# Two-Way Transmission for Low-Latency and High-Reliability 5G Cellular V2X Communications
## Appendix A Derivation of (13)  
We rewrite (12) as  
\[\mathcal{P}_{2}^{SIC} =1-\Pr\left\{\gamma_{\mathcal{D}_{1}\to R}>\gamma_{2},\gamma_{ \mathcal{D}_{2},x_{1}}>\gamma_{2}\right\} \tag{13}\] \[=1-\underbrace{\Pr\left\{\left|h_{n^{*},1}\right|^{2}>\delta \left(\left|h_{k^{*},2}\right|^{2}\rho+1\right)\right\}}_{\stackrel{{ \Delta}}{{=}\varsigma_{1}}}\underbrace{\Pr\left\{\left|g_{2,k^{*}} \right|^{2}>\chi\delta\right\}}_{\stackrel{{\Delta}}{{=} \varsigma_{2}}}.\]  
Note that \(\varsigma_{1}\) is defined and expanded as  
\[ \begin{split}\varsigma_{1}&\triangleq\Pr\left\{ \left|h_{n^{*},1}\right|^{2}>\delta\left(\left|h_{k^{*},2}\right|^{2}\rho+1 \right)\right\}\\ &=\int_{0}^{\infty}f_{\left|h_{k^{*},2}\right|^{2}}\left(x\right) \left[1-F_{\left|h_{n^{*},1}\right|^{2}}\left(\delta\left(x\rho+1\right) \right)\right]dx\\ &=\underbrace{\int_{0}^{\infty}f_{\left|h_{k^{*},2}\right|^{2}} \left(x\right)dx}_{\stackrel{{\Delta}}{{=}\varsigma_{1,1}}}- \underbrace{\int_{0}^{\infty}f_{\left|h_{k^{*},2}\right|^{2}}\left(x\right)F_ {\left|h_{n^{*},1}\right|^{2}}\left(\delta\left(x\rho+1\right)\right)dx}_{ \stackrel{{\Delta}}{{=}\varsigma_{1,2}}},\end{split}  \tag{14}\]  
where  
\[ \begin{split}\varsigma_{1,1}&=\int_{0}^{\infty}f_{ \left|h_{k^{*},2}\right|^{2}}\left(x\right)dx\\ &=\sum_{j=0}^{M-1}\bigcup_{j}\binom{M-1}{j}\frac{M\mu_{h_{2}}^{ m_{h_{2}}}(-1)^{j}\mathcal{A}_{j^{\prime}}\mathcal{B}_{j^{\prime}}}{\Gamma\left(m_{h_{2}} \right)}\int_{0}^{\infty}x^{m_{h_{2}}+\tilde{j}-1}e^{-x\mu_{h_{2}}(1+\tilde{ j})}dx\\ &=\sum_{j=0}^{K-1}\bigcup_{j^{\prime}}\binom{K-1}{j}\frac{K\mu_ {h_{2}}^{m_{h_{2}}}(-1)^{j}\mathcal{A}_{j^{\prime}}\mathcal{B}_{j^{\prime}} \Gamma\left(m_{h_{2}}+\tilde{j}\right)}{\Gamma\left(m_{h_{2}}\right)\mu_{h_{2} }^{\tilde{j}}(1+j)^{m_{h_{2}}+\tilde{j}}},\end{split}  \tag{15}\]and  
\[ \begin{split}\xi_{1,2}&=\int_{0}^{\infty}f_{\left|h_{k^{ \star},2}\right|^{2}}\left(x\right)F_{\left|h_{k^{\star},1}\right|^{2}}\left( \delta\left(x\rho+1\right)\right)dx\\ &=\sum_{j=0}^{K-1}\sum_{i=0}^{N}\bigcup_{i^{\prime}}\bigcup_{j^{ \prime}}\binom{N}{i}\binom{K-1}{j}\frac{K\mu_{h_{2}}^{m_{h_{2}}}(-1)^{j+i} \delta^{i}\mathcal{A}_{j^{\prime}}\mathcal{B}_{j^{\prime}}\mathcal{A}_{i^{ \prime}}\mathcal{B}_{i^{\prime}}e^{-h_{h_{1}}i\delta}}{\Gamma\left(m_{h_{2}} \right)}\\ &\quad\times\int_{0}^{\infty}x^{m_{h_{2}}+\tilde{j}-1}(x\rho+1)^{ \tilde{i}}e^{-x\left(h_{h_{2}}(1+\tilde{j})+\mu_{h_{1}}i\delta\rho\right)}dx. \end{split}  \tag{10}\]  
Note that the last equality of (10) is obtained by some manipulations based on (3.381.4) in [31]; and the last equality of (10) is obtained by using a trinomial expansion (1.111) in [31]. By some manipulations based on (3.381.4) in [31], (10) can be further expanded as  
\[ \begin{split}\xi_{1,2}&=\sum_{j=0}^{K-1}\sum_{i=0}^ {N}\sum_{q=0}^{\tilde{i}}\bigcup_{\tilde{r}}\bigcup_{\tilde{r}}\binom{\tilde{ i}}{q}\binom{N}{i}\binom{K-1}{j}\frac{K\mu_{h_{2}}^{m_{h_{2}}}(-1)^{j+i}\theta^{ \tilde{i}}\mathcal{A}_{j^{\prime}}\mathcal{B}_{j^{\prime}}\mathcal{A}_{i^{ \prime}}\mathcal{B}_{i^{\prime}}e^{-h_{h_{1}}i\delta}}{\Gamma\left(m_{h_{2}} \right)}\\ &\quad\times\int_{0}^{\infty}x^{m_{h_{2}}+\tilde{j}+\tilde{i}-q-1 }e^{-x\left(\mu_{h_{2}}(1+\tilde{j})+\mu_{h_{1}}i\delta\rho\right)}dx\\ &=\sum_{j=0}^{K-1}\sum_{i=0}^{N}\sum_{q=0}^{\tilde{i}}\bigcup_{ \tilde{r}}\bigcup_{\tilde{r}}\binom{\tilde{i}}{q}\binom{N}{i}\binom{K-1}{j} \frac{K\mu_{h_{2}}^{m_{h_{2}}}(-1)^{j+i}\theta^{\tilde{i}}\mathcal{A}_{j^{ \prime}}\mathcal{B}_{j^{\prime}}\mathcal{A}_{i^{\prime}}\mathcal{B}_{j^{\prime }}\Gamma\left(m_{h_{2}}+\tilde{j}+\tilde{i}-q\right)e^{-\mu_{h_{1}}i\delta}}{ \Gamma\left(m_{h_{2}}\right)\left(\mu_{h_{2}}\left(1+j\right)+\mu_{h_{1}}i \delta\rho\right)^{m_{h_{2}}+\tilde{j}+\tilde{i}-q}}.\end{split}  \tag{11}\]  
Consequently, (11) is written as  
\[