# Guaranteeing QoS for NOMA-Enabled URLLC Based on \(\kappa\)-\(\mu\) Shadowed Fading Model
## Appendix B Proof of Theorem 1  
In reference [57], the authors derived the PDF of the sum of i.n.i.d. generalized Gamma RVs. Therefore, we transform the PDF of \(\eta_{lk}\) into the PDF of generalized Gamma RVs, and its expression is as follows. \[ \begin{split} f_{\eta_{lk}}\big{(}x;\alpha,\kappa,\mu,m,\hat{ \Omega}_{\hat{l}{k}}\big{)}&=f_{\gamma_{mk}}\big{(}x;{\cal A}_{ \hat{l}{k}},{\cal K}_{\hat{l}{k}},\hat{\Omega}_{\hat{l}{k}}\big{)}\\ &=\frac{{\cal A}_{\hat{l}{k}}{\cal D}_{\hat{l}{k}}^{{\cal A}_{ \hat{l}{k}}}}}{{\hat{\Omega}}_{\hat{l}{k}}^{{\cal A}_{\hat{l}{k}}}\Gamma({\cal K }_{lk})}x^{{\cal A}_{\hat{l}{k}}}{\kappa_{\hat{l}{k}}^{-1}}\exp\left(-\left({ \cal D}_{\hat{l}{k}}\frac{x}{\hat{\Omega}_{\hat{l}{k}}}\right)^{{\cal A}_{\hat {l}{k}}}\right)\end{split}  \tag{1}\]  
where \({\cal D}_{\hat{l}{k}}=\frac{\Gamma({\cal K}_{\hat{l}{k}}+1/{\cal A}_{\hat{l}{ k}})}{\Gamma({\cal K}_{\hat{l}{k}})}\). For \(\hat{l}\in{\cal L}_{k}\) and \(k\in[1,K]\), \({\cal A}_{\hat{l}{k}}=1\) can be obtained. Therefore, when \(\hat{l}\in{\cal L}_{k},k\in[1,K]\) and \({\cal A}_{\hat{l}{k}}\) is constant, the sum of \(\hat{L}\) i.n.i.d. generalized Gamma RVs can be approximated by the generalized Puiseux series expansion. The approximation for the PDF of the sum of \(L\) i.n.i.d. \(\kappa\)-\(\mu\) shadowed RVs \(z\) is given by  
\[ \begin{split} f_{Z}(z)=&\prod_{i=1}^{L}\frac{{ \cal A}{\cal D}_{\hat{l}{k}}^{{\cal A}_{\hat{l}{k}}}}{\Omega_{i,2}^{{\cal A}_{ \hat{l}{k}}}}\Gamma({\cal K}_{i})}z^{A\sum_{l=1}^{L}{\cal K}_{i-1}}\frac{\prod_{ i=1}^{L}\Gamma({\cal A}{\cal K}_{i})}{\Gamma\Big{(}{\cal A}\sum_{l=1}^{L}{\cal K }_{i}\Big{)}}I(z;{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} )}\\ =& A_{L}z^{A\sum_{l=1}^{L}{\cal K}_{i}-1}}I(z;{{{{{{{{{{{{{{{{{{{ {{{{{{{{{{{{}}}}}}}}}}}}}}}}}}}}}}}}} })\end{split}  \tag{2}\]where  
\[A_{L}=\left(\prod_{i=1}^{L}\frac{\mathcal{A}\mathcal{D}_{i}^{\mathcal{A}\mathcal{ K}_{i}}}{\Omega_{i,2}^{\mathcal{A}\mathcal{K}_{i}}\Gamma(\mathcal{K}_{i})} \right)\frac{\prod_{i=1}^{L}\Gamma(\mathcal{A}\mathcal{K}_{i})}{\Gamma\Big{(} \mathcal{A}\sum_{i=1}^{L}\mathcal{K}_{i}\Big{)}} \tag{11}\]  
\[I(z;u_{1},u_{2},\ldots,u_{L-1}) \tag{12}\] \[=\int_{0}^{1}\int_{0}^{1}\ldots\int_{0}^{1}\prod_{k=1}^{L-1} \frac{u_{k}^{\mathcal{A}\mathcal{K}_{i}}(1-u_{k})^{\mathcal{A}\mathcal{K}_{k+1 }-1}}{B\big{(}\mathcal{A}\sum_{i=1}^{L}\mathcal{K}_{k},\mathcal{A}\mathcal{K}_ {k+1}\big{)}}\Xi^{\psi(u_{1},u_{2},\ldots,u_{L-1})}du_{1}du_{2}\ldots du_{L-1}\] (13) \[\Xi=\exp\Bigl{(}-(z)^{\mathcal{A}}\Bigr{)} \tag{14}\]  
\[\psi(u_{1},u_{2},\ldots u_{L-1}) \tag{15}\] \[=\left(\left( \begin{array}{c}\ldots\left(\left(\left(\left( \mathcal{D}_{1}\frac{u_{1}}{\Omega_{1,2}}\right)^{\mathcal{A}}+\left(\mathcal{ D}_{2}\frac{(1-u_{1})}{\Omega_{2,2}}\right)^{\mathcal{A}}\right)u_{2}^{\mathcal{A}}+ \left(\mathcal{D}_{3}\frac{(1-u_{2})}{\Omega_{3,2}}\right)^{\mathcal{A}} \right)u_{3}^{\mathcal{A}}\\ +\left(\mathcal{D}_{4}\frac{(1-u_{3})}{\Omega_{4,2}}\right)^{\mathcal{A}} \ldots\end{array} \right)u_{L-1}^{\mathcal{A}}+\left(\mathcal{D}_{L}\frac{(1- u_{L-1})}{\Omega_{4,2}}\right)^{\mathcal{A}}\right) \tag{16}\]  
On the basis of the generalized Puiseux series expansion process, \(I(z;u_{1},u_{2},\ldots,u_{L-1})\) can be approximated by the truncation order of any integer \(V\), namely,  
\[I(z;u_{1},u_{2},\ldots,u_{L-1})=\sum_{v=1}^{V}c^{\prime}{}_{v}\Xi^{\sigma_{v}} \tag{17}\]  
When \(V\to\infty\), the series can accurately approach \(I(z;u_{1},u_{2},\ldots,u_{L-1})\). Moreover, the series expansion converges very quickly. Equation (10) can be expressed as  
\[ \begin{split} f_{Z}(z)&\cong\sum_{v=1}^{V}c^{\prime }{}_{v}A_{L}z^{\mathcal{A}\sum_{i=1}^{L}\mathcal{K}_{i}-1}\Xi^{\sigma_{v}}\\ &\cong\sum_{v=1}^{V}c^{\prime}{}_{v}A_{L}z^{\mathcal{A}\sum_{i=1} ^{L}\mathcal{K}_{i}-1}\exp\Bigl{(}-\sigma_{v}z^{\mathcal{A}}\Bigr{)}\end{split}  \tag{18}\]  
To simplify the derivation, we transform the content of the cumulative sum into the PDF of the generalized Gamma distribution as follows. \[