## 2 System Model
### Reconfigurable Intelligent Surface System Model
links with the same stochastic characteristics, since it is a summation of the reflected signal of N elements of RIS (equal expected SNR, Nakagami m factor of all channel and distance), the total moment is obtained as:  
\[\prod_{i=1}^{N}M_{i}(z)=M_{i}(z)^{N}. \tag{11}\]  
For \(M_{k}(z)\), since the \(\left|h_{c,k}\right|^{2}\) is a gamma distribution gain, the total moment obtained is:  
\[M_{k}(z)=\left(\frac{1}{1+\frac{z}{m}I_{k}-\beta}\right)^{m}. \tag{12}\]  
For a random position of user, the PDF of the reference user \(U\) schemes at radius d relative to RIS\({}_{0}\) is \(p_{d}(d)=\frac{2d}{r^{2}}=\frac{2d}{\left(\frac{\beta}{\beta}\right)^{2}}\) for \(0\ \leq\ d\ \leq\ \frac{R}{3}\). Thus, new mathematical analysis for the ergodic capacity of the RIS model is as follows:  
\[C_{RIS} =log_{2}e^{\int_{0}^{\infty}\int_{0}^{\frac{R}{3}}\frac{1}{z} \left(1-\prod_{i=1}^{N}M_{i}(z)\right)\times\prod_{k=1}^{M}M_{k}(z)\ e^{-\frac{z}{ \beta}}\ \frac{2d}{\left(\frac{\beta}{\beta}\right)^{2}}dd\ dz}. \tag{13}\]  
where \(\prod\limits_{i=1}^{N}M_{i}(z)\) and \(M_{k}(z)\) are equal to (11) and (12), respectively. Equation (13) simplifies the evaluation of ergodic capacity by finding the MGF of \(M_{i}(z)\) and \(M_{k}(z)\) without having to deal with many RV. By fixing the value R, (13) simplifies the evaluation of the ergodic capacity of RIS in a single integral only. With this regard, (13) can be expressed in terms of weight and abscissa of a Laguerre orthogonal polynomial ([13], (25.4.45)), provides an efficient numerical evaluation method for the required average. The ergodic capacity in (13) computed is under the assumption that all interferences coming from all other BS channels are active; thus, the number of interferences is constant. To investigate the effect of partially loaded BSs traffic conditions on the interfering cells, \(\delta_{k}\) is a RV of on-off traffic with the expectation of MGF over the Bernoulli distribution, where we define \(q\) as the probability that the channel is active, and \(1-q\) as the probability that the channel is not active. We include the MGF of the Bernoulli distribution, where the probability of the traffic activity \(\text{P}_{\text{r}}\) (\(\delta_{k}=1)=q\) and \(\text{P}_{\text{r}}\) (\(\delta_{k}=0)=1-q\),  
From (13), the ergodic capacity is obtained as:  
\[C_{RIS} =log_{2}e^{\int_{0}^{\infty}\int_{0}^{\frac{R}{3}}\frac{1}{z} \left(1-\prod_{i=1}^{N}M_{i}(z)\right)\times\prod_{k=1}^{M}(1-\ q+q\cdot M_{k}( z))\ e^{-\frac{z}{\beta}}\ \frac{2d}{(\frac{\beta}{\beta})^{2}}dddz}. \tag{14}\]  
In order to calculate EE, the BS transmit power, \(\text{P}_{\text{T}}\), including the hardware static power consumed to operate the BS circuit, \(\text{P}_{\text{BS}}\) and RIS circuit power, \(\text{P}_{\text{RIS}}\), make up the overall power consumed to operate the analyzed RIS-based system. It is worth noting that the RIS does not absorb any transmit power because its reflectors are passive-element devices that do not amplify or reducing the magnitude of the incoming signal directly. Any amplification gain supplied by the RIS is produced by adjusting the phase shifts of the reflecting elements in such a way that the reflected signals are recombined with phase coherence. Different bit-resolution phase shifting patterns of each RIS element have different power consumption levels. The 3-, 4-, 5-, and 6-bit resolution consumed 1.5, 4.5, 6.0, and 7.8 mW of power, respectively [7]. The power dissipated on the whole surface of RIS with N identical reflecting components can be represented as \(P_{RIS}=\mathrm{NP}_{n}(b)\), where \(P_{n}(b)\) specifies the power consumption of each phase shifter with \(b\)-bit resolution. The EE of the considered RIS-based system can then be expressed as [14]:  
\[EE_{RIS}=\frac{E[C_{RIS}]}{P_{T}+P_{BS}+P_{RIS}}, \tag{15}\]  
where the nominator in (15) is (14).