# Guaranteeing QoS for NOMA-Enabled URLLC Based on \(\kappa\)-\(\mu\) Shadowed Fading Model
## 4 Simulation Results and Analyses
Furthermore, we summarize the second-order approximate parameters obtained under the different numbers of summed elements and different distributions listed in Table 2, which were obtained by using _fsolve_ in MATLAB. From Figure 3, we also observe that the generalized Puiseux series expansion process converges very quickly, so it can effectively fit the PDF without complex approximate orders. All these properties lay the foundation that appropriately characterizing the statistical characteristics of the sum of multiple i.n.i.d. shadowed RVs is the key to unsolved problems. This again verifies the effectiveness and good analytical properties of our proposed theoretical tool (Theorem 1) in complex future communication environments. For the proposed CF mMIMO system, Figure 4 depicts the PDFs of the PPSNR under different fading environments when the transmit power is 10 dBm and the distance from the user to the center of the coverage area is 200 m. We verify that the cylindrical diagram (obtained by simulation using Monte Carlo method for \(10^{7}\) iterations) and the curve with asterisk (obtained by numerical calculation (Corollary 1)) have almost the same slope. This indicates that our proposed theoretical tools based on Corollary 1 have superior flexibility, mathematical tractability and practicality for wireless communication system design. Furthermore, these findings also reveal the \(\kappa-\mu\) shadowed fading model based on our proposed theoretical tools, which form a reliable and mathematically tractable channel fading model can be extended to other common future 6G wireless communication systems such as CF mMIMO systems, enabling this model as a more applicable approach to complex 6G applications in the future. \begin{table}
\begin{tabular}{c c c c c} \hline  & \(\mathbf{c_{1}}\) & \(\mathbf{c_{2}}\) & \(\mathbf{\omega_{1}}\) & \(\mathbf{\omega_{2}}\) \\ \hline \multicolumn{5}{c}{Case 1:} \\ \(\kappa=4,\mu=1,m\rightarrow\infty\) & 0.9998 & 0.0002 & 0.9976 & 16.6538 \\ \hline \multicolumn{5}{c}{Case 2:} \\ \(\kappa=4,\mu=1,m\rightarrow\infty\) & & & & \\ \(\kappa=4,\mu=1,m=3\) & 0.9985 & 0.0015 & 0.9923 & 5.9651 \\ \(\kappa=6,\mu=1,m=3\) & & & & \\ \hline \multicolumn{5}{c}{Case 3:} \\ \(\kappa=4,\mu=1,m\rightarrow\infty\) & & & & \\ \(\kappa=4,\mu=1,m=3\) & 0.8605 & 0.1395 & 0.9355 & 1.3979 \\ \(\kappa=4,\mu=2,m\rightarrow\infty\) & & & & \\ \hline \multicolumn{5}{c}{Case 4:} \\ \(\kappa=4,\mu=1,m\rightarrow\infty\) & & & & \\ \(\kappa=4,\mu=1,m=3\) & & & & \\ \(\kappa=6,\mu=1,m=3\) & 0.8902 & 0.1098 & 0.9486 & 1.4166 \\ \(\kappa=4,\mu=2,m\rightarrow\infty\) & & & & \\ \hline \end{tabular}
\end{table} 
Table 2: The second-order approximate parameters in different cases shown in Figure 3.  
Figure 4: PDFs of the PPSNR for CF mMIMO system. (**a**) under the Rician fading channel; (**b**) under the Rayleigh fading channel; (**c**) under the One-sided fading channel; (**d**) under the Nakagami-\(m\) fading channel; (**e**) under the Rician shadowed fading channel; (**f**) under the \(\kappa\)–\(\mu\) shadowed fading channel. Figure 3: PDFs of the sum of i.n.i.d. \(\kappa\)–\(\mu\) shadowed RVs. (**a**) Case 1; (**b**) Case 2; (**c**) Case 3; (**d**) Case 4. Figure 5 plots the UB-QDVPs obtained with different target delays over a \(\kappa\)-\(\mu\) shadowed fading model for the CF mMIMO system when the transmit power is 10 dBm and the distance from the user to the center of the coverage area is 200 m. The UB-QDVP decreases as the target delay increases. For the same target delay, the users have inconsistent UB-QDVPs under different fading environments. In Figure 5c, under Nakagami-\(m\) fading (\(\kappa\to 0,\mu=3,m\rightarrow\infty\)) and a target delay of 0.05 ms, the UB-QDVP is \(10^{-7}\). However, when the UB-QDVP is \(10^{-7}\) under One-sided Gaussian fading (\(\kappa\to 0,\mu=0.5,m\rightarrow\infty\)), the target delay exceeds 0.05 ms. For the performance evaluation, analysis and design of the system, it is necessary to consider the impacts of different fading environments on the system performance.