# Reconfigurable Intelligent Surface in Wireless-Powered Interference-Limited Communication Networks
## 2 System and Channel Models
In this paper, it is assumed that the RIS link is subjected to an approximated generalised K-distribution and the PDF of the sum of multiple generalised K random variables of \(h_{R}\) can be defined as [23]:  
\[f_{h_{R}}(y)=\frac{2\Psi k_{q}+m_{q}y^{((k_{q}+m_{q})/2-1)}}{\Gamma\big{(}k_{ q}\big{)}\Gamma\big{(}m_{q}\big{)}\big{(}\overline{\gamma}_{R}/D_{R}\big{)}^{(k_{ q}+m_{q})/2}}K_{k_{q}-m_{q}}\left(2\Psi\sqrt{\frac{yD_{R}}{\overline{\gamma}_{R}}} \right), \tag{7}\]  
where \(k_{q}\) and \(m_{q}\) represent the distribution parameters and \(\Psi=\sqrt{k_{q}m_{q}/\Omega_{q}}\) with \(\Omega_{q}\) denoting the mean power. \(K_{v}(. )\) signifies the modified Bessel function of the second kind with zero order and \(\Gamma(. )\) is the Gamma function. \(\overline{\gamma}_{R}\) is the average SNR of the RIS link. By converting the PDF to a Meijer-G function using the identity stated in [24], Equation (14), the (7) can be further expressed as:  
Figure 1: System model for an RIS aided interference-limited wireless powered network. \[f_{Z}(z)=\frac{\Psi^{k_{q}+m_{q}z^{((k_{q}+m_{q})/2-1)}}}{\Gamma(k_{q}) \Gamma(m_{q})\Gamma(m_{l}N_{l})\Gamma(m_{p}N_{P})(\overline{\gamma}_{R}/D_{R})^ {(k_{q}+m_{q})/2}}\left(\frac{m_{l}}{\overline{\gamma}_{I}}\right)^{-(k_{q}+m_{ q})/2}\left(\frac{m_{P}}{\overline{\gamma}_{P}}\right)^{m_{p}N_{P}} \tag{15}\] \[\times\int\limits_{0}^{\infty}x^{m_{p}N_{P}-((k_{q}+m_{q})/2)-1} exp\left(-\frac{m_{p}}{\overline{\gamma}_{p}}x\right)G_{2,1}^{1,2}\left(\frac{m_{l} \overline{\gamma}_{R}x}{\overline{\
u}^{2}\overline{\gamma}_{I}D_{R}Z}\right| \begin{array}{c}1-\left(k_{q}+m_{q}\right)/2\,1-\left(m_{q}-k_{q}\right)/2\\ \left(k_{q}+m_{q}\right)/2+m_{I}N_{I}\end{array} \right)dx\,\]  
By utilizing the integral identity detailed in ([27], Equation (7.813(1))), (15) can be solved as:  
\[f_{Z}(z)=\frac{\Psi^{k_{q}+m_{q}}}{\Gamma(k_{q})\Gamma(m_{q}) \Gamma(m_{l}N_{l})\Gamma(m_{p}N_{P})(\overline{\gamma}_{R}/D_{R})^{(k_{q}+m_ {q})/2}}\left(\frac{m_{P}\overline{\gamma}_{I}}{m_{l}\overline{\gamma}_{P}} \right)^{(k_{q}+m_{q})/2}z^{((k_{q}+m_{q})/2-1)} \tag{16}\] \[\times G_{3,1}^{1,3}\left(\frac{m_{l}\overline{\gamma}_{R} \overline{\gamma}_{P}}{\overline{\
u}^{2}\overline{\gamma}_{I}m_{p}D_{R}Z} \right| \begin{array}{c}1+\left(\left(k_{q}+m_{q}\right)/2\right)-m_{l}N_{P},\ 1-\left(k_{q}+m_{q}\right)/2,\ 1-\left(m_{q}-k_{q}\right)/2\\ \left(k_{q}+m_{q}\right)/2+m_{I}N_{I}\end{array} \right)\,\]  
By inverting the Meijer-G function in (16) using the identity detailed in ([27], Equation (9.31(5))), (16) can be further expressed as:  
\[f_{Z}(z)=\frac{\Psi^{k_{q}+m_{q}}}{\Gamma(k_{q})\Gamma(m_{q}) \Gamma(m_{l}N_{l})\Gamma(m_{p}N_{P})(\overline{\gamma}_{R}/D_{R})^{(k_{q}+m_ {q})/2}}\left(\frac{m_{P}\overline{\gamma}_{I}}{m_{l}\overline{\gamma}_{P}} \right)^{(k_{q}+m_{q})/2}z^{((k_{q}+m_{q})/2-1)} \tag{17}\] \[\times G_{1,3}^{3,1}\left(\frac{\Psi^{2}\overline{\gamma}_{I}m_{ p}D_{R}Z}{m_{l}\overline{\gamma}_{R}\overline{\gamma}_{P}}\right| \begin{array}{c} \left(k_{q}+m_{q}\right)/2+m_{I}N_{I}\\ m_{P}N_{P}-\left(\left(k_{q}+m_{q}\right)/2\right),\left(k_{q}+m_{q}\right)/2, \left(m_{q}-k_{q}\right)/2\end{array} \right)\,\]  
Through (6) and transformation of random variable, (17) can be expressed as:  
\[f_{\gamma}(\gamma)=\frac{\Psi^{k_{q}+m_{q}}\xi^{-(k_{q}+m_{q})/2}}{\Gamma(k_{ q})\Gamma(m_{q})\Gamma(m_{l}N_{l})\Gamma(m_{p}N_{P})(\overline{\gamma}_{R}/D_{R}) ^{(k_{q}+m_{q})/2}}\left(\frac{m_{P}\overline{\gamma}_{I}}{m_{l}\overline{ \gamma}_{P}}\right)^{(k_{q}+m_{q})/2}\gamma^{((k_{q}+m_{q})/2-1)} \tag{18}\] \[\times G_{1,3}^{3,1}\left(\frac{\Psi^{2}\overline{\gamma}_{I}m_{ p}D_{R}\gamma}{m_{l}\overline{\gamma}_{R}\overline{\gamma}_{P}\xi}\right| \begin{array}{c}\left(k_{q}+m_{q}\right)/2+m_{I}N_{I}\\ m_{P}N_{P}-\left(\left(k_{q}+m_{q}\right)/2\right),\left(k_{q}+m_{q}\right)/2, \left(m_{q}-k_{q}\right)/2\end{array} \right)\.\]