## 3 Performance Evaluation  
### Outage Probability  
Generally, the e2e OP of the protocol U, U \(\in\left\{\text{H-PRS},\text{B-ORS},\text{C-ORS}\right\},\) can be expressed as follows:  
\[ \begin{array}{ll}\text{OP}_{\text{U}}&=\Pr\left(\min\left(\Psi_{1l},\Psi_{2 l}\right)<\theta\right)=1-\Pr\left(\min\left(\Psi_{1l},\Psi_{2l}\right)\geq \theta\right)\\ &=1-\Pr\left(\Psi_{1l}\geq\theta,\Psi_{2l}\geq\theta\right),\end{array}  \tag{30}\]  
where \(l\in\left\{a_{1},a_{2},b,c\right\}\) and  
\[\theta=2^{\frac{2C_{\text{th}}}{(1-\alpha)\text{T}}}-1. \tag{31}\]  
Moreover, substituting (15) and (16) into (30), yields  
\[\text{OP}_{\text{U}}=1-\Pr\left(\left(1-\tau_{\text{D}}^{2}\theta\right) \Delta\min\left(\kappa X_{0}^{\text{sum}},\frac{\delta}{\tau_{0}^{\text{max} }}\right)\gamma_{\text{SR}_{l}}\geq\theta,\left(1-\tau_{\text{D}}^{2}\theta \right)\Delta\min\left(\kappa X_{l}^{\text{sum}},\frac{\delta}{\tau_{l}^{\text {max}}}\right)\gamma_{\text{R}_{\text{U}}\text{D}}\geq\theta\right). \tag{32}\]  
It is obvious from (32) that \(\text{OP}_{\text{U}}=1,\) if \(1-\tau_{\text{D}}^{2}\theta\leq 0\). In the case that \(1-\tau_{\text{D}}^{2}\theta>0\), Equation (32) can be expressed under the following form:  
\[\text{OP}_{\text{U}}=1-\Pr\left(\min\left(\kappa X_{0}^{\text{sum}},\frac{ \delta}{\tau_{0}^{\text{max}}}\right)\gamma_{\text{SR}_{l}}\geq\frac{\rho}{ \Delta},\min\left(\kappa X_{l}^{\text{sum}},\frac{\delta}{\tau_{l}^{\text{max} }}\right)\gamma_{\text{R}_{\text{I}}\text{D}}\geq\frac{\rho}{\Delta}\right), \tag{33}\]  
where \(\rho=\theta/\left(1-\tau_{\text{D}}^{2}\theta\right).\)  
**Lemma 1**. : _As \(1-\tau_{\text{D}}^{2}\theta>0\), exact closed-form expressions of \(\text{OP}_{\text{PRS1}}\) and \(\text{OP}_{\text{PRS2}}\) can be given, respectively, as \[\text{OP}_{\text{PRS1}}=1- \left[ \begin{array}{c}\sum\limits_{t=0}^{K-1}\sum\limits_{m=0}^{M-1 }(-1)^{m}\frac{2C_{M-1}^{m}M}{t!}(m+1)^{\frac{t+1}{2}}\left(\frac{\lambda_{\text {BS}}\lambda_{\text{SR}}\rho}{\kappa\Delta}\right)^{\frac{t+1}{2}}K_{1-t} \left(2\sqrt{\frac{(m+1)\,\lambda_{\text{BS}}\lambda_{\text{SR}}\rho}{\kappa \Delta}}\right)\\ -\sum\limits_{t=0}^{K-1}\sum\limits_{n=1}^{N}\sum\limits_{m=0}^{M-1}(-1)^{n+ m+1}C_{M-1}^{m}C_{N}^{n}\frac{2M\lambda_{\text{SR}}}{t!}\left(\frac{\rho}{n \lambda_{\text{SR}}\delta\Delta+(m+1)\,\lambda_{\text{SR}}\rho}\right)^{\frac {1-t}{2}}\left(\frac{\lambda_{\text{BS}}\rho}{\kappa\Delta}\right)^{\frac{t+1 }{2}}\\ \times K_{1-t}\left(2\sqrt{\frac{\lambda_{\text{BS}}}{\kappa\Delta}\left(n \lambda_{\text{SP}}\delta\Delta+(m+1)\,\lambda_{\text{SR}}\rho\right)}\right) \end{array} \right] \tag{34}\] \[\times\left[ \begin{array}{c}\sum\limits_{t=0}^{K-1}\frac{2}{t!} \left(\frac{\lambda_{\text{BR}}\lambda_{\text{RD}}\rho}{\kappa\Delta}\right)^ {\frac{t+1}{2}}K_{1-t}\left(2\sqrt{\frac{\lambda_{\text{BR}}\lambda_{\text{RD} }\rho}{\kappa\Delta}}\right)\\ -\sum\limits_{t=0}^{K-1}\sum\limits_{n=1}^{N}(-1)^{n+1}C_{N}^{n}\frac{2\lambda _{\text{RD}}}{t!}\left(\frac{\rho}{n\lambda_{\text{RP}}\delta\Delta+\lambda_{ \text{RD}}\rho}\right)^{\frac{1-t}{2}}\left(\frac{\lambda_{\text{BR}}\rho}{ \kappa\Delta}\right)^{\frac{t+1}{2}}\\ \times K_{1-t}\left(2\sqrt{\frac{\lambda_{\text{BR}}}{\kappa\Delta}\left(n \lambda_{\text{RP}}\delta\Delta+\lambda_{\text{RD}}\rho\right)}\right)\end{array}  \right],\]  
\[\text{OP}_{\text{PRS2}}=1- \left[