# User Pairing for Delay-Limited NOMA-Based Satellite Networks with Deep Reinforcement Learning
## 2 System Model  
Consider a downlink NOMA-based satellite system that is designed to serve \(m\) (\(m\geq 2\)) users with the help of the NOMA scheme. These \(m\) users are randomly deployed in an area approximated as a circle of radius \(R\) with different channel statistical prosperities and delay QoS requirements. (In this paper, channel estimation errors, co-channel interference, complexity, and mobility constraints are not taken into consideration in the proposed system model; the influences of these parameters on user selection and system performance will be a focus in our future works, based on the contributions in the current work.) Without loss of generality, users are ordered based on their link budgets, i.e., \(Q_{1}\leq Q_{2}\cdots\leq Q_{m}\), where \(Q_{j}\) is the link budget of User \(j\) (\(j=1,2,\)\(\cdots,\)\(m\)). For simplicity, we further assume only the \(c^{th}\) and \(t^{th}\) users (\(1\leq c<t\leq m\)) are selected to form a NOMA group, and each user in the proposed model is equipped with a single antenna. Thus, the received signal at User \(j\) (\(j=c,t\)) is  
\[y_{j}=Q_{j}x+w_{j}, \tag{1}\]  
where \(w_{j}\) denotes the noise at User \(j\) with zero mean and \(\delta^{2}\) variance, \(x=\sum_{j=c,t}\sqrt{\alpha_{j}^{p}P_{s}}x_{j}\) is the superposed signal (with \(\alpha_{j}^{p}\) being a fraction of the transmission power \(P_{s}\) allocated to User \(j\) and \(x_{j}\) (\(E[|x_{j}|^{2}]=1\)) being the signal for User \(j\)), \(Q_{j}\) (including FSL, antenna gain, beam gain, and fading model) is the entire link budget from satellite to User \(j\), which can be described as follows:  
\[Q_{j}=\Phi_{j}G_{s}\big{(}\varphi_{j}\big{)}\big{|}g_{j}\big{|}^{2}, \tag{2}\]  
where \(\Phi_{j}=L_{j}G_{j}\), with \(L_{j}\) and \(G_{j}\) being the FSL and antenna gain at User \(j\), respectively. \(G_{s}\big{(}\varphi_{j}\big{)}\), which is the beam gain of User \(j\), with \(\varphi_{j}\) denoting the angle between User \(j\) and beam center with respect to the satellite, can be approximated as [5]  
\[G_{s}\big{(}\varphi_{j}\big{)}\approx G_{\max}\Bigg{(}\frac{J_{1}\big{(}ad_{j }\big{)}}{2 ad_{j}}+36\frac{J_{3}\big{(}ad_{j}\big{)}}{a^{3}d_{j}^{3}}\Bigg{)}^{ 2}=G_{s}\big{(}d_{j}\big{)}, \tag{3}\]  
with \(G_{\max}\) representing the maximum antenna gain, \(J_{n}(\cdot)\) being the Bessel function of first kind and \(n\)-th order, \(d_{j}\) being the distance from the beam center to User \(j\), and \(a=2.07123/R\). \(\big{|}g_{j}\big{|}^{2}\) is the channel power gain of the satellite link, which is assumed to follow a widely applied Shadowed Rician fading model [26; 27; 28; 29; 30]. According to [31], the probability density function (PDF) of \(\big{|}g_{j}\big{|}^{2}\) is  
\[f_{\big{|}g_{j}\big{|}^{2}}(x)=\alpha_{j}e^{-\beta_{j}x}{}_{1}F_{1}\big{(}m_{ j};1;\delta_{l}x\big{)}, \tag{4}\]  
where \(\alpha_{j}=\frac{\big{(}2b;m_{j}\big{)}^{mj}}{2b\big{(}2b;m_{j}+\Omega_{j} \big{)}^{m_{j}}}\), \(\delta_{j}=\frac{\Omega_{j}}{2b_{j}\big{(}2b;m_{j}+\Omega_{j}\big{)}}\), \(\delta_{j}=\frac{1}{2b_{j}}\) with \(2b_{j}\) and \(\Omega_{j}\), respectively, being the average power of the multipath and the LoS components, \(m_{j}\) (\(m_{j}>0\)) denoting the Nakagami-\(m\) fading parameter, and \({}_{1}F_{1}(a;b;c)\) representing the confluent hypergeometric function ([32], Equation (9.14.1)). Based on the principle of the downlink NOMA scheme, decoding order is decided by users' channel qualities, i.e., the user with a worse link condition decodes its own information firstly and directly. Thus, the signal-to-interference-plus-noise ratio (SINR) of User \(c\) is  
\[\gamma_{c}^{\rm N}=\frac{\alpha_{c}^{p}\gamma\Phi_{c}G_{s}(d_{c})\big{|}g_{c} \big{|}^{2}}{\alpha_{t}^{p}\gamma\Phi_{c}G_{s}(d_{c})\big{|}g_{c}\big{|}^{2}+ 1}=\frac{\alpha_{c}^{p}\gamma Q_{c}}{\alpha_{t}^{p}\gamma Q_{c}+1}, \tag{5}\]  
where \(\alpha_{c}^{p}+\alpha_{t}^{p}=1\) and \(\gamma=P_{s}/\delta^{2}\) is the average transmission SNR.