# Guaranteeing QoS for NOMA-Enabled URLLC Based on \(\kappa\)-\(\mu\) Shadowed Fading Model
## 2 System Model  
We consider an uplink CF mMIMO system in which \(L\) single-antenna APs and \(K\) single-antenna users are randomly uniformly distributed in a circular area with \(L\gg K\), as shown in Figure 1. Some conventional fading can be obtained by adjusting the parameters of the \(\kappa-\mu\) shadowed fading model: Rayleigh fading (\(\kappa\to 0,\mu=1,m\rightarrow\infty\)), Rician fading (\(\kappa\ =\ \kappa,\mu=1,m\rightarrow\infty\)), Rician shadowed fading (\(\kappa=\kappa,\mu=1,m=\m\)), and other fading (\(\kappa=\kappa,\mu=\underline{\mu},m=\underline{m}\), where \(\kappa\), \(\underline{\mu}\) and \(\underline{m}\) are nonnegative real numbers. When \(\underline{m}\rightarrow\infty\), it denotes that no shadow effect is observed). All APs simultaneously serve all users at the same time-frequency resource and are connected to a baseband unit (BBU) for centralized signal processing via ideal backhaul links [19; 49] (There is no delay in the ideal backhaul links between APs and the BBU [50]). The transmission from the APs to the users (downlink transmission) and the transmission from the users to the APs (uplink transmission) proceed by a time-division-duplexing operation. Each coherence interval is divided into three phases: uplink training, downlink payload data transmission, and uplink payload data transmission. In the uplink training phase, the users send pilot sequences to the APs and the BBU estimates the channel for all users. In this paper, the QoS delay for URLLC is characterized by the queuing delay (Considering ideal backhaul links and tail behaviors based on [35; 36], we consider the queuing delay as the delay of the URLLC QoS requirement). Each user transmits a short packet with \(D\) information bits through \(N\) CUs, which are spread over \(B\) MHz of bandwidth and \(t_{f}\) milliseconds of time (\(N=Bt_{f}\)). The channel coefficient \(g_{lk}(t)\) between the \(l\)th AP and the \(k\)th user can be characterized as  
\[g_{lk}(t)=\sqrt{\beta_{lk}(t)}h_{lk}(t), \tag{1}\]  
where \(\beta_{lk}(t)\) is a large-scale fading coefficient that changes very slowly with time and is assumed to be constant over many coherence conditions. \(h_{lk}(t)\) is the small-scale fading coefficient of the corresponding links, which is assumed to be static during a finite-sized time-frequency coherence interval, and it changes independently from one coherence interval to the next [51]. The channel response is assumed to follow the \(\kappa\)-\(\mu\) shadowed fading model. The gain of small-scale fading between the \(k\)th (\(k\in[1,K]\)) user and the \(l\)th AP (\(l\in[1,L]\)) can be modeled as  
\[|h_{lk}|^{2}=\sum\limits_{i=1}^{\mu_{lk}}(\mathcal{X}_{lk,i}+\xi_{lk}\beta_{ lk,i})^{2}+(\mathcal{Y}_{lk,i}+\xi_{lk}\beta_{lk,i})^{2}, \tag{2}\]  
where \(\mu_{lk}\) is a natural number. Each multipath cluster is modeled by one term of the sum; thus, \(\mu_{lk}\) is the number of multipath clusters. Each cluster has scattered components with the same power and dominant components with a certain arbitrary power. \(\mathcal{X}_{lk,i}\) and \(\mathcal{Y}_{lk,i}\) are independent and identically distributed (i.i.d.) Gaussian RVs with means of zero and variances of \(\sigma^{2}\), and \(\mathcal{X}_{lk,i}+j\mathcal{Y}_{lk,i}\) denotes the scattered components of the \(i\)th cluster. \(\beta_{lk,i}\) and \(\beta_{lk,i}\) are real numbers and the mean values of the in-phase and quadrature parts of the multipath components of the \(i\)th cluster, respectively. \(\xi_{lk}\) is a Nakagami-\(m\) RV with shaping parameter and \(\mathbb{E}\left[\xi_{lk}^{2}\right]=1\), where \(\xi_{lk}^{2}\sim\Gamma(\mu_{lk},1/m_{lk})\); \(m_{lk}\) represents the fluctuation degree of dominant components caused by the shadow [28]. For each cluster, the total power of the scattered components is \(2\sigma^{2}\). Furthermore, the ratio of dominant components to scattered components in small-scale fading is \(\kappa_{lk}=\frac{\sum\limits_{i=1}^{\mu}\beta_{lk,i}^{2}+\beta_{lk,i}^{2}}{2 \mu_{lk}\sigma^{2}}\).