# Delay-Sensitive NOMA-HARQ for Short Packet Communications
## 2 System Model and Preliminaries  
We consider an uplink power-domain NOMA scenario, where \(N_{u}\) users can simultaneously send their messages to the base station (BS). Similar to [38], we consider time division duplex system (TDD) so that BS synchronizes the uplink transmission of each user by sending a beacon signal at the beginning of each time slot. The channel between the \(i\)-th user, \(1\leq i\leq N_{u}\), and the BS, denoted by \(g_{i}\), is modeled by large-scale path-loss and small-scale Rayleigh fading [39]. We assume that BS knows the channel state information (CSI) of each user perfectly. The \(i\)-th user encodes and modulates its \(k_{i}\)-bit message into a packet of length \(n\) symbols and sends it to the BS. Let \(y(t)\) denote the received signal at the BS at time \(t\) given as:  
\[y(t)=\sum_{i=1}^{N_{u}}g_{i}x_{i}(t)+w(t), \tag{1}\]  
where \(x_{i}(t)\in\mathbb{C}\) is the transmitted complex symbols from the \(i\)-th user and \(w(t)\sim\mathcal{CN}(0,1)\) is the additive white Gaussian noise (AWGN). We assume that \(\mathbb{E}[|x_{i}(t)|^{2}]=1\). Let \(P_{i}\) be the received power of user \(i\) at the BS given as  
\[P_{i}=|g_{i}|^{2}P_{i}, \tag{2}\]  
where \(P_{i,i}\) is the transmit power of user \(i\), \(|g_{i}|^{2}=h_{i}r_{i}^{-\rho}\), \(h_{i}\) is the small-scale fading with exponential distribution, i.e., \(h_{i}\sim\exp(1)\), \(r_{i}\) is the distance between the \(i\)-th user and the BS, and \(\rho\) is the path-loss exponent. We assume block fading channel model, such that the channel remains constant over a time block and changes independently between the blocks. The BS can pair users according to their CSIs and SNR levels to meet the desired level of reliability. BS usually pairs near and far users to exploit their power difference for better SIC decoding. We assume that paired users have certain finite channel gains, so that their transmit power does not exceed the maximum energy budget. Let \(0<w_{i}<1\) denote the ratio of powers between the paired users, such that \(P_{i}=w_{i}P_{c}\), where \(P_{c}\) denotes the total received power at the BS from paired users, where \(w_{1}+w_{2}=1\) (We use parameter \(w_{i}\) to simplify presentation regarding effect of power difference in the total received power. Otherwise, if the total transmit power constraint on each user is used we need to specify channel gains while presenting the results. ), when \(N_{u}=2\). We assume that \(w_{1}>w_{2}\) to treat user \(1\) as a near user and user \(2\) as a far user. In practical settings, there may be channel estimation errors that could lead to degradation in optimal user pairing as well as the SIC performance of multiple access systems (In future publications, we would incorporate channel estimation errors and its impact on the performance.). The receiver first decodes user \(1\) while treating the message of other users as noise. If user \(1\)'s signal is successfully recovered, it is then removed from the received signal and user \(2\)'s signal is then decoded and removed from the received signal. This continues to decode all \(N_{u}\) users. Each user reports its decoding status using an instantaneous ACK. Upon receiving an ACK, the user sends a new packet; otherwise, upon receiving a NACK, it retransmits the previous packet, through either CC-HARQ or IR-HARQ, in the next time slot. Generally, user \(1\) is decoded first due to its higher received power at the BS, unless other users have more copies due to retransmissions. Figure 1a shows packet transmission with NOMA and standard HARQ (S-NOMA-HARQ) [22] when \(N_{u}=2\). We use normal approximation [10], to characterize the PER in the finite blocklength regime.