# Two-Way Transmission for Low-Latency and High-Reliability 5G Cellular V2X Communications
## 3 Analysis of Outage Probability and Throughput of NOMA-V2X System
### Outage probability of \(\mathrm{D}_{2}\)  
We let \(Z\) denote a random variable representing \(f_{1}\) and \(f_{2}\). Then, the CDF and pdf of \(Z\) are defined by  
\[F_{Z}\left(x\right)=1-e^{-\mu_{z}x}\sum\limits_{s=0}^{m_{z}-1}\frac{\left(\mu_{ z}x\right)^{s}}{s!} \tag{6}\]  
and  
\[f_{Z}\left(x\right)=\frac{\mu_{z}^{m_{z}}x^{m_{z}-1}e^{-\mu_{z}x}}{\Gamma\left( m_{z}\right)}, \tag{7}\]  
respectively, where \(\mu_{z}=\frac{m_{z}}{\lambda_{z}}\) is a parameter of multipath fading associated with \(f_{1}\) and \(f_{2}\). Let \(Y\) denote the channel gain corresponding to the best antenna of \(\mathrm{D}_{1}\) or \(\mathrm{D}_{2}\) selected for communication in V2X systems. The CDF and pdf of \(Y\) are defined by  
\[F_{Y}\left(x\right)=\left[1-e^{-\mu_{y}x}\sum\limits_{t=0}^{m_{y}-1}\frac{ \left(\mu_{y}x\right)^{t}}{t!}\right]^{G} \tag{8}\]  
and  
\[f_{Y}\left(x\right)=\frac{G\mu_{y}^{my}x^{my-1}e^{-\mu_{y}x}}{\Gamma\left(m_{ y}\right)}\times\underbrace{\left[1-e^{-\mu_{y}x}\sum\limits_{t=0}^{m_{y}-1} \frac{\left(\mu_{y}x\right)^{t}}{t!}\right]^{G-1}}_{\stackrel{{ \Delta}}{{=}}\Theta}, \tag{9}\]  
respectively, where \(\mu_{y}=\frac{m_{y}}{\lambda_{y}}\) and \(\Theta\triangleq\left(1-e^{-\mu_{y}x}\sum\limits_{t=0}^{m_{y}-1}\frac{\left( \mu_{y}x\right)^{t}}{t!}\right)^{G-1}\). Note that \(G=N\) and \(G=K\) if \(Y\) is associated with \(\mathrm{D}_{1}\) and \(\mathrm{D}_{2}\), respectively. By applying a successive binomial expansion to (8), we obtain  
\[F_{Y}\left(x\right)=\sum\limits_{i=0}^{G}\bigcup\limits_{i^{\prime}}\binom{G} {i}(-1)^{i}\mathcal{A}_{i^{\prime}}\mathcal{B}_{i^{\prime}}x^{\tilde{i}}e^{- \mu_{y}ix}, \tag{10}\]  
where \(\bigcup\limits_{i^{\prime}}\stackrel{{\Delta}}{{=}}\sum\limits_{ i_{1}=0}^{i}\sum\limits_{i_{2}=0}^{i-i_{1}}\cdots\sum\limits_{i_{m_{z}-1}=0}^{i-i_{1} -\cdots-i_{m_{z}-2}}\), \(\mathcal{A}_{i^{\prime}}=\binom{i}{i_{1}}\binom{i-i_{1}}{i_{2}}\cdots\binom{ i-i_{1}-\cdots-i_{m_{z}-2}}{i_{m_{z}-1}}\), \(\mathcal{B}_{i^{\prime}}=\prod\limits_{t=0}^{m_{z}-2}\left(\frac{\mu_{t}^{t} }{t! }\right)^{i_{1}+i}\left(\frac{\mu_{m_{z}-1}^{m_{z}-1}}{\Gamma\left(m_{z} \right)! }\right)^{i-i_{1}-i_{2}-\cdots-i_{m_{z}-1}}\) and \(\tilde{i}=\left(m_{z}-1\right)\left(i-i_{1}\right)-\left(m_{z}-2\right)i_{2}- \cdots-i_{m_{z}-1}\). We also apply the successive binomial expansion to \(\Theta\) to obtain  
\[\Theta=\sum\limits_{j=0}^{G-1}\bigcup\limits_{j^{\prime}}\binom{G-1}{j}(-1)^ {j}\mathcal{A}_{j^{\prime}}\mathcal{B}_{j^{\prime}}x^{\tilde{j}}e^{-\mu_{y}ix}. \tag{11}\]  
Let us consider the link \(\mathrm{D}_{1}\rightarrow\mathrm{RSU}\rightarrow\mathrm{D}_{2}\) and suppose that \(\mathrm{RSU}\) decodes \(x_{1}\) correctly in the first phase and \(D_{2}\) also decodes \(x_{1}\) correctly in the second phase. Note that \(\mathrm{RSU}\) can decode \(x_{2}\) after decoding \(x_{1}\) correctly. Since \(x_{1}\) is decoded first at \(\mathrm{RSU}\) before using \(\mathrm{SIC}\), the outage probability for \(\mathrm{D}_{1}\rightarrow\mathrm{RSU}\rightarrow\mathrm{D}_{2}\) link does not depend on \(\mathrm{SIC}\). Thus, the outage probabilities of \(\mathrm{D}_{2}\) obtained by using \(\mathrm{pSIC}\) and \(\mathrm{ipSIC}\) for \(\mathrm{D}_{1}\rightarrow\mathrm{RSU}\rightarrow\mathrm{D}_{2}\) link are the same and they are defined as  
\[\mathcal{P}_{2}^{\mathrm{_{_{RSU}}}}=\mathcal{P}_{2}^{\mathrm{_{RSU}}}\triangleq \mathcal{P}_{2}^{\mathrm{_{RSU}}}=1-\Pr\left\{\gamma_{\mathrm{D}_{1}\to R }>\gamma_{2},\gamma_{\mathrm{D}_{2}\times x_{1}}>\gamma_{2}\right\}, \tag{12}\]which is expanded as  
\[