# User Pairing for Delay-Limited NOMA-Based Satellite Networks with Deep Reinforcement Learning
## 3 Effective Capacity and Power Allocation
### Power Allocation Strategy  
To ensure the capacity achieved by the user with a critical delay QoS requirement using the NOMA scheme is always better than that with the TDMA scheme, the power allocation coefficient should be further constrained. In this section, a power allocation scheme is investigated for two cases, i.e., User \(c\) in Case 1 and User \(t\) in Case 2 are assumed to be delay-sensitive users. For Case 1, \(\theta_{c}>\theta_{t}\) is assumed and the power allocation factor is limited by \(C_{c}^{\rm N}(\theta_{c})\geq C_{c}^{\rm T}(\theta_{c})\), where  
\[C_{c}^{\rm N}(\theta_{c})=\frac{1}{\psi_{c}\mathrm{ln}2}\ln\biggl{(}\mathcal{E }\biggl{[}\Bigl{(}1+\gamma_{c}^{\rm N}\Bigr{)}^{\psi_{c}}\biggr{]}\biggr{)}, \tag{9}\]  
and  
\[C_{c}^{\rm T}(\theta_{c})=\frac{1}{\psi_{c}\mathrm{ln}2}\ln\biggl{(}\mathcal{E }\biggl{[}\Bigl{(}1+\gamma_{c}^{\rm T}\Bigr{)}^{0.5\psi_{c}}\biggr{]}\biggr{)}, \tag{10}\]  
with \(\gamma_{j}^{\rm T}=\gamma\Phi_{j}G_{\rm s}(d_{j})\bigl{|}g_{j}\bigr{|}^{2}= \gamma Q_{j}\) being the SINR of User \(j\) (\(j=c,t\)) achieved with the TDMA scheme, and 0.5 owes to the loss in multiplexing in the TDMA system. By substituting (5) into (8), along with some manipulations, \(\alpha_{c}^{p}\) can be derived as  
\[\alpha_{c}^{p}\geq 1-\frac{1}{\sqrt{1+\gamma Q_{c}}+1}, \tag{11}\]  
which means that the value of \(\alpha_{c}^{p}\) is decided by \(\gamma\), location information, and fading severity of User \(c\). For Case 2, \(\theta_{t}>\theta_{c}\) is considered, and factor \(\alpha_{c}^{p}\) is limitied by restriction condition \(C_{t}^{\rm N}(\theta_{t})\geq C_{t}^{\rm T}(\theta_{t})\), with  
\[C_{t}^{\rm N}(\theta_{t})=\frac{1}{\psi_{t}\mathrm{ln}2}\ln\biggl{(}\mathcal{E }\biggl{[}\Bigl{(}1+(1-\alpha_{c}^{p})\gamma Q_{t}\Bigr{)}^{\psi_{t}}\biggr{]} \biggr{)}, \tag{12}\]and  
\[C_{t}^{\rm T}(\theta_{t})=\frac{1}{\psi_{t}{\rm ln}2}\ln\Bigl{(} \mathcal{E}\Bigl{[}(1+\gamma Q_{t})^{0.5\varphi_{t}}\Bigr{]}\Bigr{)}. \tag{13}\]  
Then, we can obtain  
\[\alpha_{c}^{p}\leq 1-\frac{1}{\sqrt{1+\gamma Q_{t}+1}}. \tag{14}\]  
Based on the power allocation coefficient obtained in (11) for Case 1 or (14) for Case 2, the effective capacity of User \(c\) can be given by  
\[C_{c}^{\rm N}(\theta_{c}) = \frac{1}{\psi_{c}{\rm ln}2}\ln\biggl{(}\mathcal{E}\Bigl{[}\left(1 +\gamma_{c}^{\rm N}\right)^{\psi_{c}}\Bigr{]}\biggr{)} \tag{15}\] \[= \frac{1}{\psi_{c}{\rm ln}2}\ln\Biggl{(}\int_{R_{jm}}^{R_{jm}}\int _{0}^{\infty}\left(1+\gamma_{c}^{\rm N}\right)^{\varphi_{c}}f_{|g_{c}|^{2}}(x) f_{d_{c}}(y)dxdy\Biggr{)},\]  
where \(f_{d}(y)=\frac{2y}{{\rm R}_{jm}^{2}-{\rm R}_{jm}^{2}}\) is the PDF of User \(j\)'s location [4] if it distributes in an annular area with inner radius \(R_{jm}\) and outer radius \(R_{jf}\). To evaluate (15), we first express \({}_{1}F_{1}\bigl{(}m_{j};1;\delta_{j}x\bigr{)}\) in (4) and \((1+x)^{d}\) in terms of the Meijer G-functions from Equation (9.34.8) in [32] and binominals represented by Equation (1.11) in [32], as  
\[{}_{1}F_{1}\bigl{(}m_{j};1;\delta_{j}x\bigr{)}=\frac{1}{\Gamma(m_{j})}G_{1,2} ^{1,1}\biggl{[}-\delta_{j}x\biggr{|} \begin{array}{c}1-m_{j}\\ 0,0\end{array} \biggr{]}, \tag{16}\]  
and  
\[(1+x)^{d}=\sum_{k=0}^{\infty}\frac{\Gamma(a+k)}{k!\Gamma(a)}x^{k}, \tag{17}\]  
where \(G_{1,2}^{1,1}[\cdot|\cdot]\) ([32], Equation (9.301)) is the Meijer-G function and \(\Gamma(\cdot)\) ([32], Equation (8.310.1)) is the Gamma function.