# Hybrid NOMA Protocol with Relay Adaptive AF/DF Collaboration and Its Modeling Analysis in NB-IoT
## 3 Performance Modeling Analysis of H-NOMA Protocol
### Relay-Based Adaptive Collaboration and H-NOMA Signal Transmission Process
#### 3.1.2 Collaboration Stage
In terms of the formula, Equation (4) represents the outage probability of eNB to user 1, \(P_{BU_{1}}=1-e^{-\frac{r}{\lambda_{b_{1}}(\alpha_{1}\rho_{S}-\alpha_{2}\rho_{S })}}\), and it can be seen that \(\rho_{s}\) in the formula, i.e., denotes the SNR, and since \(\alpha_{1}-\tau\alpha_{2}\) is greater than \(0,\rho_{s}\) is monotonically decreasing about the outage probability, i.e., its outage probability decreases as the SNR is increasing. Equation (5) represents the outage probability of link \(eNB\to\UU_{2}\),

\[P_{BU_{2}}= \begin{cases}1-e^{-\frac{r}{\lambda_{b_{2}}\alpha_{2}\rho_{S}}},& \alpha_{2}<\frac{1}{\tau+2},\\ 1-e^{-\frac{r}{\lambda_{b_{2}}(\alpha_{1}\rho_{S}-\alpha_{2}\rho_{S})}},&\alpha _{2}>\frac{1}{\tau+2},\end{cases} \]

and regardless of the relationship between \(\alpha_{2}\) and \(\frac{1}{\tau+2}\), it can be seen that \(\rho_{s}\) is monotonically decreasing with \(P_{BU_{2}}\). From the figure, we know that the effect of the number of retransmissions on the outage probability is also evident, with increasing retransmissions resulting in a smaller outage probability, especially when the SNR is greater than about 5, the advantage of H-NOMA with a lower outage probability becomes apparent. From the above figure, it is clear that by increasing the SNR value or by increasing the number of retransmissions, both can reduce the system outage probability. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline Distance between nodes & \(d_{1}=3,\ d_{2}=2,\ d_{r}=1\) \\ Fading factor of Rayleigh slow-fading channel & \(m=4\) \\ Relay transmission power & \(P_{r}=15\)dB \\ AWGN Power & \(\sigma^{2}=1\) \\ Target rate & \(R_{th}=0.5\) bps/HZ \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter table. When the system allows multiple retransmissions, the number of retransmissions also determines the magnitude of the system outage probability. It can be seen in Figure 6 that as the number of allowed retransmissions becomes larger, the outage probability value decreases. This indicates that the retransmission mechanism can effectively improve the quality of information transmission and increase transmission efficiency. When the transmit power reaches 23 dB, it can effectively increase the data rate to reduce the outage probability. After multiple retransmissions, the outage probabilities of two users show little difference in the figure, which reflects the fairness of the users. The throughput of the user after two and four retransmissions of the eNB is shown in Figure 7. It is clear that the case when the eNB can retransmit up to four times is better than the case when the number of times is two. Both the increase in transmission power and the allocation of more power to user 2 make the system throughput larger. Figure 5: Outage probability of two users in the broadcast phase for different SNR in H-NOMA (\(\alpha_{2}=0.3\)). Figure 6: Outage probability of two users in the broadcast phase with different retransmission numbers (\(P_{\mathrm{s}}=23\) dB). The Figure 8 illustrates the effect of user distance on throughput. From the above figure, it can be obtained that at Case 2, in the Table 2, the average distance is the farthest, and therefore, the throughput obtained by the system is the smallest. The throughput of Case 3 is better than that of Case 1 for the same average distance, although the channel condition of user1 is worse. Shortening the distance from the sending while allocating more power to it eventually improves the system performance. At the same time, it can be seen that the throughput results of Case 2 and Case 3 are similar, indicating that the system throughput results are more dependent on the user with better channel conditions. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Case** & \(\mathbf{\xi_{1}}\) & \(\mathbf{\xi_{2}}\) & **(\(\mathbf{\xi_{1}+\xi_{2}}\))/2** \\ \hline Case 1 & 5 m & 1 m & 3 m \\ Case 2 & 5 m & 5 m & 5 m \\ Case 3 & 1 m & 5 m & 3 m \\ \hline \hline \end{tabular}
\end{table}