## 3 Performance Evaluation
### Outage Probability
\begin{array}{l}{\rm OP_{C-ORS}} ={\rm Pr}\left(\min\left(\min\left(\kappa X_{0}^{\rm sum},\frac{ \delta}{Y_{0}^{\rm sum}}\right)\gamma_{\rm SR_{c}},\min\left(\kappa X_{c}^{\rm sum },\frac{\delta}{Y_{c}^{\rm sum}}\right)\gamma_{\rm R_{c}D}\right)<\frac{\rho}{ \Delta}\right)\\ ={\rm Pr}\left(\min\left(Z_{1}\gamma_{\rm SR_{c}},Z_{2}\gamma_{\rm R_{c}D}\right)< \frac{\rho}{\Delta}\right)\\ =1-{\rm Pr}\left(\gamma_{\rm SR_{c}}\geq\frac{\rho}{\Delta Z_{1}},\gamma_{\rm R_{c }D}\geq\frac{\rho}{\Delta Z_{2}}\right),\end{array}  \tag{57}\]  
where \(Z_{2}=\min\left(\kappa X_{c}^{\rm sum},\frac{\delta}{Y_{0}^{\rm sum}}\right)\), and the CDF of \(Z_{2}\) is given, similar to \(Z_{1}\) in (44):  
\[ \begin{array}{lcl}F_{Z_{2}}\left(x\right)&=1-\sum\limits_{t=0}^{K-1}\frac{1}{ \Omega!}\left(\frac{\lambda_{\rm BB}}{\kappa}\right)^{t}x^{t}\exp\left(-\frac{ \lambda_{\rm BB}}{\kappa}x\right)\\ &-\sum\limits_{t=0}^{K-1}\sum\limits_{n=1}^{N}\left(-1\right)^{n}\frac{C_{N}^{m} }{\Omega!}\left(\frac{\lambda_{\rm BB}}{\kappa}\right)^{t}x^{t}\exp\left(-\frac{ \lambda_{\rm BB}}{\kappa}x-\frac{n\lambda_{\rm RP}\delta}{x}\right).\end{array}  \tag{58}\]  
Since \(\gamma_{\rm SR_{c}}\) and \(\gamma_{\rm R_{c}D}\) are not independent, the method in [49] can be used to calculate \({\rm OP_{C-ORS}}\). At first, using ([49], Equation (D.2)), we have  
\[{\rm Pr}\left(\gamma_{\rm SR_{c}}\geq u_{1},\gamma_{\rm R_{c}D}\geq u_{2} \right)=\int_{0}^{+\infty}\frac{\partial G\left(z\right)}{\partial z}\frac{f_{ \rm T_{max}}\left(z\right)}{f_{\rm T_{min}}\left(z\right)}dz, \tag{59}\]  
where \(T_{\rm m}=\min\left(\gamma_{\rm SR_{m}},\gamma_{\rm R_{m}D}\right),\)\(T_{\rm max}=\max\limits_{m=1,2,\sim,M}\left(T_{m}\right),\) and \(G\left(z\right)={\rm Pr}(\gamma_{\rm SR_{m}}\geq u_{1},\gamma_{\rm R_{m}D}\geq u_{2},\)\(\min\left(\gamma_{\rm SR_{m}},\gamma_{\rm R_{m}D}\right)<z)where \(\Omega=\lambda_{\text{SR}}+\lambda_{\text{RD}}\). Then, the PDF of \(T_{max}\) can obtained, similar to (40), as  
\[f_{\tau_{\text{max}}}\left(z\right)=\sum\limits_{m=0}^{M-1}\left(-1\right)^{m}C_{ M-1}^{m}M\Omega\exp\left(-\left(m+1\right)\Omega z\right). \tag{61}\]  
Considering the probability \(G\left(z\right)\) in (59); using ([49], Equation (D7)-(D8)), we have  
* **Case 1:**\(u_{1}\geq u_{2}\) \[\frac{\partial G\left(z\right)}{\partial z}=\left\{ \begin{array}{ll}0,&\text{ if }z<u_{2},\\ \lambda_{\text{RD}}\exp\left(-\lambda_{\text{SR}}u_{1}\right)\exp\left(-\lambda _{\text{RD}}z\right),&\text{if }u_{2}\leq z<u_{1},\\ \Omega\exp\left(-\Omega z\right),&\text{if }u_{1}\leq z.\end{array} \right.\] (62)  
Plugging (59)-(62) together, and after some manipulations, the following can be obtained:  
\[ \begin{array}{l}\Pr\left(\gamma_{\text{SR}_{c}}\geq u_{1},\gamma_{\text{R},\text{D}}\geq u_{2}|u_{1}\geq u_{2}\right)\\ =\sum\limits_{m=0}^{M-1}\left(-1\right)^{m}C_{M-1}^{m}M\left[\begin{array}{ l}\int_{u_{2}}^{u_{1}}\lambda_{\text{RD}}\exp\left(-\lambda_{\text{SR}}u_{1} \right)\exp\left(-\lambda_{\text{RD}}z\right)\exp\left(-m\Omega z\right)dz\\ +\int_{u_{1}}^{+\infty}\Omega\exp\left(-\left(m+1\right)\Omega z\right)dz\\ \frac{\lambda_{\text{RD}}}{\lambda_{\text{RD}}+m\Omega}\exp\left(-\lambda_{ \text{SR}}u_{1}\right)\exp\left(-\left(\lambda_{\text{RD}}+m\Omega\right)u_{2 }\right)\\ +\frac{m}{m+1}\frac{\lambda_{\text{SR}}}{\lambda_{\text{RD}}+m\Omega}\exp \left(-\left(m+1\right)\Omega u_{1}\right)\end{array}\right].\end{array}  \tag{63}\]
* **Case 2:**\(u_{1}<u_{2}\) \[\frac{\partial G\left(z\right)}{\partial z}=\left\{ \begin{array}{ll}0,&\text{ if }z<u_{1},\\ \lambda_{\text{SR}}\exp\left(-\lambda_{\text{RD}}u_{2}\right)\exp\left(-\lambda _{\text{SR}}z\right),&\text{if }u_{1}\leq z<u_{2},\\ \Omega\exp\left(-\Omega z\right),&\text{if }u_{2}\leq z.\end{array} \right.\] (64)  
Similarly, we obtain  
\[