# Delay-Sensitive NOMA-HARQ for Short Packet Communications
## 3 Delay-Sensitive NOMA-HARQ
### Generalized \(N_{u}\) User Setup  
We can extend the model for general \(N_{u}\) users. When single retransmission is allowed, each \(i\)-th user, \(1\leq i\leq N_{u}\), can have \(J_{i}\in\{0,1,e\}\), packet states. Consequently, there will be maximum \(3^{N_{u}}\) vector states denoted as \(\mathbf{J}_{u}=[J_{1},J_{2},\cdots,J_{N_{u}}]\), \(u=[1,\cdots,3^{N_{u}}]\). We use SIC decoding, where users are decoded based on their level of received powers. We assume user \(i\) is relatively closer to the BS than user \(j\), when \(j>i\). For example user \(1\) is considered near user, and user \(2\) is considered far user. We assume that with an equal number of received packets, the total received power of user \(i\) is always higher than user \(j\). Therefore, the transmit power constraint of each user is implemented as \(w_{j}\leq w_{i}\) and \(\alpha_{i}\leq\alpha_{j}\). \(w_{i}\) is the fraction of total received power from user \(i\), i.e., \(\sum_{i=1}^{N_{u}}w_{i}=1\) and \(\alpha_{i}\) is the power of retransmitting signal, i.e., \(\alpha_{i}\in[0,1]\). Therefore, when the \(i\)-th user is in error, i.e., \(J_{i}=e\) the \(j\)-th user also cannot be decoded, i.e., \(J_{j}=e\), where \(i<j\leq N_{u}\). As a result, if the BS failed to decode user \(1\), then all the subsequent users would also fail the decoding due to their less received power than user \(1\). Similarly, if a user is successfully decoded only after retransmission, no subsequent user can succeed with single transmission. This assumption reduces the total number of states in the Markov model. The reduction in the maximum number of Markov model states denote as \(U\) is significant, for example, from 9 to 6 and 27 to 10 for \(N_{u}=2\) and \(N_{u}=3\), respectively. Lemma 1 can be extended to the general \(N_{u}\) users case. Lets consider a Markov model whose states are denoted as \(\mathbf{J}_{u}\), for \(u=[1,\cdots,U]\), where \(U\) is the total number of states. Let \(\mathcal{P}_{\ell}(i,u)\) denotes the probability of user \(i\) transits to state \(\ell\), i.e., \(J_{i}=\ell\), where \(\ell\in\{0,1,e\}\), when the system state is \(\mathbf{J}_{u}\). Note that each \(\mathbf{J}_{u}\) state corresponds to a specific packet state of user \(i\), i.e., \(\mathbf{J}_{u}=[J_{i},\cdots,J_{N_{u}}]\). For example when \(N_{u}=2\), \(\mathbf{J}_{1}=[0,0]\) and \(\mathbf{J}_{6}=[e,e]\). The state transition probability for states \(\mathbf{J}_{u}\), is the product of marginal probabilities of each \(i\)-th user's state \(J_{i}=\ell\). We can define the state transition probabilities of a general \(N_{u}\) user setup. When a user \(i\), ( \(1\leq i\leq N_{u}\)), is decoded with a single transmission, its probability is given as \(\mathcal{P}_{0}(i,u)=1-\epsilon(\gamma_{u}^{(i)})\). Consequently, user \(j\), that comes next in decoding order can be decoded with probabilities \(\mathcal{P}_{0}(j,u)=1-\epsilon(\gamma_{u}^{(j)})\), \(\mathcal{P}_{1}(j,u)=\epsilon(\gamma_{u}^{(j)})-\epsilon(\gamma_{u}^{(j)}, \tilde{\gamma}_{2}^{(j)})\) and \(\mathcal{P}_{e}(j,u)=\epsilon(\gamma_{u}^{(j)},\tilde{\gamma}_{2}^{(j)})\), respectively, for being in state \(J_{j}=0\), \(J_{j}=1\) and \(J_{j}=e\). If the \(i\)-th user is recovered after retransmission, i.e., \(J_{i}=1\), whose probability is \(\mathcal{P}_{1}(i,u)=\epsilon(\gamma_{u}^{(i)}))-\epsilon(\gamma_{u}^{(i)}, \tilde{\gamma}_{2}^{(i)})\), the subsequent user \(j\) can only have two states, i.e., \(J_{j}=1\) and \(J_{j}=e\), with probabilities \(\mathcal{P}_{1}(j,u)=1-\epsilon(\gamma_{u}^{(j)},\tilde{\gamma}_{2}^{(j)})\) and \(\mathcal{P}_{e}(j,u)=\epsilon(\gamma_{u}^{(j)},\tilde{\gamma}_{2}^{(j)})\), respectively. Finally, when decoding of user \(i\) is failed, i.e., \(J_{i}=e\), which happens with probability \(\mathcal{P}_{e}(i,u)=\epsilon(\gamma_{u}^{(i)},\tilde{\gamma}_{2}^{(i)})\), subsequent users would definitely fail the decoding, i.e., \(\mathcal{P}_{e}(j,u)=1\). At \(N_{u}=3\) when all the users are successfully decoded with single transmission, the system transits to stat \(\mathbf{J}_{v}=[0,0,0]\).