## 2 System Model
We note that \(g_{\text{SS}_{i}\text{RIS}_{k}}=\bigl{|}h_{\text{SS}_{i}\text{RIS}_{k}} \bigr{|}^{2}\) and \(g_{\text{RIS}_{k}\text{SS}_{j}}=\bigl{|}h_{\text{RIS}_{k}\text{SS}_{j}} \bigr{|}^{2}\), where \(\bigl{|}h_{\text{SS}_{i}\text{RIS}_{k}}\bigr{|}\) and \(\bigl{|}h_{\text{RIS}_{k}\text{SS}_{j}}\bigr{|}\) are amplitude of \(h_{\text{SS}_{i}\text{RIS}_{k}}\) and \(h_{\text{RIS}_{k}\text{SS}_{j}}\), respectively. Moreover, we can express \(h_{\text{SS}_{i}\text{RIS}_{k}}\) and \(h_{\text{RIS}_{k}\text{SS}_{j}}\) in exponential form as follows: \(h_{\text{SS}_{i}\text{RIS}_{k}}=\bigl{|}h_{\text{SS}_{i}\text{RIS}_{k}} \bigr{|}\exp\bigl{(}-j_{\text{SS}_{i}\text{RIS}_{k}}\bigr{)}\) and \(h_{\text{RIS}_{k}\text{SS}_{j}}=\bigl{|}h_{\text{RIS}_{k}\text{SS}_{j}}\bigr{|} \exp\Bigl{(}-j_{\text{GRIS}_{k}\text{SS}_{j}}\bigr{)}\), where \(\zeta_{\text{SS}_{i}\text{RIS}_{k}}\) and \(\zeta_{\text{RIS}_{k}\text{SS}_{j}}\) are phases of \(h_{\text{RS}_{i}\text{RIS}_{k}}\) and \(h_{\text{RIS}_{k}\text{SS}_{j}}\), respectively. Similarly to [37], we can express \(r_{\text{RIS}_{k}}\) as \(r_{\text{RIS}_{k}}=\exp\bigl{(}j_{\text{\text{\text{\text{\text{\text{\text{ \text{\text{\text{\text{\text  
where \(\Gamma(x)=\int_{0}^{+\infty}t^{x-1}\exp(-t)dt\) and \(\gamma(x,y)=\int_{0}^{y}t^{x-1}\exp(-t)dt\) are gamma and lower incomplete gamma functions [41], respectively, and  
\[\theta=\frac{K\pi^{2}}{16-\pi^{2}},\phi=\frac{16-\pi^{2}}{4\pi\sqrt{\lambda_{ \text{SS}_{i}}\text{RIS}\lambda_{\text{SS}_{j}}\text{RIS}}}. \tag{9}\]  
From Equations (7) and (8), we can see that \(Z_{1,\text{sum}}\) and \(Z_{2,\text{sum}}\) have the same CDF. Hence, we can omit the subscripts \(i\) and \(j\), i.e., \(F_{Z_{i,\text{Sum}}}(x)=F_{Z_{i,\text{sum}}}(x)=F_{Z_{\text{Sum}}}(x)\). Then, we can obtain CDF of \((Z_{1,\text{sum}})^{2}\) and \((Z_{2,\text{sum}})^{2}\) as  
\[F_{(Z_{\text{Sum}})^{2}}(x)=F_{Z_{\text{Sum}}}\big{(}\sqrt{x}\big{)}=\frac{1} {\Gamma(\theta)}\gamma\bigg{(}\theta,\frac{\sqrt{x}}{\phi}\bigg{)}. \tag{10}\]  
Differentiating Equation (9) with respect to \(x\), we obtain PDF of \(Z_{1,\text{sum}}\) and \(Z_{2,\text{sum}}\) as  
\[f_{(Z_{\text{Sum}})^{2}}(x)=\frac{1}{2\Gamma(\theta)(\phi)^{\theta}}x^{\frac{ \theta}{2}-1}\exp\bigg{(}-\frac{\sqrt{x}}{\phi}\bigg{)}. \tag{11}\]  
Next, setting \(Y_{i}=P_{\text{SS}_{i}}/\sigma_{0}^{2}\), we see that \(Y_{i}\) is also a random variable whose CDF can be formulated as  
\[ \begin{array}{ll}F_{Y_{i}}(x)&=\text{Pr}\bigg{(}\min\bigg{(}\frac{\lambda F _{\text{SR}_{i}}}{\sigma_{0}^{2}}S_{\text{SS}_{i}},\frac{I_{\text{PU}}}{ \sigma_{0}^{2}S_{\text{PU}}}\bigg{)}<x\bigg{)}\\ &=1-\text{Pr}\bigg{(}S_{\text{SSBS}_{i}}\geq\frac{\sigma_{0}^{2}x}{\chi\text{ RIS}_{i}}\bigg{)}\text{Pr}\bigg{(}S_{\text{SS}_{i}}\text{PU}<\frac{I_{\text{PU}}}{ \sigma_{0}^{2}x}\bigg{)}\\ &=1-\bigg{[}1-F_{S_{\text{SSSS}_{i}}}\bigg{(}\frac{\sigma_{0}^{2}x}{\chi\text{ RIS}_{\text{SR}}}\bigg{)}\bigg{]}F_{S_{\text{SS}_{i}}\text{PU}}\bigg{(}\frac{I_{ \text{PU}}}{\sigma_{0}^{2}x}\bigg{)}.\end{array}  \tag{12}\]  
Substituting Equation (1) into Equation (12), we obtain  
\[ \begin{array}{ll}F_{Y_{i}}(x)=1-\exp\bigg{(}-\frac{\lambda_{\text{SSSS}_{i}} \sigma_{0}^{2}}{\chi\text{ RIS}_{i}}x\bigg{)}\bigg{[}1-\exp\bigg{(}-\frac{\lambda_{\text{SS}_{i}} \text{PU}I_{\text{PU}}}{\sigma_{0}^{2}x}\bigg{)}\bigg{]}\\ &=1-\exp\bigg{(}-\frac{\lambda_{\text{SSSS}_{i}}\sigma_{0}^{2}}{\chi\text{ RIS}_{i}}x\bigg{)}+\exp\bigg{(}-\frac{\lambda_{\text{SS}_{i}}\sigma_{0}^{2}}{\chi\text{ RIS}_{i}}x-\frac{\lambda_{\text{SS}_{i}}\text{PU}I_{\text{PU}}}{\sigma_{0}^{2}x} \bigg{)}\\ &=1-\exp(-\kappa_{i}x)+\exp\big{(}-\kappa_{i}x-\frac{\mu_{i}}{x}\big{)},\end{array}  \tag{13}\]  
where \(\kappa_{i}=\frac{\lambda_{\text{SSS}_{i}}\sigma_{0}^{2}}{\chi\chi\text{ RIS}_{i}}\), \(\mu_{i}=\frac{\lambda_{\text{SS}_{i}}\
u_{i}I_{\text{PU}}}{\sigma_{0}^{2}}\).