# Hybrid Cooperative Spectrum Sharing Protocols Based on OMA and NOMA in Cognitive Radio Networks
## 2 System Model and Protocol Descriptions
### Throughput of PUs
Hence the state transition probabilities are expressed by:  
\[P_{1}=1-p_{\mathrm{PT}\rightarrow\mathrm{PR}}^{out}=exp\bigg{(}-\frac{ \varphi_{P}}{\rho_{P}\lambda_{PP}}\bigg{)}, \tag{24}\] \[P_{2}=1-p_{\mathrm{PT}\rightarrow\mathrm{PR}}^{out,I}=exp\Big{(} -\frac{\varphi_{P}}{\lambda_{PP}\rho_{P}}-\frac{\varphi_{P}}{\lambda_{PP}\rho _{P}}\bigg{)}\cdot exp\Big{(}-\frac{\varphi_{P}}{\lambda_{SP}\rho_{1}}\bigg{)},\] \[+\frac{\varphi_{P}\lambda_{SP}}{\varphi_{P}\rho_{1}\lambda_{SP} +\varphi_{P}\lambda_{PP}}exp\bigg{(}-\frac{\varphi_{P}}{\rho_{P}\lambda_{PP}} \bigg{)}\Big{(}1-exp\Big{(}-\frac{\varphi_{P}\rho_{P}\lambda_{P}}{\rho_{P} \lambda_{PP}}-\frac{\varphi_{P}}{\rho_{1}\lambda_{SP}}\bigg{)}\bigg{)},\] \[P_{3}=p_{\mathrm{PT}\rightarrow\mathrm{PR}}^{out}\cdot p_{ \mathrm{PT}\rightarrow\mathrm{ST}}^{out}=\Big{(}1-exp\Big{(}-\frac{\varphi_{P}}{ \rho_{P}\lambda_{PP}}\Big{)}\Big{)}\cdot(1-exp\Big{(}-\frac{\varphi_{P}}{ \rho_{P}\lambda_{PP}}\Big{)}\Big{)},\] \[P_{4}=p_{\mathrm{PT}\rightarrow\mathrm{PR}}^{out}\cdot(1-p_{ \mathrm{PT}\rightarrow\mathrm{ST}}^{out})=\Big{(}1-exp\Big{(}-\frac{\varphi_{P}}{ \rho_{P}\lambda_{PP}}\Big{)}\Big{)}\cdot exp\Big{(}-\frac{\varphi_{P}}{\rho_{P} \lambda_{P}\lambda_{P}}\Big{)},\] \[P_{5}=1-p_{\mathrm{ST}\rightarrow\mathrm{PR}}^{out,\mathrm{NOMA}}= \begin{cases}exp\Big{(}-\frac{\varphi_{P}}{\alpha_{P}\rho_{S}\lambda_{SP}- \varphi_{P}\alpha_{S}\rho_{S}\lambda_{SP}}\Big{)},&\frac{\alpha_{P}}{\alpha_{S} }\geq\varphi_{P}\\ 0,&\frac{\alpha_{P}}{\alpha_{S}}<\varphi_{P}\end{cases} .\]The PU's packet is successfully transmitted with the probability \(P_{1}\), \(P_{2}\), \(P_{5}\), and \(P_{1}\) at the state S\({}_{1}\), S\({}_{2}\), S\({}_{3}\), and S\({}_{4}\), respectively. The PU's throughput for the hybrid cooperative CR network base on NOMA and OMA can be written as:  
\[ \begin{array}{l}\Omega_{PU}^{\frac{{\rm OMA}}{{\rm NOMA}}}=P_{1}\pi_{{\rm S} _{1}}+P_{2}\pi_{{\rm S}_{2}}+P_{5}\pi_{{\rm S}_{3}}+P_{1}\pi_{{\rm S}_{4}}\\ =\frac{P_{1}}{1+2P_{1}+P_{2}+P_{3}-P_{1}P_{2}}+\frac{P_{1}P_{2}}{1+2P_{1}+P_{4} +P_{3}-P_{1}P_{2}}\\ +\frac{P_{2}P_{1}}{1+2P_{1}+P_{4}+P_{3}-P_{1}P_{2}}+\frac{P_{1}P_{2}-P_{2}^{2}} {1+2P_{1}+P_{4}+P_{3}-P_{1}P_{2}}\\ =\frac{P_{1}+P_{1}P_{2}+P_{3}P_{4}+P_{3}P_{4}+P_{3}-P_{1}P_{2}^{2}-P_{1}P_{2}} {1+2P_{1}+P_{4}+P_{3}-P_{1}P_{2}}=\frac{B}{A}.\end{array}  \tag{25}\]  
Subsiting Equation (24) into Equation (25), we can obtain:  
\[ \begin{array}{l}A=2+exp\Big{(}-\frac{\varphi_{p}}{\rho_{{\rm P}}\lambda_{P} }\Big{)}-\frac{\rho_{{\rm P}}\lambda_{P}}{\varphi_{{\rm P}}\lambda_{{\rm S}} \varphi_{{\rm P}}\lambda_{{\rm S}}\varphi_{{\rm P}}\lambda_{{\rm P}}}exp\Big{(} -\frac{2\varphi_{p}}{\rho_{{\rm P}}\lambda_{{\rm P}}\lambda_{{\rm P}}}\Big{)} \Big{(}1-exp\Big{(}-\frac{\varphi_{{\rm P}}\lambda_{{\rm P}}\varphi_{{\rm P}} }{\rho_{{\rm P}}\lambda_{{\rm P}}}-\frac{\varphi_{{\rm P}}\lambda_{{\rm P}}}{ \rho_{{\rm P}}\lambda_{{\rm P}}}\Big{)}\Big{)}\\ -exp\Big{(}-\frac{2\varphi_{p}}{\rho_{{\rm P}}\lambda_{{\rm P}}\lambda_{{\rm P }}}\Big{)}exp\Big{(}-\frac{\varphi_{p}\varphi_{{\rm P}}\lambda_{{\rm P}}}{ \rho_{{\rm P}}\lambda_{{\rm P}}}-\frac{\varphi_{{\rm P}}\lambda_{{\rm P}}}{ \rho_{{\rm P}}\lambda_{{\rm P}}}\Big{)}.\end{array}  \tag{26}\]  
and the closed-form of B is related to the relationship between \(\alpha_{P}/\alpha_{S}\) and \(\varphi_{P}\), i.e.,  
\[