## 3 Outage Probability Performance Analysis
### Outage Probability of D2D User 2
\begin{array}{l}\Psi_{1}=\left(1-\sum\limits_{k=1}^{K}\left(\begin{array}[ ]{c}K\\ k\end{array}\right)(-1)^{k-1}\frac{\lambda_{2}}{\theta k\alpha\rho A_{f_{2}}+ \lambda_{2}}\exp\left(-\frac{\theta k}{\lambda_{2}}\right)\right)\\ \\ \times\left(1-\sum\limits_{k=1}^{K}\left(\begin{array}{c}K\\ k\end{array}\right)(-1)^{k-1}v_{1}\right).\end{array}  \tag{18}\]Next, \(\Psi_{2}\) can be calculated by  
\[ \begin{array}{l}\Psi_{2}=\Pr\left(\gamma_{SD2,k*}^{PDMA}<\varepsilon_{2}\cup \max\left\{\gamma_{SD1\gets 2,k*}^{PDMA}\chi_{D2}^{PDMA}\right\}< \varepsilon_{1},\gamma_{SD1,k*}^{PDMA}>\varepsilon_{1}\right)\\ \\ =\left[\Pr\left(\gamma_{SD2,k*}^{PDMA}<\varepsilon_{2}\right)+\underbrace{\Pr \left(\max\left\{\gamma_{SD1\gets 2,k*}^{PDMA}\chi_{D2}^{PDMA}\right\}< \varepsilon_{1}\right)}_{D_{21}}\right.\\ \\ \left.-\underbrace{\Pr\left(\gamma_{SD2,k*}^{PDMA}<\varepsilon_{2}\cup\max \left\{\gamma_{SD1\gets 2,k*}^{PDMA}\chi_{D2}^{PDMA}\right\}<\varepsilon_{1} \right)}_{D_{22}}\right]\\ \\ \times\Pr\left(\gamma_{SD1,k*}^{PDMA}>\varepsilon_{1}\right).\end{array}  \tag{19}\]  
Interestingly, we have the following result:  
\[ \begin{array}{l}\Pr\left(\gamma_{SD2,k*}^{PDMA}<\varepsilon_{2}\right)=1- \Pr\left(\gamma_{SD2,k*}^{PDMA}\geq\varepsilon_{2}\right)\\ \\ =1-\Pr\left(\left|g_{2,k}\right|^{2}\geq\frac{\varepsilon_{2}\omega\rho|f_{2}|^ {2}+\varepsilon_{2}}{\alpha_{2}\rho}\right)\\ \\ =1-\sum\limits_{k=1}^{K}\left(\begin{array}{c}K\\ k\end{array}\right)(-1)^{k-1}\frac{\alpha_{2}\rho\lambda_{2}}{\alpha_{2} \rho\lambda_{2}+\varepsilon_{2}\alpha\rho\lambda_{1}}\exp\left(-\frac{ \varepsilon_{2}k}{\alpha_{2}\rho\lambda_{2}}\right),\end{array}  \tag{20}\]  
and  
\[ \begin{array}{l}\Pr\left(\gamma_{SD1,k*}^{PDMA}>\varepsilon_{1}\right)=\Pr \left(\left|g_{1,k}\right|^{2}\geq\frac{\varepsilon_{1}\omega\rho|f_{1}|^{2}+ \varepsilon_{1}}{\alpha_{1}\rho-\varepsilon_{1}\alpha_{2}\rho}\right)\\ \\ =\frac{1}{\lambda_{f_{1}}}\int_{0}^{\infty}\sum\limits_{k=1}^{K}\left( \begin{array}{c}K\\ k\end{array}\right)(-1)^{k-1}\exp\left(-\frac{(\varepsilon_{1}\omega\rho+ \varepsilon_{1})k}{(\alpha_{1}\rho-\varepsilon_{1}\alpha_{2}\rho)\lambda_{1}}- \frac{x}{\lambda_{f_{1}}}\right)dx\\ \\ =\sum\limits_{k=1}^{K}\left(\begin{array}{c}K\\ k\end{array}\right)(-1)^{k-1}\vartheta_{1}.\end{array}  \tag{21}\]  
After this step, two lemmas as shown below need to be considered. **Lemma 1**. : _The closed-form expression of \(D_{21}\) is calculated as_  
\[D_{21}=\left(1-\frac{\rho\lambda_{h_{2}}}{\rho\lambda_{h_{2}}+\varepsilon_{1} \alpha\rho\lambda_{f_{2}}}\exp\left(-\frac{\varepsilon_{1}}{\rho\lambda_{h_{2} }}\right)\right)\left(1-\sum\limits_{k=1}^{K}\left( \begin{array}{c}K\\ k\end{array} \right)(-1)^{k-1}\vartheta_{2}\right). \tag{22}\]  
Proof. : See in Appendix B. **Lemma 2**. : \(D_{22}\) _is computed in closed-form by_  
\[ \begin{array}{l}D_{22}=\left(1-\sum\limits_{k=1}^{K}\left( \begin{array}{c}K\\ k\end{array}\right)(-1)^{k-1}\frac{\alpha_{2}\rho\lambda_{2}}{\alpha_{2}\rho \lambda_{2}+\varepsilon_{2}\alpha\rho\lambda_{f_{2}}}\exp\left(-\frac{ \varepsilon_{2}k}{\alpha_{2}\rho\lambda_{2}}\right)\right)\\ \\ \times\left(1-\sum\limits_{k=1}^{K}\left(\begin{array}{c}K\\ k\end{array}\right)(-1)^{k-1}\vartheta_{2}\right)\left(1-\frac{\rho\lambda_{h _{2}}}{\rho\lambda_{h_{2}}+\varepsilon_{1}\alpha\rho\lambda_{f_{2}}}\exp \left(-\frac{\varepsilon_{1}}{\rho\lambda_{h_{2}}}\right)\right).\end{array}  \tag{23}\]Proof. : See in Appendix C.  
From (20)-(23) we find the expression of \(\Psi_{2}\) as below:  
\[\Psi_{2}=\left[\left(1-\sum\limits_{k=1}^{K}\left( \begin{array}{c}K\\ k\end{array} \right)\left(-1\right)^{k-1}\frac{a_{2}\rho\lambda_{2}}{a_{2}\rho \lambda_{2}+\varepsilon_{2}\alpha\rho k\lambda_{f_{2}}}\exp\left(-\frac{ \varepsilon_{2}k}{a_{2}\rho\lambda_{2}}\right)\right)\right.\]  
\[+\left(1-\frac{\rho\lambda_{h_{2}}}{\rho\lambda_{h_{2}}+\varepsilon_{1}\rho \lambda_{f_{2}}}\exp\left(-\frac{\varepsilon_{1}}{\rho\lambda_{h_{2}}}\right) \right)\left(1-\sum\limits_{k=1}^{K}\left( \begin{array}{c}K\\ k\end{array}