# Delay-Sensitive NOMA-HARQ for Short Packet Communications
## 4 Numerical Results
### Effect of SNR and Rate on PER, Throughput, and Delay  
Figure 3 shows error rate performance comparison of proposed D-NOMA-HARQ and S-NOMA-HARQ with different SNRs and rates. We fix the packet length \(n=100\) and vary \(k=100\) and \(k=50\) to model rate \(R=1\) and \(R=0.5\), respectively. As shown in this figure, the PER performance of D-NOMA-HARQ is worse than S-NOMA-HARQ, and there exists an SNR performance gap. More specifically, the SNR gaps for target PER of \(10^{-4}\) at rate \(R=1\) are about 6 dB and 8 dB for user 1 and 2, respectively. However, by reducing the rate to \(R=0.5\), the PER performance gap reduces to about 3 dB and 1 dB for user 1 and 2, respectively, for the target PER performance range \(10^{-4}\) to \(10^{-6}\). This is because D-NOMA-HARQ is designed for target delay performance and uses less power and time slots by serving retransmission requests non-orthogonally with new arriving packets, whereas in S-NOMA-HARQ the retransmission is conducted with full power in the new time slot. Thus overall, D-NOMA-HARQ conducts its transmissions with efficient power and bandwidth utilization. Another cause of PER performance loss is that in D-NOMA-HARQ, non-orthogonal sharing of packets causes interference if SIC is unsuccessful. On the other hand, S-NOMA-HARQ allocates full resources, i.e., time slots,  
and the SNR is same for its transmissions and retransmissions. Fortunately, with a lower rate, i.e., \(R=0.5\), the non-orthogonal retransmission and new transmission can be decoded with a higher success rate using SIC; therefore, the SNR gap between D-NOMA-HARQ and S-NOMA-HARQ decreases. Figure 4 shows the throughput versus SNR performance comparison between S-NOMA-HARQ and the proposed D-NOMA-HARQ at two different rates, i.e., \(R=1\) and \(R=0.5\). As can be seen in the figure, S-NOMA-HARQ achieves higher throughput than D-NOMA-HARQ, when the SNR is low. With D-NOMA-HARQ at low SNRs, SIC decoding of packets is more erroneous because of overlapping transmission and retransmission packets. Whereas, the reliability of S-NOMA-HARQ is superior to D-NOMA-HARQ because HARQ retransmissions are conducted independently to regular transmission with maximum power and separate time slot. However, as the SNR increases and the packet can be recovered using SIC more reliably, excessive retransmission of S-NOMA-HARQ results in throughput saturation. On the other hand, D-NOMA-HARQ maintains a steady gain in throughput with SNR and eventually attains the same level of throughput as achieved with S-NOMA-HARQ. Note that D-NOMA-HARQ throughput performance is better than S-NOMA-HARQ at a specific rate point and SNR region, such as \(R=1\) and SNRs from 8 dB to 15 dB. This throughput gain of D-NOMA-HARQ over S-NOMA-HARQ can be seen in Figure 4 clearly, at SNR = 10 dB and \(R=1\). Because at this SNR and rate, the maximum number of retransmitted packets can be recovered using SIC with very high reliability. Finally, the throughput performance of D-NOMA-HARQ and S-NOMA-HARQ eventually becomes similar, when the SNR is very high, such that no retransmission is required and all the packets are successful with only a single transmission. We present the delay performance comparison between proposed D-NOMA-HARQ and S-NOMA-HARQ in Figure 5. We assume that \(N=1000\) packets are scheduled to be transmitted, and each packet has a specific deadline to reach the receiver. We assume feedback, decoding, and other processing delays to be zero and only account for the retransmission delay. The overall delay of 1000 packets is normalized to 1. Therefore if all the packets are delivered by their respective deadline, the delay overhead is zero. However, if a packet is received after one unit of delay due to retransmission, it incurs a delay overhead.