## 2 System Model
### Reconfigurable Intelligent Surface System Model
By using the Shannon capacity formula, the ergodic capacity by an arbitrary user positioned at \(U\), \(C_{RIS}\) can be estimated as:  
\[C_{RIS}=E\left[log_{2}\left(1+\frac{\sum_{i=1}^{N}\left|h_{s,i}h_{r,i}\right|^ {2}\left(rd\right)^{-\beta}}{\sum_{k=1}^{M}\left|h_{c,k}\right|\left|l_{k}{}^{- \beta}\delta_{k}+\frac{1}{\rho}\right.}\right)\right]. \tag{3}\]  
Equations (2) and (3) consist of a variety of RV, e.g., \(\left|h_{s,i}\right|^{2}\), \(\left|h_{c,i,k}\right|^{2}\), \(d\), \(\delta_{k}\) etc. ; therefore the closed-form expression of their probability density functions (PDFs) for \(SINR_{RIS}\) is generally difficult to obtain, particularly for large values of N and M. A direct method to compute the average \(SINR_{RIS}\) in this case would require at least \(\mathcal{L}(\text{N}+\text{M})\)-fold integrations, where \(\mathcal{L}\) is the total number of RV. Instead of performing time-consuming integration with respect to a large number RV in (3), a non-direct method using the following lemma is used to reduce the complexity [6]:  
\[ln\left(1+\frac{u}{v+a}\right)=\int_{0}^{\infty}\frac{1}{z}\left(1-e^{-zu} \right)e^{-zv}e^{-za}\;dz. \tag{4}\]  
Using (4), the expression in (3) can be re-expressed as  
\[C_{RIS}=log_{2}e\int_{0}^{\infty}\frac{1}{z}\left(1-\prod_{i=1}^{N}E\left[e^{ -z\left|h_{s,i}h_{r,i}\right|^{2}\left(rd\right)^{-\beta}}\right]\right)\times \prod_{k=1}^{M}E\left[e^{-z\left|h_{c,k}\right|l_{k}{}^{-\beta}\delta_{k}} \right]e^{-\frac{z}{\beta}}\;dz. \tag{5}\]  
Assuming an independent and identically distributed (i.i.d.) of RVs, for a fully loaded traffic activity, (5) is reduces into:  
\[C_{RIS}=log_{2}e\int_{0}^{\infty}\frac{1}{z}\left(1-\prod_{i=1}^{N}M_{i}(z) \right)\times\prod_{k=1}^{M}M_{k}(z)e^{-\frac{z}{\beta}}\;dz\;, \tag{6}\]  
Nakagami-m can be represented by the gamma distribution with m as the Nakagami fading index for useful signals from the base station and m\({}_{\text{i}}\) as the Nakagami fading index from co-channel interferers, where both m and \(m_{i}\gg 1/2\). The Nakagami-m statistical model can be used to represent many fading conditions, such as the one-sided Gaussian distribution (\(m=\frac{1}{2}\)), and Rayleigh distribution when \(m=1\) as a special case. Rician distribution can also be closely approximated using the Nakagami-m when \(m>1\). As \(m\rightarrow\infty\), the Nakagami fading channels converge to an AWGN channel. To simplify, assume \(m_{i}=m\), then using the MGF of gamma distribution for the useful signals,  
\[M_{i}(z)=\int_{0}^{\infty}\left(\frac{1}{1+\frac{z}{m}x(rd)^{-\beta}}\right)^ {m}\frac{m^{m}x^{m-1}}{\Gamma(m)}\;e^{-mx}\;dx=\left(\frac{m^{2}}{z(rd)^{- \beta}}\right)^{m}U\left(m,1,\frac{m^{2}}{z(rd)^{-\beta}}\right), \tag{7}\]  
where \(\Gamma(. )\) Is the gamma function and \(U(. )\) is the Triconomi confluent hypergeometric function. In the special case wherein Nakagami \(m=1\) (i.e., Rayleigh fading), with the help of ([12], (3.310)) and ([12], (3.353.5)), (7) becomes  
\[M_{i}(z)=\frac{1}{z(rd)^{-\beta}}\;e^{\frac{1}{z(rd)^{-\beta}}}\;\;\Gamma\left( 0,\frac{1}{z(rd)^{-\beta}}\right), \tag{8}\]where \(\Gamma(x,y)\) is in complete gamma function. For \(x=0\), the incomplete gamma function is closely related to the exponential integral \(\text{Ei}(z)\) ([12], (8.211/1)) as follows:  
\[\Gamma(0,y) =-\text{Ei}(-y)+\frac{1}{2}\left[ln(-y)-ln\left(-\frac{1}{y} \right)\right]-ln(y). \tag{9}\]  
Therefore, the MGF of \(M_{i}(z)\) can be simplified as  
\[ \begin{array}{c}M_{i}(z)=\frac{-1}{z(rd)^{-\beta}}\,\,e^{\frac{1}{z(rd)^{- \beta}}}\,\,\left[\text{Ei}\left(\frac{-1}{z(rd)^{-\beta}}\right)\right.\\ \left.-\frac{1}{2}\left[ln\left(\frac{-1}{z(rd)^{-\beta}}\right)-ln\left(-z(rd )^{-\beta}\right)\right]+ln(\frac{1}{z(rd)^{-\beta}})\right].\end{array}  \tag{10}\]  
Assuming independent identically distributed (i.i.d.)