# Delay-Sensitive NOMA-HARQ for Short Packet Communications
## 3 Delay-Sensitive NOMA-HARQ
### Generalized \(N_{u}\) User Setup
In general, the system transits from state \(\mathbf{J}_{u}\) to \(\mathbf{J}_{v}\) with probabilities as follows:  
\[\pi_{u\to v}=\prod_{i=1}^{N_{u}}\mathcal{P}_{J_{i}}(i,u),\quad[J_{1},\cdots,J_ {N_{u}}]=\mathbf{J}_{v}, \tag{9}\]  
where \(\gamma_{u}^{(i)}\) and \(\tilde{\gamma}_{2}^{(i)}\) are the SINR during first transmission and retransmission. Now, we provide general guidelines to calculate the associated SINRs of the state transition probabilities for general \(N_{u}\) users. The SINRs can be calculated based on the SIC decoding order. In general, a user with the highest received power is decoded first under the interference of other users with less power. Furthermore, if a user's message is decoded successfully, its interference is removed; otherwise, it causes interference. Furthermore, when BS receives more copies due to the retransmission of a user, it is given priority over usual decoding order. This is because, with more copies, a weaker user could be decoded with better quality than a strong user. For example, when user \(j\) is retransmitting and has one more packet due to single retransmission than user \(i\), it is decoded prior to user \(i\). Consequently, if user \(j\) is decoded successfully its interference is eliminated, and user \(i\) experience less interference. These different states of user refer to the typical state in the Markov model dented as \(\mathbf{J}_{u}\). In general, SINR of user \(i\) during its first transmission is given as:  
\[\gamma_{u}^{(i)}=\frac{\bar{\alpha}_{i}P_{i}}{\alpha_{i}P_{i}+\sum_{j}P_{j}- \sum_{k}\alpha_{k}P_{k}+1},\quad\forall j\in\mathcal{I}_{j},k\in\mathcal{I}_ {k} \tag{10}\]  
where \(1\leq i\leq N_{u}\) and \(\mathcal{I}_{j}\in\{i+1,\cdots,N_{u}+1\}\), \(P_{N_{u}+1}=0\), \(\alpha_{N_{u}+1}=0\) for notation consistency and \(\mathcal{I}_{k}\subseteq\mathcal{I}_{j}\) is the set of indices referring to users that are successfully decoded after retransmission. Moreover, \(P_{j}-\alpha_{j}P_{j}=\delta_{j}P_{j}\). The \(\alpha_{i}P_{i}\) term in the denominator of (10) is the interference caused due to \(i\)-th user's previously retransmitting packet. This would be zero if the packet has been decoded successfully. For example, when \(N_{u}=2\)\(\gamma_{4}^{(1)}\) and \(\gamma_{5}^{(1)}\) shows this case in Lemma 1. If there is no retransmitting packet, then SINR changes due to \(\bar{\alpha}_{i}=1\), as shown in \(\gamma_{2}^{(1)}\) in Lemma 1 for \(N_{u}=2\) case. Moreover, \(\sum_{k}\alpha_{k}P_{k}\) is the interference removed due to successfully decoding users with index set \(\mathcal{I}_{k}\). However, if the decoding failed then the interference cannot be removed. Consequently, the SINR could change to \(\gamma_{u}^{(i)}=\frac{P_{i}}{\sum_{j}P_{i}+1}\ \forall j\in\mathcal{I}_{j}\), which shows that the BS decodes the message of the \(i\)-th user under the interference of subsequent \(j\) users with maximum power. User \(i\) always conducts its retransmission under the interference of its own new arriving packet and packets of subsequent users. The general expression of SINR of user \(i\) during retransmission can be written as:  
\[\tilde{\gamma}_{z}^{(i)}=\frac{\alpha_{i}P_{i}}{\tilde{\alpha}_{i}P_{i}+\sum_{ j=i+1}^{N_{u}}P_{j}+1},\ \ z\in\{0,1,2\}, \tag{11}\]  
where \(1\leq i\leq N_{u}\). Upon successful decoding of user \(i\) after retransmission, BS can remove its interference amounts to \(\alpha_{i}P_{i}\), \(\forall i<j\) for decoding user \(j\). With all the state transition probabilities, the stationary distribution corresponding to the MM can be calculated using standard methods. Finally, accumulating the stationary probabilities corresponding to the erroneous packet states of each user (\(J_{i}=e\)), the PER and throughput can be obtained for \(N_{u}\) users similar to Equations (5) and (6). **Remark 3**: _Note that, in the general \(N_{u}\) users case, we keep the maximum allowed retransmission of failing packet to one. In URLLC, the number of retransmissions is kept small to minimize the delay.