## 3 Performance Evaluation
### Outage Probability
\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{BR}}\lambda_{\text{RD}}\rho}{\kappa\Delta}\right)^{\frac{t+1}{2}}K_{1-t} \left(2\sqrt{\frac{(m+1)\,\lambda_{\text{BR}}\lambda_{\text{RD}}\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{RD}}}{t!}\left(\frac{\rho}{n \lambda_{\text{RP}}\delta\Delta+(m+1)\,\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+(m+1)\,\lambda_{\text{RD}}\rho\right)}\right) \end{array} \right] \tag{35}\] \[\times\left[ \begin{array}{c}\sum\limits_{t=0}^{K-1}\frac{2}{t!} \left(\frac{\lambda_{\text{BS}}\lambda_{\text{SR}}\rho}{\kappa\Delta}\right)^ {\frac{t+1}{2}}K_{1-t}\left(2\sqrt{\frac{\lambda_{\text{BS}}\lambda_{\text{SR} }\rho}{\kappa\Delta}}\right)\\ -\sum\limits_{t=0}^{K-1}\sum\limits_{n=1}^{N}(-1)^{n+1}C_{N}^{n}\frac{2\lambda _{\text{SR}}}{t!}\left(\frac{\rho}{n\lambda_{\text{SP}}\delta\Delta+\lambda_{ \text{SR}}\rho}\right)^{\frac{1-t}{2}}\left(\frac{\lambda_{\text{BS}}\rho}{ \kappa\Delta}\right)^{\frac{t+1}{2}}\\ \times K_{1-t}\left(2\sqrt{\frac{\lambda_{\text{BS}}}{\kappa\Delta}\left(n \lambda_{\text{SP}}\delta\Delta+\lambda_{\text{SR}}\rho\right)}\right)\end{array}  \right].\]  
As mentioned in (24), we have \(\text{OP}_{\text{H}-\text{PRS}}=\min\left(\text{OP}_{\text{PRS1}},\text{OP}_{ \text{PRS2}}\right).\) In addition, the operation of the H-PRS protocol can be realized as follows. At first, we assume that the source (S) and the destination (D) can know the statistical information of the data links (i.e., \(\lambda_{\text{SR}},\lambda_{\text{RD}}\)), the interference links (i.e., \(\lambda_{\text{SP}},\lambda_{\text{RP}}\)) and the EH links (i.e., \(\lambda_{\text{BS}},\lambda_{\text{BR}}\)). In practice, the statistical CSIs can be easily obtained by averaging the instantaneous CSI [51; 52], and they can be known by all of the nodes via control messages. Next, the source and destination nodes can calculate \(\text{OP}_{\text{PRS1}},\text{OP}_{\text{PRS2}}\) by using (34) and (35), respectively. Finally, by comparing \(\text{OP}_{\text{PRS1}}\) and \(\text{OP}_{\text{PRS2}}\), the source (or the destination) can decide to use the scheme in [35] or in [37] for the source-destination data transmission. **Proof**. Firstly, we calculate the outage probability \(\text{OP}_{\text{PRS1}}\). Due to the independence between \(\boldsymbol{\Psi}_{1a_{1}}\) and \(\boldsymbol{\Psi}_{1a_{2}}\), we can rewrite (33) as  
\[ \begin{array}{rl}\text{OP}_{\text{PRS1}}&=1-\text{Pr}\left(\min\left(\kappa X_{ 0}^{\text{sum}},\frac{\delta}{Y_{0}^{\text{max}}}\right)\gamma_{\text{SR}_{a_{1} }}\geq\frac{\rho}{\Lambda}\right)\text{Pr}\left(\min\left(\kappa X_{a_{1}}^{ \text{sum}},\frac{\delta}{Y_{1}^{\text{max}}}\right)\gamma_{\text{TR}_{a_{1}} \text{D}}\geq\frac{\rho}{\Lambda}\right)\\ &=1-\left(1-\text{A}_{1}\right)\left(1-\text{A}_{2}\right),\end{array}  \tag{36}\]  
where \(A_{1}\) and \(A_{2}\) are outage probability at the first and second hops, respectively, given as  
\[ \begin{array}{rl}\text{A}_{1}&=\text{Pr}\left(\min\left(\kappa X_{0}^{\text{sum}}, \frac{\delta}{Y_{0}^{\text{max}}}\right)\gamma_{\text{SR}_{a_{1}}}<\frac{\rho}{ \Lambda}\right),\\ \text{A}_{2}&=\text{Pr}\left(\min\left(\kappa X_{a_{1}}^{\text{sum}},\frac{\delta}{Y _{1}^{\text{max}}}\right)\gamma_{\text{TR}_{a_{1}}\text{D}}<\frac{\rho}{ \Lambda}\right).\end{array}