# A Novel Multi-Cell Interference-Aware Cooperative QoS-Based NOMA Group D2D System
## Appendix A Outage Probability in Phase II
As such, based on (11), \(A\) is given as follows:  
\[A= \int_{\tilde{\gamma}_{t}}^{\infty}\int_{0}^{\infty}\!\!f_{|| \mathrm{H}||}(z+\eta_{1}) \tag{14}\] \[\cdot F_{|h_{R_{x}}|}\bigg{(}\frac{\eta_{2}(1+x\xi_{I})}{z}\bigg{)} f_{I}(x)dzdx\] \[= \int_{\tilde{\gamma}_{t}}^{\infty}\int_{0}^{\infty}e^{-(z+\eta_{ 1})}\sum_{l=1}^{L}b_{m_{2}}e^{\frac{-\gamma_{t}\eta_{2}(1+x\xi_{I})}{z}}f_{I} (x)dzdx\] \[= \int_{0}^{\infty}\sum_{l=1}^{L}b_{m_{2}}f_{I}(x)\underbrace{ \int_{\tilde{\gamma}_{t}}^{\infty}e^{-(z+\eta_{1})}e^{\frac{-\gamma_{t}\eta_{2 }(1+x\xi_{I})}{z}}}_{A_{1}}dzdx\]  
For ease of notation, let us denote \(\big{(}\lambda=\psi_{l}\eta_{2}(1+x\xi_{I})\big{)}\) in (14). Therefore, \(A_{1}\) in (14) can be expressed as follows:  
\[A_{1}=e^{-\eta_{1}}\int_{\tilde{\gamma}_{t}}^{\infty}e^{-z}e^{\frac{-\lambda}{ z}}dz \tag{15}\]  
Using (1.211) of [32], the expression \(A_{1}\) can be written as:  
\[ \begin{split} A_{1}&=e^{-\eta_{1}}\sum_{k_{1}=0}^{N_ {1}}\int_{\tilde{\gamma}_{t}}^{\infty}e^{-z}\cdot\frac{(-\lambda)^{k_{1}}}{k_{1 }!}\bigg{(}\frac{1}{z}\bigg{)}^{k_{1}}dz\\ A_{1}&=\sum_{k_{1}=0}^{N_{1}}e^{-\eta_{1}}\frac{(- \lambda)^{k_{1}}}{k_{1}!}\int_{\tilde{\gamma}_{t}}^{\infty}e^{-z}\bigg{(}\frac {1}{z}\bigg{)}^{k_{1}}dz\end{split}  \tag{16}\]  
where \(N_{1}=\infty\). Using (3.381.3) of [32], (16) becomes:  
\[A_{1}=\sum_{k_{1}=0}^{N_{1}}e^{-\eta_{1}}\frac{(-\lambda)^{k_{1}}}{k_{1}!} \Gamma(1-k_{1},\tilde{\gamma}_{t}) \tag{17}\]Substituting the value of (A11) into (A8) would provide the following expression:  
\[ \begin{split} A&=\int_{0}^{\infty}\sum_{l=1}^{L}b_{m}g _{l}(x)\sum_{k_{1}=0}^{N_{1}}e^{-\eta_{1}}\frac{(-\lambda)^{k_{1}}}{k_{1}!} \Gamma(1-k_{1},\tilde{\gamma_{t}})dx\\ &=\sum_{l=1}^{L}\sum_{k_{1}=0}^{N_{1}}b_{m}e^{-\eta_{1}}\frac{(-1 )^{k_{1}}}{k_{1}!}\Gamma(1-k_{1},\tilde{\gamma_{t}})(\psi_{l}\eta_{2})^{k}\\ &\cdot\underbrace{\int_{0}^{\infty}(1+x_{\xi}^{\gamma_{t}})^{k_ {l}}f_{l}(x)dx}_{A_{2}}\end{split} \] (A12)  
In order to solve \(A_{2}\) in (A12), the PDF of interference at the receiver is required. As such, the Laplace transform of the interference PDF at the receiver is given by [18]:  
\[\mathcal{L}_{I}(s)=e^{-2\pi\lambda_{\mathcal{C}T}}\sum_{p=1}^{P}\Omega_{p} \frac{a(1-X_{1}(\gamma_{p})+s\gamma_{p}^{-a}}{1+s\gamma_{p}^{-a}}.\Lambda_{2} (c)\] (A13)  
where \(\Omega_{p}=\omega_{p}e_{p}^{\gamma}\), \(\omega_{p}=\frac{\Gamma(p+2)\gamma_{p}}{p(p+1)^{2}(L_{p+1}(\gamma_{p}))^{2}}\), \(L_{p}(. )\) is the Laguerre polynomial of degree P\({}_{l}\) and \(\gamma_{p}\) are the roots of \(L_{p}(.)\). \(\Lambda_{2}(. )\) and \(\mathcal{X}_{1}(. )\) are calculated using (A-7) and (A-10) of [2]. Based on (A13), the Laplace inverse of the interference \(I\), denoted by \(f_{I}(x)\), can be expressed as follows, from [33]:  
\[f_{I}(x)= \frac{e^{vx}}{T}\sum_{k_{2}=0}^{2N_{2}}\,{}^{q}\Re\Big{[}\mathcal{L}_{I} (s=\mathfrak{s})\exp\Big{(}\frac{i\tau tx}{T}\Big{)}\Big{]}\] (A14)  
where \(\mathfrak{s}=v_{0}+\frac{i\tau k_{2}}{T}\), \(v_{0}=v_{1}-\frac{\log(c)}{T}\), \(v,v_{1}>0\) are real numbers, \(\varsigma\) is the desired relative accuracy, \(T\) is a scaling parameter, \(t=\sqrt{-1}\), \(N_{2}\) is the number of terms used to invert Laplace transform, and the prime indicates that \(k_{2}=0\) summation term is halved. Now, based on (A14), \(A_{2}\) in (A12) can be expressed as follows:  
\[ \begin{split} A_{2}&=\frac{1}{T}\sum_{k_{2}=0}^{2N _{2}}\int_{0}^{\infty}\Re\Big{[}(1+x_{\xi}^{\gamma_{t}})^{k_{1}}e^{\left(v+ \frac{\omega}{T}\right)x}\mathcal{L}_{I}(\mathfrak{s})\Big{]}dx\\ &=\frac{1}{T}\sum_{k_{2}=0}^{2N_{2}}\Re\left[\mathcal{L}_{I}( \mathfrak{s})\underbrace{\int_{0}^{\infty}(1+x_{\xi}^{\gamma_{t}})^{k_{1}}e^{ \left(v+\frac{\omega}{T}\right)x}dx}_{A_{3}}\right]\end{split} \] (A15)  
It is very challenging to solve the integral in \(A_{3}\). Therefore, we use Gauss-Laguerre quadrature to approximate \(A_{3}\) as follows:  
\[