## 5 Results and Discussion  
In this section, Monte-Carlo simulations are used to substantiate the outage probability and sum capacity performances of the proposed blind-RIS-NOMA uplink. The simulations are repeated for \(10^{6}\) channel realizations and the average results are presented. The simulation parameters are listed in Table 1.  
For the blind-RIS-assisted NOMA uplink, the closed-form probability of outage expressions is developed in this work. The related works have emphasized that NOMA does not necessarily outperform OMA. RIS is introduced to enhance the performance of conventional NOMA. Most classic works take an ideal RIS with an infinite resolution phase shifter into consideration. A low-resolution phase shifter RIS is taken into consideration in several works. Here, we sought to demonstrate that even a blind-RIS will have a considerable impact on the effectiveness of uplink outages and sum capacity. For a classic NOMA uplink, the probability of outage expressions is derived in [18]. The probability of outage expressions for user 1 of the traditional NOMA system is given by [18]  
\[P_{out\_NOMA}^{User\ 1}=1-\frac{exp\Big{\{}-\frac{D_{1}}{\mu_{1}\ \kappa_{1}^{ 2}}\Big{\}}}{1+\frac{\mu_{2}D_{1}\xi_{1}^{2}}{\mu_{1}\ \xi_{1}^{2}}} \tag{38}\]  
where \({\zeta_{1}}^{2}\) and \({\zeta_{2}}^{2}\) are the average channel gains of user 1 and user 2 of a traditional NOMA system. The probability of outage expression for user 2 of a traditional NOMA system is given by [18]  
\[P_{out\_NOMA}^{User\ 2}=1-\frac{exp\Big{(}-\Big{(}\frac{D_{1}}{\mu_{1}\ \kappa_{1}^{ 2}}+\frac{D_{2}}{\mu_{2}\ \kappa_{2}^{2}}+\frac{D_{1}D_{2}}{\mu_{1}\ \kappa_{1}^{2}}\Big{)}\Big{)}}{1+\frac{\mu_{2}D_{1}\xi_{2}^{2}}{\mu_{1}\ \xi_{1}^{2}}} \tag{39}\]  
Hence, the proposed model is compared to the classic NOMA discussed in [18]. The outage performance of the proposed blind-RIS-SR-NOMA and traditional NOMA is shown in Figure 3. The power fractions assigned to users 1 and 2 are 0.9 and 0.1, respectively. The number of RIS elements is 32. Due to higher channel gain and more power allocation for user 1, the outage probability of user 1 of traditional NOMA is better than user 2. As per (38), a larger \(\mu_{1}\) makes larger \(exp\Big{\{}-\frac{D_{1}}{\mu_{1}\ \kappa_{1}^{2}\ }\Big{\}}\). This results in a lower outage probability compared to user 2. The introduction of blind-RIS elements in the environment increases the array gain, thereby decreasing the outage probability. RIS adds \(N\) channel paths to create stronger combined channels. In accordance with (14) and (21), \(N\) is introduced to the probability of outage expressions for the RIS-assisted NOMA system. Higher \(\mu_{1}\) and \(N\) make \(exp\Big{\{}-\frac{D_{1}}{\mu_{1}\ \kappa\ \mathrm{N}\
u_{1}\tau^{2}}\Big{\}}\) in (14) higher. As a result, user 1 of the proposed system is less likely to experience an outage than in the conventional NOMA system. Similarly, \(N\) is introduced into the denominator of \(exp\Big{(}-\frac{D_{1}}{N}\Big{(}\frac{D_{1}}{\mu_{1}\ \kappa_{1}^{2}}+\frac{D_{2}}{\mu_{2}\ \kappa_{2}^{2}}+\frac{D_{1}D_{2}}{\mu_{1}\ \kappa_{1}^{2}}\Big{)}\Big{)}\) of (21). When compared to a traditional NOMA system, a higher \(N\) introduces a significant reduction in the outage probability of user 2. At the SNR of 10 dB, user 1\({}^{\prime}\)s outage probability in the blind-RIS-SR-NOMA and traditional NOMA is 0.0224 and 0.0432, respectively. User \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline Number of reflecting elements & 32, 64, 128, 256, 512 \\ Number of users & 2 \\ Average channel gain of user 1 & 5 \\ Average channel gain of user 2 & 1 \\ The desired rate of users (b/s/Hz) & 1 \\ Power allocated to user 1 & 0.9, 0.7 \\ Power allocated to user 2 & 0.1, 0.3 \\ Block size & \(10^{6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters [18; 21; 25; 30; 31]. 2's outage probability in the blind-RIS-SR-NOMA and traditional NOMA is 0.0532 and 0.6558, respectively, for 10 dB SNR.