## 3 Performance Evaluation
### Outage Probability
 \tag{37}\]Next, we can rewrite A\({}_{1}\) as  
\[\text{A}_{1}=\int_{0}^{+\infty}F_{Z_{1}}\left(\frac{\rho}{\Delta x}\right)f_{ \gamma_{\text{SR}_{\text{A}_{1}}}}\left(x\right)dx, \tag{38}\]  
where \(Z_{1}=\min\left(\kappa X_{0}^{\text{sum}},\frac{\delta}{\gamma_{0}^{\text{max} }}\right)\), \(F_{Z_{1}}\left(.\right)\) and \(f_{\gamma_{\text{SR}_{\text{A}_{1}}}}\left(.\right)\) are CDF and PDF of \(Z_{1}\) and \(\gamma_{\text{SR}_{\text{A}_{1}}}\), respectively. Combining Equations (1) and (20), \(F_{\gamma_{\text{SR}_{\text{A}_{1}}}}\left(.\right)\) can be given as  
\[ \begin{array}{ll}F_{\gamma_{\text{SR}_{\text{A}_{1}}}}\left(x\right)&=\Pr \left(\max_{m=1,2,\ldots,M}\left(\gamma_{\text{SR}_{m}}\right)<x\right)=\left( F_{\gamma_{\text{SR}_{m}}}\left(x\right)\right)^{M}\\ &=\left(1-\exp\left(-\lambda_{\text{SR}}x\right)\right)^{M}.\end{array}  \tag{39}\]  
From (39), the corresponding PDF can be obtained as  
\[ \begin{array}{ll}f_{\gamma_{\text{SR}_{\text{A}}}}\left(x\right)&=M\lambda_ {\text{SR}}\exp\left(-\lambda_{\text{SR}}x\right)\left(1-\exp\left(-\lambda_{ \text{SR}}x\right)\right)^{M-1}\\ &=\sum\limits_{m=0}^{M-1}\left(-1\right)^{m}C_{M-1}^{m}M\lambda_{\text{SR}} \exp\left(-\left(m+1\right)\lambda_{\text{SR}}x\right),\end{array}  \tag{40}\]  
where \(C_{M-1}^{m}\) is a binomial coefficient. Considering the RV \(Z_{1}\), its CDF can be formulated by  
\[ \begin{array}{ll}F_{Z_{1}}\left(x\right)&=\Pr\left(\min\left(\kappa X_{0}^ {\text{sum}},\frac{\delta}{\gamma_{0}^{\text{max}}}\right)<x\right)\\ &=1-\left(1-F_{\gamma_{0}^{\text{sum}}}\left(\frac{x}{x}\right)\right)F_{ \gamma_{0}^{\text{max}}}\left(\frac{\delta}{x}\right).\end{array}  \tag{41}\]  
From (9), since \(X_{0}^{\text{sum}}\) is the sum of i.i.d. exponential RVs, CDF \(F_{X_{0}^{\text{sum}}}\left(x/\kappa\right)\) in (41) can be given as  
\[F_{X_{0}^{\text{sum}}}\left(\frac{x}{\kappa}\right)=1-\exp\left(\frac{\lambda _{\text{SR}}}{\kappa}x\right)\sum\limits_{t=0}^{K-1}\frac{1}{t!}\left(\frac{ \lambda_{\text{SR}}}{\kappa}\right)^{t}\!\!x^{t}. \tag{42}\]  
Next, from (12), we can obtain CDF \(F_{Y_{0}^{\text{max}}}\left(\delta/x\right)\), similar to (39), as  
\[F_{Y_{0}^{\text{max}}}\left(\frac{\delta}{x}\right)=1+\sum\limits_{n=1}^{N} \left(-1\right)^{n}C_{N}^{n}\exp\left(-\frac{n\lambda_{\text{SR}}\delta}{x} \right). \tag{43}\]  
Substituting (42) and (43) into (41), we have  
\[ \begin{array}{ll}F_{Z_{1}}\left(x\right)=1&-\sum\limits_{t=0}^{K-1}\frac{1} {n!}\left(\frac{\lambda_{\text{RS}}}{\kappa}\right)^{t}\!\!x^{t}\exp\left(- \frac{\lambda_{\text{RS}}}{\kappa}x\right)\\ &-\sum\limits_{t=0}^{K-1}\sum\limits_{n=1}^{N}\left(-1\right)^{n}\frac{C_{N}^{ n}}{t!}\left(\frac{\lambda_{\text{RS}}}{\kappa}\right)^{t}\!\!x^{t}\exp\left(- \frac{\lambda_{\text{RS}}}{\kappa}x-\frac{n\lambda_{\text{SR}}\delta}{x} \right).\end{array}  \tag{44}\]  
Then, substituting (40) and (44) into (38), we arrive at  
\[ \begin{array}{ll}\text{A}_{1}&=1-\sum\limits_{t=0}^{K-1}\sum\limits_{m=0}^{M- 1}(-1)^{m}C_{M-1}^{m}\frac{M\lambda_{\text{SR}}}{t!}\left(\frac{\lambda_{\text{ RS}}\rho}{\kappa\kappa}\right)^{t}\int_{0}^{+\infty}\frac{1}{x^{t}}\exp\left(- \frac{\lambda_{\text{RS}}\rho}{\kappa\kappa}\frac{1}{x}\right)\exp\left(-\left( m+1\right)\lambda_{\text{SR}}x\right)dx\\ &-\sum\limits_{t=0}^{K-1}\sum\limits_{n=1}^{N}\frac{M-1}{m-1}(-1)^{n+m}C_{N}^{m} C_{M-1}^{m}\frac{M\lambda_{\text{RS}}}{t!}\left(\frac{\lambda_{\text{RS}}\rho}{ \kappa\kappa}\right)^{t}\int_{0}^{+\infty}\frac{1}{x^{t}}\exp\left(-\frac{ \lambda_{\text{SR}}\rho}{\kappa\kappa}\frac{1}{x}\right)\\ &\times\exp\left(-\left(\frac{n\lambda_{\text{SR}}\rho}{\rho}+\left(m+1\right) \lambda_{\text{SR}}\right)x\right)dx.\end{array}  \tag{45}\]Using ([53], Equation (3.471.9)) for the corresponding integrals in (45), we obtain  
\[