l\in\{f_{1},f_{2}\}\). We assume that RF propagation over any transmitter-receiver pair is an independent stationary Rayleigh flat-fading channel and FSO link is an independent stationary Gamma-Gamma atmospheric turbulence-induced fading.  
Let \(x_{i}^{j}\) denote a transmit signal, with \(E[|x_{i}^{j}|^{2}]=1\), between communication pair \(j\in\{\texttt{RD},\texttt{HY}\}\) over frequency channel \(i\in\{f_{0},f_{1},f_{2}\}\). And let \(P_{i}^{j}\) denote transmitting power of \(x_{i}^{j}\). Let \(\theta_{i}^{j}\) be a binary variable that indicates whether an interfering signal exits at the receiver of \(j\) (\(\theta_{i}^{j}=1\)) or not (\(\theta_{i}^{j}=0\)). Since the FSO channel is not rented, \(\theta_{i}^{j}=0\) if \(i=f_{0}\). Then, the received signal for pair \(j\) over channel \(i\) is  
\[y_{i}^{j}=\sqrt{p_{i}^{j}}h_{i}^{j}x_{i}^{j}+\theta_{i}^{j}\sqrt{p_{i}^{k}}g_ {i}^{j}x_{i}^{k}+n_{i}^{j}, \tag{1}\]  
where \(h_{i}^{j}\) and \(g_{i}^{j}\) are independently and identically distributed according to \(CN(0,\sigma_{i}^{j})\) and \(CN(0,\sigma_{i}^{j})\), respectively. \(k\) (\(k\
eq j,k\in\{\texttt{RO},\texttt{HY}\}\)) denotes an interfering transmitter, and  
Figure 1: The heterogeneous hybrid FSO/RF systems considered in this study.  
is an additive white Gaussian noise (AWGN) at the receiver of \(j\). Hereafter, we assume that the AWGN at each receiver is independently and identically distributed according to \(CN(0,N_{0})\).  
Then, the SNR or signal-to-interference-plus-noise ratio (SINR) is given by  
\[\gamma_{i}^{j}=\frac{P_{i}^{j}|h_{i}^{j}|^{2}}{N_{0}+\theta_{i}^{j}P_{i}^{k}| \delta_{i}^{j}|^{2}}. \tag{2}\]  
We assume that a receiving node can decode a packet successfully if the received SNR or SINR is greater than or equal to a certain threshold \(\delta_{i}^{j}\). Let \(s_{i\in\{f_{1},f_{2}\},j\in\{\text{ RD},\text{HY}\}}\) denote the probability of successfully decoding a packet without interference(\(\theta_{i}^{j}=0\)) through the RF link \(i\in\{f_{1},f_{2}\}\), and it is given by  
\[s_{i,j}=\text{Pr}(\gamma_{i}^{j}>\delta_{i}^{j}|\theta_{i}^{j}=0)=exp\left(- \frac{N_{0}\delta_{i}^{j}}{P_{i}^{j}\sigma_{i}^{j}}\right). \tag{3}\]  
And let \(s_{i,j}^{l}\) denote the probability of successfully decoding a packet with interference (\(\theta_{i}^{j}=1\)), and it given by  
\[s_{i,j}^{l}=\text{Pr}(\gamma_{i}^{j}>\delta_{i}^{j}|\theta_{i}^{j}=1)=\frac{P _{i}^{j}\sigma_{i}^{j}}{P_{i}^{j}\sigma_{i}^{j}+P_{i}^{k}\sigma_{i}^{j}}exp \left(-\frac{N_{0}\delta_{i}^{j}}{P_{i}^{j}\sigma_{i}^{j}}\right). \tag{4}\]  
Let \(s_{f_{0}}^{HY}\) denote the probability of successfully decoding a packet through an FSO link, and it is given by  
\[s_{f_{0},HY}=\frac{\xi^{2}}{\Gamma(\alpha)\Gamma(\beta)}G_{2,4}^{4,0}\left[ \alpha\beta\omega\delta_{f_{0}}^{HY}/\eta\right]\begin{array}{ccc}1&\xi^{2}+ 1\\ 0&\xi^{2}&\alpha&\beta\end{array}\bigg{]}, \tag{5}\]  
where \(\alpha\), \(\beta\), and \(\xi\) are the turbulence parameters, and \(\delta_{f_{0}}^{HY}\) and \(\eta\) are the target SNR of FSO link and average SNR of FSO link and \(o=\xi^{2}/(\xi^{2}+1)\) and \(\Gamma(\cdot)\) and \(G_{a,b}^{c,d}(\cdot)\) are gamma function and Meijer's G-function, respectively.