# Reconfigurable Intelligent Surface in Wireless-Powered Interference-Limited Communication Networks
## 4 System Performance Analysis
### Outage Probability Analysis  
Outage probability is one of the key performance metrics for quantifying the performance of a wireless communication system under the effect of fading. It is defined as the probability that the system's instantaneous SNR falls below the preset threshold value \(\gamma_{th}\). It can thus be obtained as [28]:  
\[P_{out}(\gamma_{th})=\int\limits_{0}^{\gamma_{th}}f_{\gamma}(\gamma_{th})d\gamma, \tag{19}\]  
where \(\gamma_{th}=2^{R}-1\) and \(R\) denotes the fixed rate.  
Substituting (18) into (19) and applying the integral identity detailed in [24], Equation (26), the outage probability of the system can be expressed as:  
\[P_{out}(\gamma_{th})=\frac{\Psi^{k_{q}+m_{q}}\xi^{-(k_{q}+m_{q})/2}}{\Gamma(k_{ q})\Gamma(m_{q})\Gamma(m_{l}N_{l})\Gamma(m_{p}N_{P})(\overline{\gamma}_{R}/D_{R})^{(k_{q}+m_ {q})/2}}\left(\frac{m_{P}\overline{\gamma}_{I}}{m_{l}\overline{\gamma}_{P}} \right)^{(k_{q}+m_{q})/2}\gamma_{th}^{(k_{q}+m_{q})/2}} \tag{20}\] \[\times G_{2,4}^{3,2}\left(\frac{\Psi^{2}\overline{\gamma}_{I}m_{ p}D_{R}\gamma_{th}}{m_{l}\overline{\gamma}_{R}\overline{\gamma}_{P}\xi}\right| \begin{array}{c}\left(k_{q}+m_{q}\right)/2+m_{I}N_{I},\ 1-\left(k_{q}+m_{q}\right)/2\\ m_{P}N_{P}-\left(\left(k_{q}+m_{q}\right)/2\right),\left(k_{q}+m_{q}\right)/2, \left(m_{q}-k_{q}\right)/2-\left(k_{q}+m_{q}\right)/2\end{array}\right)\.\]  
\[P_{b} =\frac{\Psi^{k_{q}+m_{q}}\xi^{-(k_{q}+m_{q})/2}\bigg{(}\frac{m_{p} \overline{\gamma}_{I}}{m_{1}\overline{\gamma}_{P}}\bigg{)}^{(k_{q}+m_{q})/2}}{2 \Gamma(\psi)\Gamma(k_{q})\Gamma\left(m_{q}\right)\Gamma\left(m_{I}N_{I}\right) \Gamma\left(m_{P}N_{P}\right)(\overline{\gamma}_{R}/D_{R})^{(k_{q}+m_{q})/2} \int\limits_{0}^{\infty}\gamma^{((k_{q}+m_{q})/2-1)}\Gamma(\psi,\mu x)} \tag{27}\] \[\times G_{1,3}^{3,1}\Bigg{(}\frac{\Psi^{2}\overline{\gamma}_{I}m_ {P}D_{R}\gamma}{m_{1}\overline{\gamma}_{R}\overline{\gamma}_{P}\xi}\Bigg{|} \begin{array}{c}\left(k_{q}+m_{q}\right)/2+m_{I}N_{I}\\ m_{P}N_{P}-\big{(}\left(k_{q}+m_{q}\right)/2\big{)},\big{(}k_{q}+m_{q}\big{)}/2,\big{(}m_{q}-k_{q}\big{)}/2\end{array}\Bigg{)}d\gamma,\]  
By converting the \(\Gamma(\psi,\mu x)\) to the Meijer-G function using the identity detailed in ([30], Equation (8.4.16(2))), then (27) can be expressed as:  
\[P_{b} =\frac{\Psi^{k_{q}+m_{q}}\xi^{-(k_{q}+m_{q})/2}\bigg{(}\frac{m_{ p}\overline{\gamma}_{I}}{m_{1}\overline{\gamma}_{P}}\bigg{)}^{(k_{q}+m_{q})/2}}{2 \Gamma(\psi)\Gamma\left(k_{q}\right)\Gamma\left(m_{q}\right)\Gamma\left(m_{ I}N_{I}\right)\Gamma\left(m_{P}N_{P}\right)(\overline{\gamma}_{R}/D_{R})^{(k_{q}+m_{ q})/2}}\stackrel{{\infty}}{{\to}}\gamma^{((k_{q}+m_{q})/2-1)}G_{1,2}^{2,0} \bigg{(}\mu x\left|\begin{array}{c}1\\ 0,\psi\end{array}\right.\bigg{)} \tag{28}\] \[\times G_{1,3}^{3,1}\Bigg{(}\frac{\Psi^{2}\overline{\gamma}_{I} m_{P}D_{R}\gamma}{m_{1}\overline{\gamma}_{R}\overline{\gamma}_{P}\xi}\Bigg{|} \begin{array}{c}\left(k_{q}+m_{q}\right)/2+m_{I}N_{I}\\ m_{P}N_{P}-\big{(}\left(k_{q}+m_{q}\right)/2\big{)},\big{(}k_{q}+m_{q}\big{)}/2,\big{(}m_{q}-k_{q}\big{)}/2\end{array}\Bigg{)}d\gamma\ \,\]  
Using the integral identity given in [24], Equations (21) and (28) can be solved as:  
\[P_{b} =\frac{\Psi^{k_{q}+m_{q}}\xi^{-(k_{q}+m_{q})/2}\bigg{(}\frac{m_{ p}\overline{\gamma}_{I}}{m_{1}\overline{\gamma}_{P}}\bigg{)}^{(k_{q}+m_{q})/2}}{2 \Gamma(\psi)\Gamma\left(k_{q}\right)\Gamma\left(m_{q}\right)\Gamma\left(m_{I}N _{I}\right)\Gamma\left(m_{P}N_{P}\right)(\overline{\gamma}_{R}/D_{R})^{(k_{q}+ m_{q})/2}} \tag{29}\] \[\times G_{3,4}^{3,3}\Bigg{(}\frac{\Psi^{2}\overline{\gamma}_{I} m_{P}D_{R}\gamma}{\mu m_{1}\overline{\gamma}_{R}\overline{\gamma}_{P}\xi} \Bigg{|}\begin{array}{c}\left(k_{q}+m_{q}\right)/2+m_{I}N_{I},1-\big{(}\left( k_{q}+m_{q}\right)/2\big{)},-\big{(}\left(k_{q}+m_{q}\right)/2\big{)}-\psi\\ m_{P}N_{P}-\bigg{(}\frac{\left(k_{q}+m_{q}\right)}{2}\bigg{)},\big{(}k_{q}+m_{q} \big{)}/2,\big{(}m_{q}-k_{q}\big{)}/2,-\big{(}\left(k_{q}+m_{q}\right)/2\big{)} \end{array}\Bigg{)}\ \.\]