## 3 Performance Evaluation
### Outage Probability
\begin{array}{c}f_{\gamma_{\text{SR}_{c}}}(y)&=\sum\limits_{m=1}^{M}\left(-1 \right)^{m-1}C_{M}^{m}\frac{m\lambda_{\text{SR}}\lambda_{\text{BD}}}{m\left( \lambda_{\text{SR}}+\lambda_{\text{RD}}\right)-\lambda_{\text{SR}}}\exp\left(- \lambda_{\text{SR}}y\right)\\ &+\sum\limits_{m=1}^{M}\left(-1\right)^{m-1}C_{M}^{m}\frac{m\left(m-1\right) \lambda_{\text{SR}}\left(\lambda_{\text{SR}}+\lambda_{\text{RD}}\right)}{m \left(\lambda_{\text{SR}}+\lambda_{\text{RD}}\right)-\lambda_{\text{SR}}}\exp \left(-m\left(\lambda_{\text{SR}}+\lambda_{\text{RD}}\right)y\right),\\ f_{\gamma_{\text{R}_{c}\text{D}}}(y)&=\sum\limits_{m=1}^{M}\left(-1\right)^{m-1}C_{M} ^{m}\frac{m\lambda_{\text{SR}}\lambda_{\text{RD}}}{m\left(\lambda_{\text{SR}}+ \lambda_{\text{RD}}\right)-\lambda_{\text{RD}}}\exp\left(-\lambda_{\text{RD}}y \right)\\ &+\sum\limits_{m=1}^{M}\left(-1\right)^{m-1}C_{M}^{m}\frac{m\left(m-1\right) \lambda_{\text{SR}}\left(\lambda_{\text{SR}}+\lambda_{\text{RD}}\right)}{m \left(\lambda_{\text{SR}}+\lambda_{\text{RD}}\right)-\lambda_{\text{RD}}}\exp \left(-m\left(\lambda_{\text{SR}}+\lambda_{\text{RD}}\right)y\right).\end{array}  \tag{74}\]Combining (44) and (74), which yields  
\[\begin{array}{rl}\Pr\left(Z_{1}\geq\frac{\rho}{\Delta_{\rm{TSR_{c}}}}\right)&= \sum\limits_{t=0}^{K-1}\sum\limits_{m=1}^{M}{(-1)^{m-1}\frac{C_{\rm{M}}^{m}}{ \Gamma! }\left(\frac{\lambda_{\rm{BS}}\rho}{\kappa\Delta}\right)^{t}\frac{m \lambda_{\rm{SB}}\lambda_{\rm{BD}}}{m(\lambda_{\rm{SR}}+\lambda_{\rm{BD}})- \lambda_{\rm{SR}}}}\\ &\times\int_{0}^{+\infty}{\frac{1}{y^{\prime}}\exp\left(-\frac{\lambda_{\rm{BS }}\rho}{\kappa\Delta}\frac{1}{y}\right)\exp\left(-\lambda_{\rm{SR}}y\right)dy} \\ &+\sum\limits_{t=0}^{K-1}\sum\limits_{m=1}^{M}{(-1)^{m-1}\frac{C_{\rm{M}}^{m}} {\Gamma! }\left(\frac{\lambda_{\rm{BS}}\rho}{\kappa\Delta}\right)^{t}\frac{m(m- 1)\lambda_{\rm{SB}}(\lambda_{\rm{SR}}+\lambda_{\rm{BD}})}{m(\lambda_{\rm{SR}}+ \lambda_{\rm{BD}})-\lambda_{\rm{SR}}}}\\ &\times\int_{0}^{+\infty}{\frac{1}{y^{\prime}}\exp\left(-\frac{\lambda_{\rm{BS }}\rho}{\kappa\Delta}\frac{1}{y}\right)\exp\left(-m\left(\lambda_{\rm{SR}}+ \lambda_{\rm{BD}}\right)y\right)dy}\\ &+\sum\limits_{t=0}^{K-1}\sum\limits_{m=1  
results (Sim), the exact theoretical results (Exact) and the asymptotically theoretical results (Asym) are denoted by markers, solid line, and dash line, respectively. In Figure 2, we present outage probability (OP) of the conventional PRS protocol [35] (denoted by PRS1), the modified PRS protocol [37] (denoted by PRS2) and the proposed H-PRS protocol as a function of \(x_{\mathrm{R}}\). This figure shows that the analytical results are in complete agreement with the simulation results. Next, we can see that as \(x_{\mathrm{R}}\) is small (the relays are close to the source but far from the destination), OP of PRS1 is higher than that of PRS2. However, as \(x_{\mathrm{R}}\) is high enough, PRS1 outperforms PRS2. As we can see, OP of H-PRS is the same as OP of PRS1, as the relays are near the destination, and is the same as OP of PRS2, as the relays are near the source. Moreover, there exists a value of \(x_{\mathrm{R}}\) (denoted \(x_{\mathrm{R}}^{*}\)) at which the OP values of PRS1 and PRS2 are same. Indeed, by solving the equation OP\({}_{\mathrm{PRS1}}\)\(=\) OP\({}_{\mathrm{PRS2}}\) (using (34) and (35)), we can find the value of \(x_{\mathrm{R}}^{*}\). Finally, it is also seen from Figure 2 that the outage performance of PRS1, PRS2 and H-PRS is better as increasing the transmit SNR (\(\Delta\)). Figure 3 presents the values of \(x_{\mathrm{R}}^{*}\) with different positions of the primary users. As mentioned above, \(x_{\mathrm{R}}^{*}\) is obtained by solving the equation OP\({}_{\mathrm{PRS1}}\)\(=\) OP\({}_{\mathrm{PRS2}}\). Moreover, \(x_{\mathrm{R}}^{*}\) is a reference distance (between the source and the relays) used in H-PRS to determine which protocol (PRS1 or PRS2) will be used to send the source data to the destination. Particularly, as \(x_{\mathrm{R}}<x_{\mathrm{R}}^{*}\), PRS2 is employed and as \(x_{\mathrm{R}}>x_{\mathrm{R}}^{*}\), the PRS1 is used.