# NOMA or OMA in Delay-QoS Limited Satellite Communications: Effective Capacity Analysis
## 2 System Model  
As shown in Figure 1, we suppose that there is a satellite simultaneously transmitting signals to multiple users with the help of the NOMA scheme. These users are uniformly deployed in the same spot beam. Specifically, users are ordered based on their link qualities, i.e., \(g_{1}\leq g_{2}\cdot\cdot\cdot\leq g_{m}\), where \(g_{j}\) is the channel coefficient from satellite to User \(j\) (\(j=1\), \(2\), \(\cdot\cdot\cdot\), \(m\)). 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, each user in the proposed model is equipped with a single antenna.  
Thus, the received signal at User \(j\) (\(j=c,t\)) is  
\[y_{j}=g_{j}x+w_{j}, \tag{1}\]  
where \(w_{j}\) denotes the noise at the User \(j\) with zero mean and \(\delta^{2}\) variance, \(x=\sqrt{\alpha P_{s}}x_{t}+\sqrt{(1-\alpha)P_{s}}x_{c}\) is the superposed signal with \(\alpha\) being a fraction of the transmission power \(P_{s}\) allocated to User \(t\) and \(x_{j}\) (\(E[|x_{j}|^{2}]=1\)) being the signal for User \(j\).  
According to the downlink NOMA principle, the user with a weaker channel condition is allocated with more power resources; thus, \(\alpha\leq 0.5\) is assumed here, and decoded directly by viewing other users' information as noise. Then, with \(\left|g_{c}\right|^{2}\leq\left|g_{t}\right|^{2}\), the signal-to-interference-plus-noise ratio (SINR) of User \(c\) can be written as  
\[\gamma_{c}^{N}=\frac{(1-\alpha)\rho\left|g_{c}\right|^{2}}{\alpha\rho\left|g_ {c}\right|^{2}+1}, \tag{2}\]  
while user with better channel condition, i.e., User \(t\), first adopts the successive interference cancellation (SIC) strategy to decode and remove the interference from User \(c\), and the decoding SINR is  
\[\gamma_{t-\tau c}^{N}=\frac{(1-\alpha)\rho\left|g_{t}\right|^{2}}{\alpha\rho \left|g_{t}\right|^{2}+1}. \tag{3}\]  
We can derive that \(\gamma_{c}^{N}<\gamma_{t-\tau c}^{N}\) since \(g_{c}<g_{t}\). Then, User \(t\) decodes its own information and the achieved SINR is  
\[\gamma_{t}^{N}=\alpha\rho\left|g_{t}\right|^{2}. \tag{4}\]  
When a Shadowed-Rician fading distribution is adopted [26; 27; 28; 29; 30], the probability density function (PDF) of \(\rho_{j}\) with \(\rho_{j}=\rho\alpha\left|g_{j}\right|^{2}\) and \(\rho=P_{s}/\delta^{2}\) being the transmission SNR, can be given by  
\[f_{\rho_{j}}(x)=\frac{\alpha_{j}}{\alpha\rho}e^{-\frac{\beta_{j}x}{4\rho}}{{}_ {1}F_{1}}\left(m_{j};1;\frac{\delta_{j}x}{\alpha\rho}\right), \tag{5}\]  
Figure 1: The proposed NOMA-based satellite networks with delay QoS constraints.  
where \(\alpha_{j}\)=\(\frac{\left(2b_{j}m_{j}\right)^{m_{j}}}{2b_{j}\left(2b_{j}m_{j}+\alpha_{j} \right)^{m_{j}}}\), \(\delta_{j}\)=\(\frac{\Omega_{j}}{2b_{j}\left(2b_{j}m_{j}+\Omega_{j}\right)}\), \(\delta_{j}\)=\(\frac{1}{2b_{j}}\) 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 \({}_{1}F_{1}(a;b;c)\) being the confluent hypergeometric function ([31], Equation (9.14.1)).  
Then, the achievable data rate of User \(j\) with the NOMA strategy can be formulated as  
\[R_{j}^{N}=\log_{2}\Bigl{(}1+\gamma_{j}^{N}\Bigr{)}. \tag{6}\]  
While if an OMA scheme, i.e., time division multiple access (TDMA), is used, the achievable SINR and data rate of User \(j\) can be, respectively, obtained as  
\[\gamma_{l}^{T}=\rho|g_{l}|^{2}, \tag{7}\]  
and  
\[R_{j}^{T}=0.5\log_{2}\Bigl{(}1+\gamma_{j}^{T}\Bigr{)}, \tag{8}\]  
where \(\gamma_{j}^{T}=\rho\bigl{|}g_{j}\bigr{|}^{2}\) and 0.5 here is due to each user only having half of the available time resources to transmit information in the TDMA scheme.