# 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  
The relay adaptively selects the way to forward and the signal to be sent. At this point, the two users choose to switch to OMA mode or remain in NOMA mode depending on their decoding of each other. The transmission protocol during the collaboration phase is as follows:  
1. The relay successfully decodes \(X_{s}\) according to the DF forwarding protocol. * If the desired signal of \(U_{1}\) is not obtained, and the desired signal of \(U_{2}\) is successfully obtained, \(U_{1}\) switches to OMA mode and the desired signal \(X_{1}\) is sent by the relay. * If the desired signal of \(U_{2}\) is not obtained, and the desired signal of \(U_{1}\) is successfully obtained, \(U_{2}\) switches to OMA mode and the desired signal \(X_{2}\) is sent by the relay. * If both \(U_{1}\) and \(U_{2}\) fail to get the required signal, the relay will use DF forwarding to resend the superimposed signal to both users, and then both users will get the final required signals \(X_{1},X_{2}\) in NOMA mode. 2. The relay is not successfully decoded according to the DF forwarding protocol. * If at least one of \(U_{1}\) and \(U_{2}\) fails to obtain the required signal, the relay uses the AF forwarding protocol to retransmit the superimposed signal to the two users after amplification, at which point the two users are in NOMA mode. Algorithm 1 shows the transmission process in the collaboration phase for a system with hybrid NOMA and hybrid collaborative transmission method. ```
1:AF/DF relay according to the superimposed signal \(X_{s}\) collaborative transmission
2:begin
3:begin
4:if DF relay successfully obtained signal \(X_{1}\) and \(X_{2}\)then
5:if\(U_{1}\) not getting signal \(X_{1}\)then
6:if\(U_{2}\) gets signal \(X_{

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and then use variable substitution and the second-class first-order modified Bessel function to get the final result:

\[ \begin{array}{ll}P^{A}_{RU_{1}}&=Pr(\log_{2}(1+\gamma^{A}_{RU_{1}})<R_{th})\\ &=Pr(|g_{1}|^{2}<\frac{\tau}{\rho_{P}\bar{\beta}_{1}-\tau\rho_{P}\bar{\beta}_{2 }})+Pr(|h_{r}|^{2}<\frac{\tau\rho_{P}|g_{1}|^{2}+\tau}{\rho_{P}\bar{\beta}_{1} \bar{\beta}_{1}^{2}(\bar{\beta}_{1}-\tau\bar{\beta}_{2})-\tau\rho_{s}},|g_{1}|^ {2}>\frac{\tau}{\rho_{P}\bar{\beta}_{1}-\tau\rho_{P}\bar{\beta}_{2}})\\ &=Pr(|g_{1}|^{2}<\frac{\tau}{\rho_{P}\bar{\beta}_{1}-\tau\rho_{P}\bar{\beta}_{2 }})+\int_{\frac{\tau}{\rho_{P}\bar{\beta}_{1}-\tau\rho_{P}\bar{\beta}_{2}}}^{ \infty}\frac{1}{\lambda_{\bar{\beta}1}}\exp(-\frac{y}{\lambda_{\bar{\beta}1}} )\int_{\frac{\tau\rho_{P}|g_{1}|^{2}+\tau}{\rho_{P}\bar{\beta}_{1}(\bar{\beta }_{1}-\bar{\beta}_{2})-\tau\rho_{s}}}^{\frac{\tau\rho_{P}|g_{1}|^{2}+\tau}{ \lambda_{\bar{\beta}1}}}\exp(-\frac{y}{\lambda_{\bar{\beta}1}})\mathrm{d}x \mathrm{d}y\\ &=1-\int_{\frac{\tau}{\rho_{P}\bar{\beta}_{1}-\tau\rho_{P}\bar{\beta}_{2}}}^{ \infty}\frac{1}{\lambda_{\bar{\beta}1}}\exp(-\frac{y}{\lambda_{\bar{\beta}1}} )\exp\left(-\frac{\tau\rho_{P}|g_{1}|^{2}+\tau}{\lambda_{\bar{\beta}1}\left( \rho_{P}\rho_{P}|g_{1}|^{2}(\bar{\beta}_{1}-\bar{\beta}_{2})-\tau\rho_{s} \right)}\right)\mathrm{d}y,\end{array} \] (22)

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