## 3 Performance Evaluation
### Outage Probability
\begin{array}{l}\Pr\left(\gamma_{\text{SR}_{c}}\geq u_{1},\gamma_{\text{R},\text{D}}\geq u_{2}|u_{2}>u_{1}\right)\\ =\sum\limits_{m=0}^{M-1}\left(-1\right)^{m}C_{M-1}^{m}M\left[\begin{array}{ l}\frac{\lambda_{\text{SR}}}{\lambda_{\text{SR}}+m\Omega}\exp\left(-\lambda_{ \text{RD}}u_{2}\right)\exp\left(-\left(\lambda_{\text{SR}}+m\Omega\right)u_{1 }\right)\\ +\frac{m}{m+1}\frac{\lambda_{\text{RD}}}{\lambda_{\text{SR}}+m\Omega}\exp \left(-\left(m+1\right)\Omega u_{2}\right)\end{array}\right].\end{array}  \tag{65}\]  
Now, with \(u_{1}=\frac{\rho}{\Delta Z_{1}}\) and \(u_{2}=\frac{\rho}{\Delta Z_{2}}\), the following are respectively obtained:  
\[ \begin{array}{l}\Pr\left(\gamma_{\text{SR}_{c}}\geq\frac{\rho}{\Delta z_{1} },\gamma_{\text{R},\text{D}}\geq\frac{\rho}{\Delta z_{2}}|z_{2}\geq z_{1}\right) \\ =\sum\limits_{m=0}^{M-1}\left(-1\right)^{m}C_{M-1}^{m}M\times\left[\begin{array}{ l}\frac{\lambda_{\text{RD}}}{\lambda_{\text{RD}}+m\Omega}\exp\left(-\frac{\lambda_{ \text{SR}}\rho}{\Delta}\frac{1}{z_{1}}\right)\exp\left(-\frac{\left(\lambda_{ \text{RD}}+m\Omega\right)\rho}{\Delta}\frac{1}{z_{2}}\right)\\ +\frac{m}{m+1}\frac{\lambda_{\text{SR}}}{\lambda_{\text{RD}}+m\Omega}\exp \left(-\frac{\left(m+1\right)\Omega\rho}{\Delta}\frac{1}{z_{1}}\right)\end{array} \right],\\ \Pr\left(\gamma_{\text{SR}_{c}}\geq\frac{\rho}{\Delta z_{1}},\gamma_{\text{R},\text{D}}\geq\frac{\rho}{\Delta z_{2}}|z_{1}>z_{2}\right)\\ =\sum\limits_{m=0}^{M-1}\left(-1\right)^{m}C_{M-1}^{m}M\left[\begin{array}{ l}\frac{\lambda_{\text{SR}}}{\lambda_{\text{SR}}+m\Omega}\exp\left(-\frac{\lambda_{ \text{RD}}\rho}{\Delta}\frac{1}{z_{2}}\right)\exp\left(-\frac{\left(\lambda_{ \text{SR}}+m\Omega\right)\rho}{\Delta}\frac{1}{z_{1}}\right)\\ +\frac{m}{m+1}\frac{\lambda_{\text{RD}}}{\lambda_{\text{SR}}+m\Omega}\exp \left(-\frac{\left(m+1\right)\Omega\rho}{\Delta}\frac{1}{z_{2}}\right)\end{array} \right].\end{array}  \tag{66}\]Moreover, \(\mathrm{OP_{C-ORS}}\) in (57) can be formulated as  
\[ \begin{array}{ll}\mathrm{OP_{C-ORS}}&=1-\int_{0}^{+\infty}\int_{0}^{2z_{2}} \mathrm{Pr}\left(\gamma_{\mathrm{SR}_{c}}\geq\frac{\rho}{\Delta z_{1}},\gamma_{ \mathrm{R}_{c}\mathrm{D}}\geq\frac{\rho}{\Delta z_{2}}|z_{2}\geq z_{1}\right) f_{Z_{1}}\left(z_{1}\right)f_{Z_{2}}\left(z_{2}\right)dz_{1}dz_{2}\\ &-\int_{0}^{+\infty}\int_{0}^{z_{1}}\mathrm{Pr}\left(\gamma_{\mathrm{SR}_{c}} \geq\frac{\rho}{\Delta z_{1}},\gamma_{\mathrm{R}_{c}\mathrm{D}}\geq\frac{ \rho}{\Delta z_{2}}|z_{1}\geq z_{2}\right)f_{Z_{1}}\left(z_{1}\right)f_{Z_{2}} \left(z_{2}\right)dz_{1}dz_{2},\end{array}  \tag{68}\]  
where the PDFs of \(Z_{1}\) and \(Z_{2}\) can be obtained from their CDFs, i.e.,  
\[f_{Z_{1}}\left(z_{1}\right) =\sum\limits_{t=0}^{K-1}\sum\limits_{n=0}^{N}\left(-1\right)^{n} \frac{C_{N}^{n}}{t!}\left(\frac{\lambda_{\mathrm{BS}}}{\kappa}\right)^{t} \left(\frac{\lambda_{\mathrm{BS}}}{\kappa}z_{1}^{t}-n\lambda_{\mathrm{SP}} \delta z_{1}^{t-2}-tz_{1}^{t-1}\right)\exp\left(-\frac{\lambda_{\mathrm{BS}}}{ \kappa}z_{1}-\frac{n\lambda_{\mathrm{SP}}\delta}{z_{1}}\right), \tag{69}\] \[f_{Z_{2}}\left(z_{2}\right) =\sum\limits_{w=0}^{K-1}\sum\limits_{q=0}^{N}\left(-1\right)^{q }\frac{C_{N}^{q}}{q!}\left(\frac{\lambda_{\mathrm{BR}}}{\kappa}\right)^{w} \left(\frac{\lambda_{\mathrm{BR}}}{\kappa}z_{2}^{w}-q\lambda_{\mathrm{RP}} \delta z_{2}^{w-2}-wz_{2}^{w-1}\right)\exp\left(-\frac{\lambda_{\mathrm{BR}}}{ \kappa}z_{2}-\frac{q\lambda_{\mathrm{RP}}\delta}{z_{2}}\right). \tag{70}\]  
Substituting (66), (67), (69), and (70) into (68), Equation (56) can be obtained to finish the proof. However, the exact expression of \(\mathrm{OP_{C-ORS}}\) is still in integral form, which is difficult to use for designing and optimizing the system. This motivates us to derive the approximate closed-form expression for \(\mathrm{OP_{C-ORS}}\). Lemma 4: _When \(1-\tau_{\mathrm{D}}^{2}\theta>0\), \(\mathrm{OP_{C-ORS}}\) can be approximated by a closed-form expression as given in (71) at the top of next page._  
\[\mathrm{OP_{C-ORS}}\approx 1-\left\{