# 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
Similar to [27], the combined signal received by the eNB after \(t\) transmissions by two users using the combination factor \(w_{t}\) is represented as follows, where the variables \(h_{1}^{t}\), \(w_{t}\), \(n_{B1}^{t}\) are related to the number of transmissions t.

\[ \begin{array}{l}y_{B1}^{t}=\sum\limits_{t=1}^{T}h_{1}^{t}w_{t}(\sqrt{\alpha_{1 }P_{s}}X_{1}+\sqrt{\alpha_{2}P_{s}}X_{2})+\sum\limits_{t=1}^{T}n_{B1}^{t}w_{t}. \\ y_{B1}^{t}=\sum\limits_{t=1}^{T}h_{2}^{t}w_{t}(\sqrt{\alpha_{1}P_{s}}X_{1}+ \sqrt{\alpha_{2}P_{s}}X_{2})+\sum\limits_{t=1}^{T}n_{B2}^{t}w_{t}.\end{array}  \tag{39}\]

According to the MRC rule, the signal-to-noise ratio at \(U_{1}\) and \(U_{2}\) after retransmission can be obtained by combining Equation (39):

\[\gamma_{BU_{2}\to U_{1}}^{t}=\frac{\frac{\alpha_{1}\rho_{s}\sum\limits_{t=1}^{T }\left|h_{2}^{t}\right|^{2}}{\alpha_{2}\rho_{s}\sum\limits_{t=1}^{T}\left|h_{2 }^{t}\right|^{2}+1}}{\gamma_{BU_{1}}^{t}=\frac{\sum\limits_{t=1}^{T}\left|h_{1 }^{t}\right|^{2}\alpha_{1}\rho_{s}}{\sum\limits_{t=1}^{T}\left|h_{1}^{t} \right|^{2}\alpha_{2}\rho_{s}+1}}. \tag{40}\]

\[\gamma_{BU_{2}}^{t}=\sum\limits_{t=1}^{T}\left|h_{2}^{t}\right|^{2}\alpha_{2} \rho_{s}. \tag{41}\]

During retransmission, the link outage probability from eNB to \(U_{1}\) is represented by the following equation:

\[P_{BU_{1}}^{tN}=\Pr(\log_{2}(1+\gamma_{BU_{1}}^{t})<R_{th})=Pr\Bigg{(}\sum \limits_{t=1}^{T}\left|h_{1}^{t}\right|^{2}<\frac{2^{R_{th}}-1}{\alpha_{1} \rho_{s}-\alpha_{2}\rho_{s}(2^{T{R_{th}}}-1)}\Bigg{)}\Bigg{(}\frac{a}{\rho_{s }\lambda_{h1}(\alpha_{1}-\alpha_{2}(2^{R_{th}}-1))}\Bigg{)}. \tag{42}\]

Here, \((a)\) is due to the random variable \(|h_{1}|^{2},|h_{2}|^{2},|h_{3}|^{2}\cdots|h_{T}|^{2}i.i.d\sim Exp(\lambda)\), so \(\sum\limits_{i=1}^{T}\left|h_{i}\right|^{2}\sim\Gamma(T,\lambda)\). The link outage probability from eNB to user 2 is also obtained in the same way:\[ \begin{split} P_{BU_{2}}^{IN}=1-Pr\Big{(}\log_{2}\Big{(}1+\gamma_{BU_{2 }\to\UU_{1}}^{t}\Big{)}>R_{th},\log_{2}\Big{(}1+\gamma_{BU_{2}}^{t}\Big{)}>R_{ th}\Big{)}\\ =Pr\bigg{(}\sum\limits_{t=1}^{T}\big{|}h_{2}^{t}\big{|}^{2}<\frac{2 ^{R_{th}}-1}{\alpha_{2}\rho_{S}},\sum\limits_{t=1}^{T}\big{|}h_{2}^{t}\big{|}^{ 2}<\frac{2^{R_{th}}-1}{\alpha_{1}\rho_{S}-\alpha_{2}\rho_{S}(2^{R_{th}}-1)} \big{)}=Pr\bigg{(}\sum\limits_{t=1}^{T}\big{|}h_{2}^{t}\big{|}^{2}<\max\Big{\{} \frac{2^{R_{th}}-1}{\alpha_{2}\rho_{S}},\frac{2^{R_{th}}-1}{\alpha_{1}\rho_{S} -\alpha_{2}\rho_{S}(2^{R_{th}}-1)}\Big{\}}\bigg{)}\\ =\begin{cases}\Gamma\Big{(}T,\frac{2^{R_{th}}-1}{\rho_{S}\lambda_{ 2}(\alpha_{1}-\alpha_{2}(2^{R_{th}}-1))}\Big{)},&\alpha_{2}\in\Big{(}\frac{1}{2 ^{R_{th}}+1},0.5\Big{)}\\ \Gamma\Big{(}T,\frac{2^{R_{th}}-1}{\alpha_{2}\lambda_{2}\rho_{S}}\big{)},& \alpha_{2}\in\Big{(}0,\frac{1}{2^{R_{th}}+1}\Big{)}.\end{cases}\end{split}  \tag{44}\]

## 6 Numerical Simulation and Analysis

In this section, to verify the accuracy of the proposed adaptive relay collaborative hybrid NOMA model and to better understand its system performance, we use Matlab for performing numerical analysis. For the values of some parameters, we refer to the NB-IoT standard, the system parameters are set as shown in Table 1:

We respectively compare the outage probabilities of user 1 and user 2 in H-NOMA and OMA modes at different SNRs and the effect of different numbers of retransmissions on them. From the Figure 5, it can be seen that the outage probability decreases with the increase of SNR about both users, and it decreases faster than OMA due to the superior performance of the H-NOMA proposed in this paper. We believe that the improvement of the signal-to-noise ratio of the system, i.e., the ratio of the signal power to the noise power, improves the signal quality, which leads to a higher success probability of the decoding and thus corresponds to the outage probability that will be reduced.