# 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
Then, the above equation is:

\[\begin{array}{ll}P^{A}_{RU_{2}}&=1-\int_{0}^{\infty}\frac{1}{\lambda_{\bar{ \beta}2}}\exp\left(-\frac{t+\frac{\tau}{\rho_{P}\bar{\beta}_{2}}}{\lambda_{ \bar{\beta}2}\lambda_{\bar{\beta}2}\lambda_{\bar{\beta}1}\rho_{P}\bar{\beta}_{2}} \right)\cdot\exp\left(-\frac{\tau(\rho_{P}t+\frac{\tau}{\rho_{P}\bar{\beta}_{2}} +1)}{\lambda_{\bar{\beta}1}\rho_{P}\rho_{P}\bar{\beta}_{2}}\right)\mathrm{d}t\\ &=1-\frac{1}{\lambda_{\bar{\beta}2}}\exp(-\frac{y}{\lambda_{\bar{\beta}2}}\exp \left(-\frac{y}{\lambda_{\bar{\beta}2}}\right)\cdot\exp\left(-\frac{\tau\rho_{P} \bar{\beta}_{2}}{\lambda_{\bar{\beta}2}\lambda_{\bar{\beta}2}\lambda_{\bar{\beta} 2}\lambda_{\bar{\beta}2}\lambda_{\bar{\beta}2}\lambda_{\bar{\beta}2}\lambda_{\bar{ \beta}2}\lambda_{\bar{\beta}2}\lambda_{\bar{\beta}2}\lambda_{\bar{\beta}2}\lambda_{ \bar{\beta}2}\lambda_{\bar{\beta}2}\lambda_{\bar{\beta}2}\lambda_{\bar{\beta}2}\lambda_{ \bar{\beta}2}\lambda_{\bar{\beta}2}\lambda_{\bar{\beta}2}\lambda_{\bar{\beta}2}\lambda_{ \bar{\beta}2}\lambda_{\bar{\beta}2}\lambda_{\bar{\beta}2}\lambda_{\bar{\beta}2}\lambda_{ \bar{\beta}2}\lambda_{\bar{\beta}2}\lambda_{\bar{\beta}2}\lambda_{\bar{\beta}2}\lambda_{ \bar{\beta}2}
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State \(S_{3}\): Relay transmits signal \(X_{2}\) to \(U_{2}\) with the OMA mode, regardless of whether User 1 receives the signal, at the next time slot, will enter a new round of transmission, i.e., transferring to the state \(S_{1}\).

State \(S_{4}\): DF relay transmits the superimposed signal to the two users with NOMA mode, and the two users enter the state after sending the ACK/NACK signal.

State \(S_{5}\): AF relay transmits superimposed signal to two users with NOMA mode, at this time the relay cannot decode correctly. The system still can use the amplify-and-forward method for cooperative transmission and transfer to the state \(S_{1}\) after the two users receive the signal.

Suppose that the state of the system at the t th time slot, denoted by \(X_{t}\), is a discrete random variable with a state space of \(\{S_{1},S_{2},S_{3},S_{4},S_{5}\}\). Represent by \(p_{ij},i,j\in\{1,2,3,4,5\}\) the probability that the system is in state \(S_{i}\) at the t th time slot and the system is in state \(S_{j}\) at the t + 1 th time slot, i.e., the state transfer probability of the system. Since the state \(S_{j}\) in the t + 1 th time slot depends only on the state \(S_{i}\) in the previous time slot and transfer probability \(p_{ij}\), and not on the previous state \(X_{t-1},X_{t-2},X_{t-3},\ldots\), the model has memorylessness, and we establish this stochastic process as a Markov model. The state transfer process of this model is analyzed below.

The state transition diagram of Markov chain is shown in Figure 4, and the state transition probability is obtained using the link outage probability, and the transition probability is as follows:

Figure 4: Markov state transition diagram.

Figure 3: Illustration of the transition of state 1 to the other states.

After the source node sends the superimposed signal through the direct transmission link connected with two users, the two users receive it successfully, and no outage occurs in both links, so the probability of the state \(S_{1}\) remains in the state \(S_{1}\):

\[p_{11}=\big{(}1-P_{BU_{1}}\big{)}\big{(}1-P_{BU_{2}}\big{)}. \tag{26}\]

In the broadcast phase, both the relay and the user \(U_{2}\) successfully get the corresponding information, and \(U_{1}\) fails to decode, so the transition probability from state \(S_{1}\) to state \(S_{2}\) is given by:

\[p_{12}=(1-P_{BR})P_{BU_{1}}\big{(}1-P_{BU_{2}}\big{)}. \tag{27}\]

In the broadcast phase, both the relay and the user \(U_{1}\) successfully get the corresponding information, and the user \(U_{2}\) fails to decode, so the transition probability from the state \(S_{1}\) to the state \(S_{3}\) is given by:

\[p_{13}=(1-P_{BR})\big{(}1-P_{BU_{1}}\big{)}P_{BU_{2}}. \tag{28}\]

In the broadcast phase, the relay succeeds in getting the corresponding information, \(U_{1}\) and \(U_{2}\) both fail to decode, so the transition probability from state \(S_{1}\) to state \(S_{4}\) is given by:

\[p_{14}=(1-P_{BR})P_{BU_{1}}P_{BU_{2}}. \tag{29}\]

In the broadcast phase, the relay fails to decode, and at least one of \(U_{1}\) and \(U_{2}\) fails to decode, so the transition probability from state \(S_{1}\) to state \(S_{5}\) is given by:

\[p_{15}=P_{BR}\big{(}P_{BU_{1}}+P_{BU_{2}}-P_{BU_{1}}P_{BU_{2}}\big{)}. \tag{30}\]

In the collaboration phase, the probability that user in a state \(S_{2}\) enters state \(S_{1}\) after decoding the signal \(X_{1}\) is:

\[p_{21}=P_{RU_{1}}+1-P_{RU_{1}}. \tag{31}\]

During the collaboration phase, the probability that a user \(U_{2}\) in state \(S_{3}\) enters state \(S_{1}\) after decoding the signal \(X_{2}\) is:

\[p_{31}=P_{RU_{2}}+1-P_{RU_{2}}. \tag{32}\]

In the cooperation phase, the probability that \(U_{1}\) and \(U_{2}\) in state \(S_{4}\) enters state \(S_{1}\) after decoding the superimposed signal is:

\[p_{41}=P_{RU_{1}}^{D}\cdot P_{RU_{2}}^{D}+\Big{(}1-P_{RU_{1}}^{D}\Big{)}P_{RU _{2}}^{D}+P_{RU_{1}}^{D}\Big{(}1-P_{RU_{2}}^{D}\Big{)}+\Big{(}1-P_{RU_{1}}^{D }\Big{)}\Big{(}1-P_{RU_{2}}^{D}\Big{)}. \tag{33}\]

In the cooperation phase, the probability that \(U_{1}\) and \(U_{2}\) in state \(S_{5}\) will enter state \(S_{1}\) after decoding the amplified superimposed signal is:

\[p_{51}=P_{RU_{1}}^{A}\cdot P_{RU_{2}}^{A}+\Big{(}1-P_{RU_{1}}^{A}\Big{)}P_{RU _{2}}^{A}+P_{RU_{1}}^{A}\Big{(}1-P_{RU_{2}}^{A}\Big{)}+\Big{(}1-P_{RU_{1}}^{A }\Big{)}\Big{(}1-P_{RU_{2}}^{A}\Big{)}. \tag{34}\]

In fact, it will be transferred to state \(S_{1}\) in the next time slot regardless of whether the user eventually succeeds in decoding the signal, so \(p_{i1}=1,i\in\{2,3,4\}\) and the rest of the state transition probability is 0.