# Guaranteeing QoS for NOMA-Enabled URLLC Based on \(\kappa\)-\(\mu\) Shadowed Fading Model
## Appendix B Proof of Theorem 1
\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}\exp\Bigl{(}-\sigma_ {v}z^{\mathcal{A}}\Bigr{)}\\ &\cong\sum_{v=1}^{V}c^{\prime}{}_{v}\biggl{(}\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})}\biggr{)}\frac{\prod_{i=1}^ {L}\Gamma(\mathcal{A}\mathcal{K}_{i})}{\Gamma\Big{(}\mathcal{A}\sum_{i=1}^{L} \mathcal{K}_{i}\Big{)}}z^{\mathcal{A}\sum_{i=1}^{L}\mathcal{K}_{i}-1}\exp \Bigl{(}-\sigma_{v}z^{\mathcal{A}}\Bigr{)}\\ &\cong\sum_{v=1}^{V}c^{\prime}{}_{v}\biggl{(}\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})}\biggr{)}\frac{\prod_{i=1}^{L }\Gamma(\mathcal{A}\mathcal{K}_{i})}{\Gamma\Big{(}\mathcal{A}\sum_{i=1}^{L} \mathcal{K}_{i}\Big{)}}\frac{(\omega_{v}\Omega_{2,2})^{\mathcal{A}\mathcal{K}_ {i}}\Gamma(\mathcal{K}_{2})}{\mathcal{A}\mathcal{D}_{z}^{\mathcal{A}\mathcal{K}_ {i}}}\frac{\mathcal{A}_{z}\mathcal{D}_{z}^{\mathcal{A}\mathcal{K}_{i}}}{(\omega_ {v}\Omega_{2,2})^{\mathcal{A}\mathcal{K}_{i}}\Gamma(\mathcal{K}_{2})}z^{ \mathcal{A}\mathcal{K}_{i}-1}\exp\left(-\Bigl{(}\frac{\mathcal{D}_{z}}{\omega_ {v}\Omega_{2,2}}z\Bigr{)}^{\mathcal{A}}\right)\\ &\cong\sum_{v=1}^{V}c_{v}\frac{\mathcal{A}_{z}\mathcal{D}_{z}^{ \mathcal{A}\mathcal{K}_{i}}}{(\omega_{v}\Omega_{2,2})^{\mathcal{A}\mathcal{K}_ {i}}\Gamma(\mathcal{K}_{2})}z^{\mathcal{A}\mathcal{K}_{i}-1}\exp\left(-\Bigl{(} \mathcal{D}_{z}\frac{z}{\omega_{v}\Omega_{2,2}}\Bigr{)}^{\mathcal{A}}\right) \end{split}  \tag{19}\]  
Additionally, according to the nature of the Gamma distribution, the corresponding CDF of the approximation can be estimated as  
\[F_{Z}(z)\cong\sum_{v=1}^{V}c_{v}\frac{\gamma\Big{(}\mathcal{K},\left(\mathcal{D }_{z}\frac{z}{\omega_{v}\Omega_{2,2}}\right)^{\mathcal{A}}\Big{)}}{\Gamma( \mathcal{K}_{z})} \tag{20}\]  
The coefficients \(c_{m}\) and \(\omega_{m}\) can be obtained by using the moment-matching method [57]. Using this approach, the moments of the approximate PDF are equal to the exact moments. For this purpose, the following equations are formulated, in which the approximate moments appear on the left and the exact moments appear on the right. \[ \begin{cases}\sum\limits_{v=1}^{V}c_{v}=1,\\ \left(\sum\limits_{v=1}^{V}c_{v}\omega_{v}\right)\Omega_{z,2}=\mathbb{E}(Z)= \mathbb{E}(X_{1}+X_{2}+\ldots+X_{L})=\sum\limits_{i=1}^{L}\mathbb{E}(X_{i})\\ \left(\sum\limits_{v=1}^{V}c_{v}\omega_{v}^{2}\right)\left(\Omega_{z,2}^{2} \left(\frac{\Gamma(\mathcal{K}_{z})}{\Gamma(\mathcal{K}_{z}+1/\mathcal{A}_{z} )}\right)^{2}\frac{\Gamma(\mathcal{K}_{z}+2/\mathcal{A}_{z})}{\Gamma(\mathcal{ K}_{z})}\right)=\mathbb{E}\Big{(}Z^{2}\Big{)}\\ \vdots\\ \left(\sum\limits_{v=1}^{V}c_{v}\omega_{v}^{2V-2}\right)\left(\Omega_{z,2}^{2V -2}\left(\frac{\Gamma(\mathcal{K}_{z})}{\Gamma(\mathcal{K}_{z}+1/\mathcal{A}_ {z})}\right)^{2V-2}\frac{\Gamma\big{(}\mathcal{K}_{z}+(2M_{z}-2)/\mathcal{A} _{z}\big{)}}{\Gamma(\mathcal{K}_{z})}\right)=\mathbb{E}\Big{(}Z^{2V-2}\Big{)} \\ \left(\sum\limits_{v=1}^{V}\frac{c_{v}}{\omega_{v}^{4,\mathcal{K}_{z}}}\right) \left(\frac{\mathcal{A}_{z}\mathcal{D}_{z}^{\mathcal{A}_{z}\mathcal{K}_{z}}}{ \Omega_{z,2}^{4,\mathcal{K}_{z}}\Gamma(\mathcal{K}_{z})}\right)=\sum\limits_ {v=1}^{V}c^{\prime}_{v}A_{L}=A_{L}\end{cases}  \tag{11}\]  
Theorem 1 is proven.