# Two-Way Transmission for Low-Latency and High-Reliability 5G Cellular V2X Communications
## Appendix B Derivation of (15)
 \tag{11}\]and  
\[\xi_{3,2} =\int_{0}^{\infty}\int_{\tilde{\delta}(x\varpi\rho+1)}^{\infty}f_{|f_ {1}|^{2}}\left(x\right)f_{\left|h_{x},2\right|^{2}}\left(y\right)F_{\left|h_{x },1\right|^{2}}\left(\delta\left(y\rho+1\right)\right)dxdym_{f_{1}}\] \[=\sum_{j=0}^{K-1}\sum_{i=0}^{N}\bigcup_{i^{\prime}}\bigcup_{j^{ \prime}}\binom{N}{i}\binom{K-1}{j}\frac{K\mu_{h_{2}}^{m_{h_{2}}}\mu_{f_{1}}^{m_ {f_{1}}}(-1)^{j+i}\delta^{\tilde{t}}\mathcal{A}_{j^{\prime}}\mathcal{B}_{j^{ \prime}}\mathcal{A}_{i^{\prime}}\mathcal{B}_{i^{\prime}}e^{-\mu_{h_{1}}i\delta}} {\Gamma\left(m_{h_{2}}\right)\Gamma\left(m_{f_{1}}\right)}\] \[\quad\times\int_{0}^{\infty}x^{m_{f_{1}}-1}e^{-\mu_{f_{1}}x}\int_ {\tilde{\delta}(x\varpi\rho+1)}^{\infty}y^{m_{h_{2}}+\tilde{j}-1}e^{-y\left[ \mu_{h_{2}}(1+j)+\mu_{h_{1}}i\delta\rho\right]}(y\rho+1)^{\tilde{t}}dxdy\] \[=\sum_{j=0}^{K-1}\sum_{i=0}^{N}\sum_{a=0}^{\tilde{i}}\bigcup_{i^ {\prime}}\bigcup_{j^{\prime}}\binom{\tilde{i}}{a}\binom{N}{i}\binom{K-1}{j} \frac{K\mu_{h_{2}}^{m_{h_{2}}}\mu_{f_{1}}^{m_{f_{1}}}\rho^{\tilde{i}-a}(-1)^{j +i}\delta^{\tilde{t}}\mathcal{A}_{j^{\prime}}\mathcal{B}_{j^{\prime}}\mathcal{ A}_{i^{\prime}}\mathcal{B}_{i^{\prime}}e^{-\mu_{h_{1}}i\delta}}{\Gamma\left(m_{h_{2}} \right)\Gamma\left(m_{f_{1}}\right)}\] \[\quad\times\int_{0}^{\infty}x^{m_{f_{1}}-1}e^{-\mu_{f_{1}}x}\int_ {\tilde{\delta}(x\varpi\rho+1)}^{\infty}y^{m_{h_{2}}+\tilde{j}+\tilde{i}-a-1}e ^{-y\left[\mu_{h_{2}}(1+j)+\mu_{h_{1}}i\delta\rho\right]}dxdy\] \[=\sum_{j=0}^{K-1}\sum_{i=0}^{N}\sum_{a=0}^{\tilde{i}}\sum_{i=0}^ {m_{h_{2}}+\tilde{j}+\tilde{i}-a-1}\bigcup_{i^{\prime}}\bigcup_{j^{\prime}} \binom{\tilde{i}}{a}\binom{N}{i}\binom{K-1}{j}\] \[\quad\times\frac{K\Gamma\left(m_{h_{2}}+\tilde{j}+\tilde{i}-a \right)\delta^{s}\mu_{h_{2}}^{m_{h_{2}}}\mu_{f_{1}}^{m_{f_{1}}}\rho^{\tilde{i} -a}(-1)^{j+i}\delta^{\tilde{t}}\mathcal{A}_{j^{\prime}}\mathcal{B}_{j^{\prime }}\mathcal{A}_{i^{\prime}}\mathcal{B}_{i^{\prime}}e^{-\mu_{h_{1}}i\delta- \delta\left[\mu_{h_{2}}(1+j)+\mu_{h_{1}}i\delta\rho\right]}}{s!\left[\mu_{h_{ 2}}\left(1+j\right)+\mu_{h_{1}}i\delta\rho\right]^{m_{h_{2}}+\tilde{j}+\tilde {i}-a-s}\Gamma\left(m_{h_{2}}\right)\Gamma\left(m_{f_{1}}\right)}\] \[\quad\times\int_{0}^{\infty}x^{m_{f_{1}}-1}e^{-x\left[\mu_{f_{1} }+\tilde{\delta}\varpi\rho\left(\mu_{h_{2}}(1+j)+\mu_{h_{1}}i\delta\rho\right) \right]}(x\varpi\rho+1)^{s}dx \tag{12}\] \[=\sum_{j=0}^{K-1}\sum_{i=0}^{N}\sum_{a=0}^{\tilde{i}}\sum_{i=0}^{ m_{h_{2}}+\tilde{j}+\tilde{i}-a-1}\sum_{d=0}^{s}\bigcup_{i^{\prime}}\bigcup_{j^{ \prime}}\binom{s}{d}\binom{\tilde{i}}{a}\binom{N}{i}\binom{K-1}{j}\] \[\quad\times\frac{\delta^{s}\mu_{h_{2}}^{m_{h_{2}}}\mu_{f_{1}}^{m_ {f_{1}}}\varpi^{s-d}(-1)^{j+i}\delta^{\tilde{t}}\mathcal{A}_{j^{\prime}} \mathcal{B}_{j^{\prime}}\mathcal{A}_{i^{\prime}}\mathcal{B}_{i^{\prime}}e^{-\mu _{h_{1}}i\delta-\delta\left[\mu_{h_{2}}(1+j)+\mu_{h_{1}}i\delta\rho\right]}}{s! \left[\mu_{h_{2}}\left(1+j\right)+\mu_{h_{1}}i\delta\rho\right]^{m_{h_{2}}+ \tilde{j}+\tilde{i}-a-s}\Gamma\left(m_{h_{2}}\right)\Gamma\left(m_{f_{1}}\right)}\] \[=\sum_{j=0}^{K-1}\sum_{i=0}^{N}\bigcup_{i^{\prime}}\bigcup_{j^{ \prime}}\binom{N}{i}\binom{K-1}{j}\frac{K\mu_{h_{2}}^{m_{h_{2}}}\mu_{f_{1}}^{m_ {f_{1}}}(-1)^{j+i}\delta^{\tilde{t}}\mathcal{A}_{j^{\prime}}\mathcal{B}_{j^{ \prime}}\mathcal{A}_{i^{\prime}}\mathcal{B}_{i^{\prime}}e^{-\mu_{h_{1}}i\delta}} {\Gamma\left(m_{h_{2}}\right)\Gamma\left(m_{f_{1}}\right)}\]0 \[=\sum_{j=0}^{K-1}\sum_{i=0}^{N}\bigcup_{i^{\prime}}\bigcup_{j^{ \prime}}\binom{N}{i}\binom{K-1}{j}\frac{K\mu_{h_{2}}^{m_{h_{2}}}\mu_{f_{1}}^{m_ {f_{1}}}(-1)^{j+i}\delta^{\tilde{t}}\mathcal{A}_{j^{\prime}}\mathcal{B}_{j^{ \prime}}\mathcal{A}_{i^{\prime}}\mathcal{B}_{i^{\prime}}e^{-\mu_{h_{1}}i\delta}} {\Gamma\left(m_{h_{2}}\right)\Gamma\left(m_{f_{1}}\right)}\]1 \[=\sum_{j=0}^{K-1}\sum_{i=0}^{N}\bigcup_{i^{\prime}}\bigcup_{j^{ \prime}}\binom{N}{i}\binom{K-1}{j}\frac{K\mu_{h_{2}}^{m_{h_{2}}}\mu_{f_{1}}^{m_ {f_{1}}}(-1)^{j+i}\delta^{\tilde{t}}\mathcal{A}_{j^{\prime}}\mathcal{B}_{j^{ \prime}}\mathcal{A}_{i^{\prime}}\mathcal{B}_{i^{\prime}}e^{-\mu_{h_{1}}i\delta}} {\Gamma\left(m_{h_{2}}\right)\Gamma\left(m_{f_{1}}\right)}\]2 \[=\sum_{j=0}^{K-1}\sum_{i=0}^{N}\bigcup_{i^{\prime}}\bigcup_{j^{ \prime}}\binom{N}{i}\binom{K-1}{j}\frac{K\mu_{h_{2}}^{m_{h_{2}}}\mu_{f_{1}}^{m_ {f_{1}}}(-1)^{j+i}\delta^{\tilde{t}}\mathcal{A}_{j^{\prime}}\mathcal{B}_{j^{ \prime}}\mathcal{A}_{i^{\prime}}\mathcal{B}_{i^{\prime}}e^{-\mu_{h_{1}}i\delta}} {\Gamma\left(m_{h_{2}}\right)\Gamma\left(m_{f_{1}}\right)}\]3For manipulation of \(\xi_{3,1}\), we adopt (3.351.2) and (3.351.3) in [31].