# A Novel Multi-Cell Interference-Aware Cooperative QoS-Based NOMA Group D2D System
## Appendix A Outage Probability in Phase II  
In Phase II, the relays transmit their data to \(G_{T_{2}}\), which is responsible for delivering it to the end receiver, i.e., \(R_{x}\). The \(SINR\) of the received signal given in (5) can be written as:  
\[ \begin{split} SINR_{R_{x}}&=\frac{h_{R_{x},2}|| \mathbf{H}||^{2}\beta_{R_{x}}P_{G_{T_{2}}}}{h_{R_{x},2}\sum_{m_{2}=j_{2}}^{M_{2} }\beta_{m_{2}}P_{G_{T_{2}}}+I+\aleph^{2}}\\ j_{2}&=1,2,\cdots,M_{2}\end{split}  \tag{18}\]  
The users in the proposed system's model are in close proximity to each other. The outage event at \(R_{x}\) (\(\varphi_{R_{x}}\)) occurs if the threshold rate of any high-order user or the threshold rate of \(R_{x}\), \(\mathrm{B}_{th}^{R_{x}}\) is not achieved and can be described as follows:  
\[ \begin{split}\varphi_{i\to R_{x}}&=\left\{\mathrm{B }_{i\to R_{x}}<\mathrm{B}_{th}^{R_{x}}\right\}\\ &=\left\{\log_{2}\!\left(1+\frac{h_{R_{x},2}||\mathbf{H}||^{2} \beta_{R_{x}}P_{G_{T_{2}}}}{h_{R_{x},2}\sum_{m_{2}=j}^{M_{2}}\beta_{m_{2}}P_{G _{T_{2}}}+I+\aleph^{2}}\right)<\mathrm{B}_{th}^{R_{x}}\right\}\\ &=\left\{\frac{h_{R_{x},2}||\mathbf{H}||^{2}\beta_{R_{x}}P_{G_{T_ {2}}}}{h_{R_{x},2}\sum_{m_{2}=j_{2}}^{M_{2}}\beta_{m_{2}}P_{G_{T_{2}}}+I+ \aleph^{2}}<\tau_{R_{x}}\right\}\\ &=\left\{h_{R_{x},2}<\frac{\tau_{R_{x}}(\xi_{I}I+1)}{\xi_{R_{x}}( ||\mathbf{H}||^{2}\beta_{R_{x}}-\tau_{R_{x}}\sum_{m_{2}=j}^{M_{2}}\beta_{m_{2}} )}\right\}\end{split}  \tag{19}\]  
where \(\xi_{I}=\frac{1}{\aleph^{2}}\tau_{R_{x}}=2\mathrm{B}_{th}^{R_{x}}-1,\xi_{R_{x }}=\frac{P_{G_{T_{2}}}}{\aleph^{2}}\). In case of \(m_{2}\)=\(M_{2}\), the outage event can be stated as:  
\[h_{R_{x},2}<\frac{\tau_{R_{x}}(\xi_{I}I+1)}{\xi_{R_{x}}(||\mathbf{H}||^{2} \beta_{R_{x}})} \tag{20}\]  
The outage probability in Phase II can be expressed as follows:  
\[ \begin{split}\mathrm{P}_{2}=&\mathrm{P}_{r}\!\left(|| \mathbf{H}||^{2}<\left|\frac{\tau_{R_{x}}}{\xi_{R_{x}}}\right|\!,\right.\\ h_{R_{x}}<\left.\frac{\tau_{R_{x}}(1+\xi_{I}I)}{\xi_{R_{x}}(|| \mathbf{H}||^{2}\beta_{R_{x}}-\tau_{R_{x}}\sum_{m_{2}=j_{2}}^{M_{2}}\beta_{m_{2 }})}\right)\end{split}  \tag{21}\]where \(\gamma_{t}=\frac{\tau_{R_{x}}}{\xi_{R_{x}}}\)  
\[ \begin{split}\mathrm{P}_{2}=&\mathrm{P}_{r}\Bigg{(}|| \mathrm{H}||^{2}<|\gamma_{t}|,\\ & h_{R_{x}}<\frac{\tau_{R_{x}}(1+\xi_{I}I)}{\xi_{R_{x}}(|| \mathrm{H}||^{2}\beta_{R_{x}}-\tau_{R_{x}}\sum_{m_{2}=j_{2}}^{M_{2}}\beta_{m_{2 }})}\Bigg{)}\\ =&\int_{0}^{\gamma_{t}}\!\!f_{||\mathrm{H}||}(y)dy+ \int_{\gamma_{t}}^{\infty}\int_{0}^{\infty}\!\!f_{||\mathrm{H}||^{2}}(y)\\ &\cdot F_{|h_{R_{x}}|}\Bigg{(}\frac{\tau_{R_{x}}(1+x\xi_{I})}{\xi_ {R_{x}}(y\beta_{R_{x}}-\tau_{R_{x}}\sum_{m_{2}=j_{2}}^{M_{2}}\beta_{m_{2}})} \Bigg{)}\\ &\cdot f_{I}(x)dxdy\end{split}  \tag{12}\]  
where the integral in first term represents the CDF of the Rayleigh fading random variable \(||\mathrm{H}||\) and is given as \(1-\exp^{\frac{-\gamma_{t}}{\Omega_{2,j}}}\)[31]; \(f_{I}(. )\) denotes the probability density function (PDF) of the interference \(I\). For ease of notation, let us assume that in (11), \(\eta_{1}=\frac{\tau_{R_{x}}}{\beta_{R_{x}}}\sum_{m_{2}=j_{2}}^{M_{2}}\beta_{m_{ 2}}\), \(\eta_{2}=\frac{\tau_{R_{x}}}{\xi_{R_{x}}\beta_{R_{x}}}\), and \(y-\eta_{1}=z\). In addition, let us denote the integral in the second term of (11) by \(A\). Therefore, the outage probability in Phase II can now be written as follows:  
\[\mathrm{P}_{2}=1-\exp^{\frac{-\gamma_{t}}{\Omega_{2,j}}}+A \tag{13}\]  
In order to obtain \(\mathrm{P}_{2}\), it is required to compute the integral in \(A\). Thus, in what follows, we will solve the integral in \(A\).