# 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
After



It can be seen that any one of the values in \(p^{2}\) is larger than 0, so this Markov chain is a regular Markov chain. According to the limit theorem for regular Markov chains [26], there exists a unique steady-state probability for the model. The system stationary distribution, \(\pi=\left(\pi_{S_{1}},\pi_{S_{2}},\pi_{S_{3}},\pi_{S_{4}},\pi_{S_{5}}\right)\), can be calculated by using Equation (35). \[\pi\cdot\left( \begin{array}{ccccc}p_{11}&p_{12}&p_{13}&p_{14}&p_{15}\\ p_{21}&0&\cdots&\cdots&0\\ p_{31}&\vdots&\ddots&&\vdots\\ p_{41}&\vdots&&\ddots&\vdots\\ p_{51}&0&\cdots&\cdots&0\\ \end{array} \right)=\pi. \tag{35}\]

\[\pi_{S_{1}}+\pi_{S_{2}}+_{S_{3}}+\pi_{S_{4}}+\pi_{S_{5}}=1.\]

The steady-state probability can be obtained as:

\[\pi_{S_{1}}=1/1+P_{BL_{1}}+P_{BL_{2}}-P_{BL_{1}}P_{BL_{2}}. \tag{36}\] \[\pi_{S_{2}}=\frac{\left(1-P_{BR}\right)P_{BL_{1}}\left(1-P_{BL_{2 }}\right)}{1+P_{BL_{1}}+P_{BL_{2}}-P_{BL_{1}}P_{BL_{2}}}.\] \[\pi_{S_{3}}=\frac{\left(1-P_{BR}\right)\left(1-P_{BL_{1}}\right) P_{BL_{2}}}{1+P_{BL_{1}}+P_{BL_{2}}-P_{BL_{1}}P_{BL_{2}}}.\] \[\pi_{S_{4}}=\frac{\left(1-P_{BR}\right)P_{BL_{1}}P_{BL_{2}}}{1+P_{ BL_{1}}+P_{BL_{2}}-P_{BL_{1}}P_{BL_{2}}}.\] \[\pi_{S_{5}}=\frac{P_{BR}\left(P_{BL_{1}}+P_{BL_{2}}-P_{BL_{1}}P_{ BL_{2}}\right)}{1+P_{BL_{1}}+P_{BL_{2}}-P_{BL_{1}}P_{BL_{2}}}.\]

In the system of this paper, the probabilities that the user can successfully receive the signal on the state \(S_{1},S_{2},S_{3},S_{4},S_{5}\) are \(\left(1-P_{BU_{1}}\right)\left(1-P_{BU_{2}}\right)\), \(\left(1-P_{RU_{1}}\right)\), \(\left(1-P_{RU_{2}}\right)\), \(\left(1-P_{RU_{1}}^{D}\right)\left(1-P_{RU_{2}}^{D}\right)\) and \(\left(1-P_{RU_{1}}^{A}\right)\left(1-P_{RU_{2}}^{A}\right)\), respectively, so the outage probability of the system of this paper can be expressed as

\[P_{out}=1-\left[\left(1-P_{BU_{1}}\right)\left(1-P_{BU_{2}}\right)\pi_{S_{1}} +\left(1-P_{RU_{1}}\right)\pi_{S_{2}}+\left(1-P_{RU_{2}}\right)\pi_{S_{3}}+ \left(1-P_{RU_{1}}^{D}\right)\left(1-P_{RU_{2}}^{D}\right)\pi_{S_{4}}+\left(1 -P_{RU_{1}}^{A}\right)\left(1-P_{RU_{2}}^{A}\right)\pi_{S_{5}}\right]. \tag{37}\]

The system throughput is obtained from the outage probability:

\[\eta=\left(1-P_{out}\right)\cdot R_{th}. \tag{38}\]

## 5 The Impact of The Number of Retransmissions on System Performance

To enhance transmission reliability, the case that eNB can transmit multiple times is considered in this section to improve the accuracy of the signal being successfully received. When the user receives the signal, it sends an ACK to the source node if the decoding is successful, otherwise, it sends a NACK to the source node. Only when the source node receives ACK responses from both users at the same time, does it send a new superimposed signal to start a new round of transmission. If not, the eNB will continue to retransmit the same superimposed signal for the two users to decode again. Assuming that the maximum number of retransmissions the eNB can make is \(T\), i.e., during the broadcast phase, the eNB stops transmission when the two users receive the signal successfully, and the number of eNB retransmissions reaches \(T\). During the retransmission process of eNB, the relay is collaborating with other units for transmission, and the signal of \(T\) retransmissions is not received at this time, so it can improve the efficiency of relay usage and reduce energy consumption. If the two users still fail to decode successfully, the system will enter the collaboration phase. Since multiple transmission signals for retransmission in this system come from the same codebook, both users use the maximum ratio combining (MRC) technique to receive signals retransmitted multiple times from the transmitter to improve the signal-to-noise ratio. The Algorithm 2 for multiple retransmissions is shown as follows.