## 3 Performance Evaluation
### Outage Probability
\begin{array}{ll}&\mathrm{OP_{B-ORS}}\\ &=1+\sum\limits_{t=0}^{K-1}\sum\limits_{m=1}^{M}(-1)^{m}\frac{2C_{M}^{m}}{t!} \bigg{(}\frac{m\lambda_{\mathrm{BS}}\lambda_{\mathrm{SR}}\rho}{\kappa\Delta} \bigg{)}^{\frac{t+1}{2}}K_{t-1}\left(2\sqrt{\frac{m\lambda_{\mathrm{BS}} \lambda_{\mathrm{SR}}\rho}{\kappa\Delta}}\right)\\ &\times\left[\begin{array}{c}\sum\limits_{t=0}^{K-1}\frac{2}{t!} \left(\frac{\lambda_{\mathrm{BR}}\lambda_{\mathrm{RD}}\rho}{\kappa\Delta} \right)^{\frac{t+1}{2}}K_{1-t}\left(2\sqrt{\frac{\lambda_{\mathrm{BR}} \lambda_{\mathrm{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_{ \mathrm{RD}}}{t!}\bigg{(}\frac{\rho}{n\lambda_{\mathrm{RP}}\delta\Delta+ \lambda_{\mathrm{RD}}\rho}\bigg{)}^{\frac{1-t}{2}}\bigg{(}\frac{\lambda_{ \mathrm{BR}}\rho}{\kappa\Delta}\bigg{)}^{\frac{t+1}{2}}K_{1-t}\left(2\sqrt{ \frac{\lambda_{\mathrm{BR}}}{\kappa\Delta}\left(n\lambda_{\mathrm{RP}}\delta \Delta+\lambda_{\mathrm{RD}}\rho\right)}\right)\end{array}\right]^{m}\\ &+\sum\limits_{t=0}^{K-1}\sum\limits_{n=1}^{N}\sum\limits_{m=1}^{M}(-1)^{m+ n}\frac{2C_{N}^{n}C_{M}^{m}}{t!}\bigg{(}\frac{m\lambda_{\mathrm{BS}}\lambda_{ \mathrm{SR}}\rho}{\kappa\Delta}\bigg{)}^{\frac{t+1}{2}}\bigg{(}1+\frac{n \lambda_{\mathrm{SP}}\delta\Delta}{m\lambda_{\mathrm{SR}}\rho}\bigg{)}^{\frac {t-1}{2}}K_{t-1}\left(2\sqrt{\frac{\lambda_{\mathrm{BS}}}{\kappa\Delta} \left(n\lambda_{\mathrm{SP}}\delta\Delta+m\lambda_{\mathrm{SR}}\rho\right)} \right)\\ &\times\left[\begin{array}{c}\sum\limits_{t=0}^{K-1}\frac{2}{t!} \left(\frac{\lambda_{\mathrm{BR}}\lambda_{\mathrm{RD}}\rho}{\kappa\Delta} \right)^{\frac{t+1}{2}}K_{1-t}\left(2\sqrt{\frac{\lambda_{\mathrm{BR}}\lambda_{ \mathrm{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_{\mathrm{RD}}}{t!}\bigg{(}\frac{\rho}{n\lambda_{\mathrm{RP}}\delta \Delta+\lambda_{\mathrm{RD}}\rho}\bigg{)}^{\frac{t-1}{2}}\left(\frac{\lambda_{ \mathrm{BR}}\rho}{\kappa\Delta}\right)^{\frac{t+1}{2}}K_{1-t}\bigg{(}2\sqrt{\frac {\lambda_{\mathrm{BR}}}{\kappa\Delta}\left(n\lambda_{\mathrm{RP}}\delta\Delta+ \lambda_{\mathrm{RD}}\rho\right)}\bigg{)}\end{array}\right]^{m}.\end{array}  \tag{48}\]  
From (26) and (30), the e2e OP of the B-ORS protocol is expressed by  
\[\mathrm{OP_{B-ORS}}=\Pr\left(\min\left(\Psi_{1b},\Psi_{2b}\right)<\theta\right)= \Pr\left(\max_{m=1,2,\ldots,M}\left(\min\left(\Psi_{1m},\Psi_{2m}\right)\right)< \theta\right). \tag{49}\]We note that the RVs \(\Psi_{1m}\left(m=1,2,...,M\right)\) have a common RV, i.e., \(Z_{1}=\min\left(\kappa X_{0}^{\text{sum}},\frac{\delta}{\Psi_{0}^{\text{max}}} \right).\) Hence, we have to rewrite (49) under the following form:  
\[ \begin{array}{ll}\text{OP}_{\text{B-ORS}}&=\int_{0}^{+\infty} \underbrace{\Pr\left(\max_{m=1,2,...,M}\left(\min\left(\Psi_{1m},\Psi_{2m} \right)\right)<\theta|Z_{1}=u\right)}_{\text{B}_{1}\left(u\right)}f_{Z_{1}} \left(u\right)du\\ &=1+\int_{0}^{+\infty}\frac{\partial\text{B}_{1}\left(u\right)}{\partial u} \left(1-F_{Z_{1}}\left(u\right)\right)du.\end{array}  \tag{50}\]  
Now, we attempt to calculate \(\text{B}_{1}\left(u\right)\) as marked in (50):  
\[\text{B}_{1}\left(u\right)=\left[\Pr\left(\min\left(\Psi_{1m},\Psi_{2m}\right)< \theta|Z_{1}=u\right)\right]^{M}=\left[\text{B}_{2}\left(u\right)\right]^{M}\!, \tag{51}\]  
where  
\[ \begin{array}{ll}\text{B}_{2}\left(u\right)&=\Pr\left(\min\left(\Psi_{1m}, \Psi_{2m}\right)<\theta|Z_{1}=u\right)=1-\Pr\left(\min\left(\Psi_{1m},\Psi_{2 m}\right)\geq\theta|Z_{1}=u\right)\\ &=1-\Pr\left(\gamma_{\text{SR}_{m}}\geq\frac{\rho}{\Delta u},\min\left(\kappa X _{m}^{\text{sum}},\frac{\delta}{\Psi_{m}^{\text{max}}}\right)\gamma_{\text{R }_{m}\text{D}}\geq\frac{\rho}{\Delta}\right)\\ &=1-\left(1-\Pr\left(\gamma_{\text{SR}_{m}}<\frac{\rho}{\Delta u}\right) \right)\left(1-\text{A}_{2}\right)\\ &=1-\left(1-\text{A}_{2}\right)\exp\left(-\frac{\lambda_{\text{SR}}\rho}{ \Delta u}\right).\end{array}  \tag{52}\]  
In (52), \(\text{A}_{2}\) is given in (47).