# Stable Throughput Analysis of Heterogeneous Hybrid FSO/RF Networks with Cognitive Radio Capability
## 4 Stable Throughput of Cooperative System (COG)
### Simultaneous Multipacket Transmission Mode  
In this system, \(N=3\) for HY and \(N=2\) for RO, and the success rates of packet transmission for \(j\in\{\texttt{RO,HY}\}\)\(v_{j,n,0}\) and \(v_{j,n,1}\) expressions are given in Appendix A.2, then we have the service rate \(\mu_{j,n}\) which is given in (8). Referring to the state transition diagram presented by Figure 3, the balance equation of \(Q_{\texttt{HY}}\) is given in (6) and (7). A simple manipulation of the above balance equations gives  
\[\pi_{\texttt{HY,}j} = \pi_{\texttt{HY,}j-1}\frac{\lambda_{\texttt{HY}}-\frac{\pi_{ \texttt{HY,2}}(\mu_{\texttt{HY,2}}+\mu_{\texttt{HY,3}})}{\pi_{\texttt{HY,1}-1} }-\frac{\pi_{\texttt{HY,1}+2\mu_{\texttt{HY,1}}}}{\pi_{\texttt{HY,2}-2}}}{\mu _{\texttt{HY,1}}+\mu_{\texttt{HY,2}}+\mu_{\texttt{HY,3}}}=\pi_{\texttt{HY,1}-1 }\frac{\lambda_{\texttt{HY}}-\psi_{\texttt{HY,1}-1}}{\mu_{\texttt{HY,1}}+\mu_{ \texttt{HY,2}}+\mu_{\texttt{HY,3}}}, \tag{26}\] \[\text{If }\psi_{\texttt{HY,0}}=\psi_{\texttt{HY,1}}=\psi_{ \texttt{HY,2},...}=\psi_{\texttt{HY}},\text{ then }\psi_{\texttt{HY}}\text{ with }0<\psi_{\texttt{HY}}<1\text{ is a solution of}\] \[(\lambda_{\texttt{HY}}-\psi_{\texttt{HY}})(\mu_{\texttt{HY,1}}+ \mu_{\texttt{HY,2}}+\mu_{\texttt{HY,3}})^{3}+(\mu_{\texttt{HY,2}}+\mu_{ \texttt{HY,3}})(\mu_{\texttt{HY,1}}+\mu_{\texttt{HY,2}}+\mu_{\texttt{HY,3}})( \lambda_{\texttt{HY}}-\psi_{\texttt{HY}})\] \[+\mu_{\texttt{HY,3}}(\lambda_{\texttt{HY}}-\psi_{\texttt{HY}})^{3} -\lambda_{\texttt{HY}}(\mu_{\texttt{HY,1}}+\mu_{\texttt{HY,2}}+\mu_{\texttt{HY,3 }})^{3}=0. \tag{27}\]  
With \(\psi_{\texttt{HY}}\) in Equation (27), the probability of \(Q_{\texttt{HY}}=0\) is obtained by  
\[\Pr(Q_{\texttt{HY}}=0)=1-\frac{\lambda_{\texttt{HY}}-\psi_{\texttt{HY}}}{\mu _{\texttt{HY,1}}+\mu_{\texttt{HY,2}}+\mu_{\texttt{HY,3}}}. \tag{28}\]  
The RO system now transmits one or two packets in a single time slot, the state transition diagram of which is given in Figure 2. The balance equations of \(Q_{\texttt{RO}}\) are the same to the equations in (14) by replacing \(\mu_{\texttt{HY,1}}\), \(\mu_{\texttt{HY,2}}\) with \(\mu_{\texttt{RO,1}}\), \(\mu_{\texttt{RO,2}}\), respectively. Figure 3: The buffer state transition diagram with up to three-packet service in a single time slot. Thus, in the first dominant system, the service rates for \(Q_{\mathtt{HY}}\) and \(Q_{\mathtt{RO}}\) are obtained by  
\[\mu^{(1)}_{\mathtt{HY},n\in\{1,2,3\}} = v_{\mathtt{HY},n,1}, \tag{29}\] \[\mu^{(1)}_{\mathtt{RO},l\in\{1,2\}} = \left(1-\frac{\lambda^{(1)}_{\mathtt{HY}}-\psi^{(1)}_{\mathtt{HY} }}{\mu^{(1)}_{\mathtt{HY},1}+\mu^{(1)}_{\mathtt{HY},2}+\mu^{(1)}_{\mathtt{HY},3}}\right)v_{\mathtt{RO},l,0}+\frac{\lambda^{(1)}_{\mathtt{HY}}-\psi^{(1)}_{ \mathtt{HY}}}{\mu^{(1)}_{\mathtt{HY},1}+\mu^{(1)}_{\mathtt{HY},2}+\mu^{(1)}_ {\mathtt{HY},3}}v_{\mathtt{RO},l,1}, \tag{30}\]  
where \(\psi^{(1)}_{\mathtt{HY}}\) is obtained by a solution of the equation given similar to (27). The equation should be reconstructed by replacing \(\mu_{\mathtt{HY},1}\), \(\mu_{\mathtt{HY},2}\), \(\mu_{\mathtt{HY},3}\), and \(\lambda_{\mathtt{HY}}\) with \(\mu^{(1)}_{\mathtt{HY},1}\), \(\mu^{(1)}_{\mathtt{HY},2}\), \(\mu^{(1)}_{\mathtt{HY},3}\), and \(\lambda^{(1)}_{\mathtt{HY}}\), respectively.