# User Pairing for Delay-Limited NOMA-Based Satellite Networks with Deep Reinforcement Learning
## 3 Effective Capacity and Power Allocation
### Problem Formulation  
Although the closed-form expression of sum rate for the considered system has been derived, we must note that the rate of User \(j\) (\(j=c,t\)) is influenced by many factors, such as delay exponent \(\theta_{j}\), transmission SNR \(\gamma\), fading severity, location information \(d_{j}\), and \(\alpha_{j}^{p}\). Thus, to expressively show the different impacts of these key parameters on the achieved performance, the normalized effective capacity of User \(j\) is plotted in Figure 1, where ILS, AS, and FHS are infrequent light shadowing, average shadowing, and frequent heavy shadowing, respectively.  
From Figure 1, we can directly observe that, when \(\theta_{j}\to 0\), effective capacity converges to the ergodic capacity, since only delay-insensitive traffic is needed. However, when \(\theta_{j}>10\), even for case \(\alpha_{j}^{p}=1\), effective capacity reduces to 0 due to the required delay QoS being too stringent. Thus, the range of User \(j\)'s delay limitation is assumed to be constrained as \(\theta_{j}\in[0.5,10]\) in this paper. In addition, an increased \(d_{j}\), i.e., a worse fading severity, or decreased \(\gamma\) can degrade the capacity curves. Moreover, all  
Figure 1: Normalized effective capacity versus delay exponent \(\theta_{j}\) for various SNR \(\gamma\), fading severity, and location information \(d_{j}\), when \(\alpha_{j}^{p}=1\).  
with increasing \(\theta_{j}\). This observation clearly indicates that the achieved performance suffers from a combination of these factors, although, in both Case 1 and Case 2, it seems like a user with the smallest delay QoS exponent, nearest location information, and best shadowing should be selected as User \(t/c\) and paired with the User \(c/t\) in Case 1/2 to maximize the sum performance of the considered system. Conversely, while in a spot beam, the user with the nearest location information or best fading condition may have a relatively large \(\theta\), or vice versa. Thus, how to select User \(t/c\) in Case 1/2 is a vital issue in a delay-limited scenario.  
For simplicity, herein, we mainly focus on the user pairing in Case 1, which means that \(\alpha_{c}^{p}\) must meet \(C_{t}^{\text{N}}(\theta_{t})\geq C_{t}^{\text{T}}(\theta_{t})\). Then, the optimization problem is to find a user who can obtain the best power utilization efficiency, after taking into account link budget and delay QoS requirement, to be the User \(t\). The mathematical formulation of this problem can be denoted by P1 and formulated as  
\[\text{P1}:\max_{\{d_{t},\rho_{t}\}}C_{t}^{\text{N}}(\theta_{t})\] \[s.t.\] \[\text{C1}:Q_{t}>Q_{c},t\in 1,2,\cdots,c-1,c+1,\cdots,m;\] \[\text{C2}:(11),\theta_{c}>\theta_{t}>0;\] \[\text{C3}:d_{c,t}\leq R. \tag{23}\]  
In the aforementioned problem, C1 ensures that the link budget of User \(t\) must be better than that of User \(c\) to successfully perform SIC; C2 denotes that, in Case 1, the resource allocation threshold in (11) must be ensured to guarantee the minimum data rate requirement of User \(c\), and C3 implies that the limited location information of Users \(c\) and \(t\).