# Delay-Sensitive NOMA-HARQ for Short Packet Communications
## 3 Delay-Sensitive NOMA-HARQ
### Reliability and Throughput Analysis of Two-User D-NOMA-HARQ
: _For the two-user D-NOMA-HARQ with a maximum of \(1\) retransmission, i.e., \(m=2\), the probability of transitioning from state \(\mathbf{J}_{u}\) to state \(\mathbf{J}_{v}\), denoted by \(\pi_{u\to v},\forall u,v\in\{1,2,3,4,5,6\}\) is given by_  
\[\pi_{u\to v}=\begin{cases}(1-\epsilon\big{(}\gamma_{u}^{(1)}\big{)})\big{(} 1-\epsilon\big{(  
\(\gamma_{3}^{(1)}=\frac{P_{1}}{P_{2}+1}\), \(\gamma_{3}^{(2)}=\frac{\alpha_{5}P_{2}}{\alpha_{2}P_{2}+1}\), \(\gamma_{4}^{(1)}=\frac{\alpha_{1}P_{1}}{\alpha_{2}P_{2}+1}\), \(\gamma_{4}^{(2)}=\gamma_{2}^{2}\), \(\gamma_{5}^{(1)}=\frac{\alpha_{1}P_{1}}{P_{2}+1}\), \(\gamma_{5}^{(2)}=\frac{\alpha_{5}P_{2}}{\alpha_{2}P_{2}+1}\), \(\gamma_{6}^{(1)}=\frac{\alpha_{1}P_{1}}{P_{2}+1}\), and \(\gamma_{6}^{(2)}=\frac{\alpha_{6}P_{2}}{\alpha_{1}P_{1}+\alpha_{2}P_{2}+1}\). Moreover, \(\epsilon(.)\) is given in (3) and (4) for CC-HARQ and IR-HARQ respectively._  
Proof.: Similar to ([41], Lemma 1), we know that \(1-\epsilon\big{(}\gamma_{u}^{(i)}\big{)}\), \(\epsilon\big{(}\gamma_{u}^{(i)}\big{)}-\epsilon\big{(}\gamma_{u}^{(i)},\tilde{ \gamma}_{1}^{(1)}\big{)}\), and \(\epsilon\big{(}\gamma_{u}^{(i)},\tilde{\gamma}_{1}^{(1)}\big{)}\) are representing the probabilities that packet of user \(i\) is decoded without retransmission, with one retransmission, or is failed the decoding, respectively. These correspond to the probabilities of user \(i\) being at states \(J_{i}=0\), \(J_{i}=1\), and \(J_{i}=e\), respectively. Secondly, the state transition probability for states \(\mathbf{J}_{u}\) (\(u=[1,\cdots,6]\)) is the product of marginal probabilities of each user's state \(J_{i}\). For example, when \(J_{1}=0\) and \(J_{2}=0,1\) or \(e\), the system state transits from \(\mathbf{J}_{u}\) to \(\mathbf{J}_{1}\), \(\mathbf{J}_{2}\) and \(\mathbf{J}_{3}\) with probabilities \(\big{(}1-\epsilon\big{(}\gamma_{u}^{(1)}\big{)}\big{)}\big{(}1-\epsilon\big{(} \gamma_{u}^{(2)}\big{)}\big{)}\), \(\big{(}1-\epsilon\big{(}\gamma_{u}^{(1)}\big{)}\big{)}\big{(}\epsilon\big{(} \gamma_{u}^{(2)}\big{)}-\epsilon\big{(}\big{[}\gamma_{u}^{(2)},\tilde{\gamma} _{1}^{(2)}\big{]}\big{)}\big{)}\) and \(\big{(}1-\epsilon\big{(}\gamma_{u}^{(1)}\big{)}\big{)}\epsilon\big{(}\big{[} \gamma_{u}^{(2)},\tilde{\gamma}_{1}^{(2)}\big{]}\big{)}\) respectively. When user \(1\) is successful after a single retransmission, i.e., \(J_{1}=1\). Then user \(2\) can only be in two states i.e., \(J_{2}=1\) or \(e\). Consequently, the marginal probabilities of \(J_{2}=1\) and \(J_{2}=e\) when \(J_{1}=1\) are \(\big{(}1-\epsilon\big{(}\gamma_{u}^{(2)},\tilde{\gamma}_{2}^{(2)}\big{]}\big{)}\)) and \(\epsilon\big{(}\big{[}\gamma_{u}^{(2)},\tilde{\gamma}_{2}^{(2)}\big{]}\big{)}\), respectively. Therefore, the system state transits from \(\mathbf{J}_{u}\) to \(\mathbf{J}_{4}\) and \(\mathbf{J}_{5}\) with probabilities \(\big{(}\epsilon\big{(}\gamma_{u}^{(1)}\big{)}-\epsilon\big{(}\big{[}\gamma_{u }^{(1)},\gamma_{1}^{(1)}\big{]}\big{)}\big{)}\big{(}1-\epsilon\big{(}\big{[} \gamma_{u}^{(1)},\tilde{\gamma}_{1}^{(1)}\big{]}\big{)}\big{)}\big{(}1- \epsilon\big{(}\big{[}\gamma_{u}^{(1)},\tilde{\gamma}_{1}^{(1)}\big{]}\big{)} \big{)}\big{(}\epsilon\big{(}\gamma_{u}^{(2)},\tilde{\gamma}_{1}^{(2)}\big{]} \big{)}\), respectively. Similarly, the system state transits form \(\mathbf{J}_{u}\) to \(\mathbf{J}_{e}\) with probability \(\epsilon\big{(}\big{[}\gamma_{u}^{(1)},\tilde{\gamma}_{1}^{(1)}\big{]}\big{)}\). This is due to the fact that when user \(1\) is in error, user \(2\) is for sure in error.  
Note that, in D-NOMA-HARQ, the \(i\)-th user conducts its retransmission at power \(\alpha_{i}P_{i}\) and the remaining \(\tilde{\alpha}_{i}P_{i}\) is dedicated to the new arriving packet. After retransmission user \(1\) always decoded under the interference of the other user. Therefore its SINR during retransmission is given as \(\tilde{\gamma}_{1}^{(1)}=\frac{\alpha_{1}P_{1}}{P_{2}+\alpha_{1}P_{1}+1}\). The retransmission SINR of user \(2\) depends on the state of user \(1\). When receiver has two copies of user \(2\) and only a single copy of user \(1\) new transmission, then user \(2\) can be decoded first considering user \(1\)'s new arriving packet as interference. In this situation, the power of interference signal of user \(1\) could be \(P_{i}\) or \(\tilde{\alpha}_{i}P_{i}\) based on the state of user \(1\), i.e., \(J_{1}=0\) or \(J_{1}=1\), respectively. Consequently, the SINR of user \(2\) during retransmission, when \(J_{1}=0\) and \(J_{1}=1\), is \(\tilde{\gamma}_{1}^{(2)}=\frac{\alpha_{2}P_{2}}{P_{1}+\tilde{\alpha}_{2}P_{2}+1}\) and \(\tilde{\gamma}_{2}^{(2)}=\frac{\alpha_{2}P_{2}}{\alpha_{1}P_{1}+\tilde{\alpha}_{2}P _{2}+1}\), respectively.  
When the system is at state \(\mathbf{J}_{1}\), the SINR of user \(1\) is \(\gamma_{1}^{(1)}=\frac{P_{1}}{P_{2}+1}\) as user \(1\) is decoded first. After removing interference of user \(1\), user \(2\) is decoded with SNR \(\gamma_{1}^{(2)}=P_{2}\). When the system is at state \(\mathbf{J}_{2}\), \(\gamma_{2}^{(1)}=\frac{P_{1}}{\tilde{\alpha}_{2}P_{2}+1}\), which indicates that the previous packet of user \(2\) was recovered after retransmission and the new packet is the only interference. After removing packet of user \(1\), the new packet of user \(2\) experiences SNR \(\gamma_{2}^{2}=\tilde{\alpha}_{2}P_{2}\). When the system is at state \(\mathbf{J}_{3}\), the previous packet of user \(2\) is not decoded so user \(1\) experiences a higher interference as \(\gamma_{3}^{(1)}=\frac{P_{1}}{P_{2}+1}\). After removing the interference due to user \(1\), user \(2\) only experiences interference from its retransmission packet, i.e., \(\gamma_{3}^{(2)}=\frac{\alpha_{2}P_{2}}{\alpha_{2}P_{2}+1}\). When the system is at state \(\mathbf{J}_{4}\), \(\gamma_{4}^{(1)}=\frac{\alpha_{1}P_{1}}{\alpha_{2}P_{2}+1}\), \(\gamma_{4}^{(2)}=\gamma_{2}^{2}\), since for both users previous packets are decoded successfully, and their interference is removed. At state \(\mathbf{J}_{5}\), user \(2\) is in error; therefore user \(1\) is decoded under its interference as \(\gamma_{5}^{(1)}=\frac{\alpha_{1}P_{1}}{P_{2}+1}\), \(\gamma_{5}^{(2)}=\frac{\alpha_{5}P_{2}}{\alpha_{2}P_{2}+1}\). Finally, when the system is at state \(\mathbf{J}_{6}\) user \(1\) is also not decoded so it causes interference for user \(2\), i.e., \(\gamma_{6}^{(1)}=\frac{\alpha_{1}P_{1}}{P_{2}+\alpha_{1}P_{1}+1}\)\(\gamma_{6}^{(2)}=\frac{\alpha_{5}P_{2}}{\alpha_{1}P_{1}+\alpha_{2}P_{2}+1}\).  
Remark 1: _Let \(\Pi=[\pi_{u\to p}]\) denotes the state transition matrix for the D-NOMA-HARQ system for \(2\) users and \(1\) retransmission. \(P_{stat}=[p_{1},\cdots,p_{6}]^{\mathrm{T}}\) denotes the stationary distribution corresponding to the MM in Figure 2. The PER of user \(i\), denoted by \(\xi_{i}\), is given by_  
\[\xi_{i}=\sum_{u\in\mathcal{E}_{i}}p_{u}, \tag{5}\]_where \(\mathcal{E}_{1}=\{6\}\) and \(\mathcal{E}_{2}=\{3,5,6\}\).