# Research on Hybrid Relay Protocol Design and Cross-Layer Performance Based on NOMA
## 3 Cross-Layer Performance Analysis of Two-Hop Relay System Based on NOMA
### Cross-Layer Performance Analysis
\tag{28}\]  
where \(P_{s}^{\text{OMA}\prime}\) represents the probability that the packet was successfully transmitted without interruption of each hop transmission link, and \(P_{s}^{\text{D}_{1}}\) represents the probability that the link between the signal and the user D\({}_{2}\) is interrupted, so only the user D\({}_{1}\) successfully receives the packet. \(P_{s}^{\text{D}_{1},\text{D}_{2}}\) represents the probability that both user D\({}_{1}\) and user D\({}_{2}\) receiving the data packet without being interrupted. Then, the duration of successful packet transmission \(\overline{T_{s}}\) is  
\[\overline{T_{s}}=P_{s}^{OMA\prime}.T_{s}^{OMA}+P_{s}^{\text{D}_{1}}.T_{s}^{\text {D}_{1}}+P_{s}^{\text{D}_{1},\text{D}_{2}}.T_{s}^{\text{D}_{1},\text{D}_{2}}. \tag{29}\]  
The duration \(T_{sk}^{OMA}\) for successful OMA transmission in the \(k\)-th hop, the duration \(T_{sk}^{\text{D}_{1}}\) for NOMA transmission in the \(k\)-th hop but only user D\({}_{1}\) successfully receives the data packet, and the duration \(T_{sk}^{\text{D}_{1},\text{D}_{2}}\) for NOMA transmission in the \(k\)-th hop and both users D\({}_{1}\) and D\({}_{1}\) successfully receive the data packet are shown below:  
\[\left\{ \begin{array}{l}T_{s1}^{OMA}=T_{s1}^{\text{D}_{1}}=T_{s1}^{\text{D}_{ 1},\text{D}_{2}}=T_{\text{H}}+T_{\text{D}}+T_{\text{ACK}_{w}}+T_{\text{SIFS}}+ T_{\text{DIFS}}+2\delta,\\ T_{s2}^{OMA}=2T_{\text{PRS}}+T_{\text{P}}+T_{\text{D}}+T_{\text{RIFS}}+T_{ \text{CIFS}}+T_{\text{ACK}_{\text{pl}}},\\ T_{s2}^{\text{D}_{1},\text{D}_{2}}=2T_{\text{PRS}}+T_{\text{P}}+T_{\text{D}}+2 T_{\text{RIFS}}+T_{\text{CIFS}}+2T_{\text{ACK}_{\text{pl}}},\\ T_{s2}^{\text{D}_{1}}=2T_{\text{PRS}}+T_{\text{P}}+T_{\text{D}}+2T_{\text{RIFS}}+ T_{\text{CIFS}}+T_{\text{ACK}_{\text{pl}}}+T_{\text{AT}_{\text{pl}}}.\end{array} \right. \tag{30}\]  
The total duration of successful system transmission in the above cases is as follows:  
\[T_{s}^{OMA}=T_{s1}^{OMA}+T_{s2}^{OMA}, \tag{31}\]  
\[T_{s}^{\text{D}_{1},\text{D}_{2}}=T_{s1}^{\text{D}_{1},\text{D}_{2}}+T_{s2}^{ \text{D}_{1},\text{D}_{2}}, \tag{32}\]  
\[T_{s}^{\text{D}_{1}}=T_{s1}^{\text{D}_{1}}+T_{s2}^{\text{D}_{1}}. \tag{33}\]  
The probability \(P_{o1}^{OMA}\) of outage in the first hop wireless transmission when the system adopts OMA transmission, the probability \(P_{o2}^{OMA}\) of outage in the second hop power line transmission when the first hop wireless transmission succeeds, the probability \(P_{o1}^{NOMA}\) of outage in the first hop wireless transmission when the system adopts NOMA transmission, and the probability \(P_{o2}^{NOMA}\) of outage in the second hop power line transmission when the first hop wireless transmission succeeds are as follows:  
\[\left\{ \begin{array}{l}P_{o1}^{OMA}=P_{tr}.P_{s}^{OMA}.P_{\text{out}}^{\text {SR}},\\ P_{o2}^{OMA}=P_{tr}.P_{s}^{NOMA}\big{(}1-P_{\text{out}}^{\text{SR}}\big{)}P_{ \text{out}}^{\text{RD}},\\ P_{o1}^{NOMA}=P_{tr}.P_{s}^{NOMA}\big{(}1-P_{r}\big{(}\gamma_{\text{R},\text{r }1}>\gamma_{\text{th}_{1}},\gamma_{\text{R},\text{r}2}>\gamma_{\text{th}_{2}} \big{)}\big{)},\\ P_{o2}^{NOMA}=P_{tr}.P_{s}^{NOMA}.P_{r}\big{(}\gamma_{\text{R},\text{r}1}> \gamma_{\text{th}_{1}},\gamma_{\text{R},\text{r}2}>\gamma_{\text{th}_{2}} \big{)}\\ \times\big{(}1-P_{r}\big{(}\gamma_{\text{D}_{2},\text{x}1}>\gamma_{\text{th}_{ 1}},\gamma_{\text{D}_{2},\text{x}2}>\gamma_{\text{th}_{2}},\gamma_{\text{D}_{ 1},\text{x}1}>\gamma_{\text{th}_{1}}\big{)}\big{)}.\end{array} \right. \tag{34}\]  
The total time \(\overline{T_{o}}\) of transmission failure due to channel outage is  
\[ \begin{array}{l}\overline{T_{o}}=P_{o1}^{OMA}.T_{o1}^{OMA}+P_{o2}^{OMA} \big{(}T_{s1}^{OMA}+T_{o2}^{OMA}\big{)}+P_{o1}^{NOMA}.T_{o1}^{NOMA}\\ +P_{o2}^{NOMA}\big{(}T_{s1}^{NOMA}+T_{o2}^{NOMA}\big{)}\end{array} . \tag{35}\]  
The duration of outage \(T_{ok}^{OMA}\) and \(T_{ok}^{\text{NOMA}}\) for the \(k\)-th hop using OMA and NOMA are as follows:  
\[\left\{