# Hybrid Cooperative Spectrum Sharing Protocols Based on OMA and NOMA in Cognitive Radio Networks
## 2 System Model and Protocol Descriptions
### Throughput of PUs  
In overlay OMA cooperative mode, ST acts as a relay to help forward the PU's packet to PR. The outage probability of the channel ST \(\rightarrow\) PR is expressed as:  
\[ \begin{array}{l}P_{\text{ST}\rightarrow\text{PR}}^{out,\text{OMA}}=Pr \Big{(}R_{\text{PT}\rightarrow\text{ST},x_{p}}^{\text{overlay,OMA}}>R_{P},R_{ \text{ST}\rightarrow\text{PR},x_{p}}^{\text{overlay,OMA}}<R_{P}\Big{)}\\ =Pr\Big{(}log_{2}\Big{(}1+\frac{|h_{PS}|^{2}P_{P}}{\sigma^{2}}\Big{)}>R_{P}, log_{2}\Big{(}1+\frac{|h_{PS}|^{2}P_{S}}{\sigma^{2}}\Big{)}<R_{P}\Big{)}\\ =Pr\Big{(}|h_{PS}|^{2}>\frac{\varphi_{P}}{\rho_{P}},|h_{PS}|^{2}<\frac{\varphi_ {P}}{\rho_{S}}\Big{)}=\int_{\frac{\varphi_{P}}{\rho_{P}}}^{\infty}\frac{\varphi _{P}}{\lambda_{PS}}e^{-\frac{1}{\lambda_{PS}}}\frac{1}{\lambda_{PS}}e^{-\frac{ 1}{\lambda_{PS}}}\frac{1}{\lambda_{SP}}e^{-\frac{1}{\lambda_{SP}}}dxdy\\ =\int_{\frac{\varphi_{P}}{\rho_{P}}}^{\infty}\frac{1}{\lambda_{PS}}e^{-\frac{ 1}{\lambda_{PS}}}\bigg{(}1-e^{-\frac{\varphi_{P}}{\lambda_{SP}}}\bigg{)}dy= exp\Big{(}-\frac{\varphi_{P}}{\rho_{P}\lambda_{PS}}\Big{)}\Big{(}1-exp \Big{(}-\frac{\varphi_{P}}{\rho_{S}\lambda_{SP}}\Big{)}\Big{)}.\end{array}  \tag{36}\]  
The PU's throughput for hybrid cooperative radio wireless network base on pure OMA is derived by the steady state probabilities and state transition probabilities of the Markov chain, given in Figure 3. The outage probabilities of the corresponding channels are given by Equations (21), (22) and (36). Hence, the state transition probabilities are expressed by:  
\[ \begin{array}{l}P_{1}=1-p_{\text{PT}\rightarrow\text{PR}}^{out}=exp\Big{(} -\frac{\varphi_{P}}{\rho_{P}\lambda_{PP}}\Big{)},\\ P_{2}=1-P_{\text{PT}\rightarrow\text{PR}}^{out,I}=exp\Big{(}-\frac{\varphi_{P} \rho_{P}}{\rho_{P}\lambda_{PP}}-\frac{\varphi_{P}}{\rho_{P}\lambda_{PP}}\Big{)} exp\Big{(}-\frac{\varphi_{P}}{\rho_{1}\lambda_{SP}}\Big{)}\\ +\frac{\varphi_{P}\lambda_{PP}}{\varphi_{P}\lambda_{SP}+\varphi_{P}\lambda_{PP} }\Big{(}-\frac{\varphi_{P}}{\rho_{P}\lambda_{PP}}\Big{)}\Big{(}1-exp\Big{(}- \frac{\varphi_{P}\rho_{P}}{\rho_{P}\lambda_{PP}}-\frac{\varphi_{P}}{\rho_{1} \lambda_{SP}}\Big{)}\Big{)},\\ P_{3}=p_{\text{PT}\rightarrow\text{PR}}^{out}\cdot P_{\text{PT} \rightarrow\text{ST}}^{out}=\Big{(}1-exp\Big{(}-\frac{\varphi_{P}}{\rho_{P} \lambda_{PP}}\Big{)}\Big{)}\Big{(}1-exp\Big{(}-\frac{\varphi_{P}}{\rho_{P} \lambda_{PS}}\Big{)}\Big{)},\\ P_{4}=p_{\text{PT}\rightarrow\text{PR}}^{out}\cdot\big{(}1-p_{\text{PT} \rightarrow\text{ST}}^{out}\big{)}=\Big{(}1-exp\Big{(}-\frac{\varphi_{P}}{\rho_ {P}\lambda_{PP}}\Big{)}\Big{)}exp\Big{(}-\frac{\varphi_{P}}{\rho_{P}\lambda_{ P}}\Big{)},\\ P_{5}^{\prime}=1-p_{\text{ST}\rightarrow\text{PR}}^{out,\text{OMA}}=1-exp\Big{(}-\frac{ \varphi_{P}}{\rho_{P}\lambda_{PS}}\Big{)}\Big{(}1-exp\Big{(}-\frac{\varphi_{P} }{\rho_{S}\lambda_{SP}}\Big{)}\Big{)}.\end{array}  \tag{37}\]  
The PU's packet is successfully transmitted with the probabilities \(P_{1}\),\(P_{2}\), \(P_{5}^{\prime}\), and \(P_{1}\) at the state \(S_{1}\), \(S_{2}\), \(S_{3}\), and \(S_{4}\), respectively. The PU's throughput for hybrid cooperative radio wireless network base on pure OMA can be written as:  
\[ \begin{array}{l}\pi_{PI}^{OMA}=P_{1}\pi_{S_{1}}+P_{2}\pi_{S_{2}}+P_{5}^{ \prime}\pi_{S_{3}}+P_{1}\pi_{S_{4}}\\ =\frac{P_{1}}{1+2P_{1}+P_{3}-P_{1}P_{2}}+\frac{P_{1}P_{2}}{1+2P_{1}+P_{4}+P_{3} -P_{1}P_{2}}\\ +\frac{P_{5}^{\prime}P_{4}}{1+2P_{1}+P_{4}+P_{3}-P_{1}P_{2}}+\frac{P_{3}P_{1}+ P_{2}^{\prime}-P_{1}^{\prime}P_{2}}{1+2P_{1}+P_{4}+P_{3}-P_{1}P_{2}}\\ =\frac{P_{1}+P_{1}P_{2}+P_{5}^{\prime}P_{4}+P_{3}P_{1}+P_{2}^{\prime}-P_{1}^{ \prime}P_{2}}{1+2P_{1}+P_{4}+P_{3}-P_{1}P_{2}}=\frac{B^{\prime}}{\lambda}\,\end{array}  \tag{38}\]  
where the closed-form of \(A\) is the same as Equation (26), and subsisting Equations (37) into Equation (38), we can obtain the closed-form of \(B^{\prime}\):\[