# Guaranteeing QoS for NOMA-Enabled URLLC Based on \(\kappa\)-\(\mu\) Shadowed Fading Model
## 2 System Model
Then, the PDF, CDF and MGF of \(\zeta_{k}\) are obtained from Theorem 1. **Theorem 1**. : _Considering a \(\kappa-\mu\) distribution with parameters \((\mathcal{A}_{k},\mathcal{K}_{k},\hat{\Omega}_{\hat{lk}})\), the PDF can be approximately obtained by a \(V\)-order generalized Puiseux series expansion process as follows._  
\[f_{\zeta_{k}}(z;\mathcal{A}_{k},\mathcal{K}_{k},\hat{\Omega}_{k})\cong\sum \limits_{v=1}^{V}c_{v}\frac{\mathcal{A}_{k}\mathcal{D}_{k}^{\mathcal{A}_{k} \mathcal{K}_{k}}}{(\omega_{v}\Omega_{k})^{\mathcal{A}_{k}\mathcal{K}_{k}} \Gamma(\mathcal{K}_{k})}z^{\mathcal{A}_{k}\mathcal{K}_{k}-1}\exp\left(-\bigg{(} \mathcal{D}_{k}\frac{z}{\omega_{v}\hat{\Omega}_{k}}\bigg{)}^{\mathcal{A}_{k} }\right), \tag{8}\]  
_where \(\mathcal{A}_{k}=\mathcal{A}_{\hat{lk}}=1\), \(\mathcal{K}_{k}=\sum\limits_{\hat{l}\in\mathcal{L}_{k}}\mathcal{K}_{\hat{lk}}\), \(\mathcal{K}_{k}=\sum\limits_{\hat{l}\in\mathcal{L}_{k}}\mathcal{K}_{\hat{lk}}\) and \(\mathcal{D}_{k}=\frac{\Gamma(\mathcal{K}_{k}+1/\mathcal{A}_{k})}{\Gamma( \mathcal{K}_{k})}\)._  
_We can obtain approximate parameters \(c_{v}\) and \(\omega_{v}\) via the following formula:_  
\[ \begin{cases}\sum\limits_{v=1}^{V}c_{v}=1,\\ \left(\sum\limits_{v=1}^{V}c_{v}\omega_{v}\right)=1,\\ \left(\sum\limits_{v=1}^{V}c_{v}\omega_{v}^{2}\right)\left(\Omega_{k,2}^{2V-2} \bigg{(}\frac{\Gamma(\mathcal{K}_{k})}{\Gamma(\mathcal{K}_{k}+1/\mathcal{A}_{k} )}\bigg{)}^{2V-2}\frac{\Gamma(\mathcal{K}_{k}+(2M_{k}-2)/\mathcal{A}_{k})}{ \Gamma(\mathcal{K}_{k})}\right)=\mathbb{E}\Big{(}\zeta_{k}^{2V-2}\Big{)},\\ \left(\sum\limits_{v=1}^{V}\frac{c_{v}}{\omega_{v}^{\mathcal{A}_{k}\mathcal{K}_ {k}}}\right)\left(\frac{\mathcal{A}_{k}\mathcal{D}_{k}^{\mathcal{A}_{k} \mathcal{K}_{k}}}{\Omega_{k,2}^{\mathcal{A}_{k}\mathcal{K}_{k}}\Gamma( \mathcal{K}_{k})}\right)=A_{\mathcal{L}_{k}},\end{cases}  \tag{9}\]\[\text{where }A_{\mathcal{L}_{k}}=\left(\prod\limits_{l\in\mathcal{L}_{k}}\frac{ \mathcal{A}\mathcal{D}_{lk}^{\mathcal{A}_{k}\mathcal{K}_{k}}}{\Omega_{k,2}^{ \mathcal{A}_{k}\Gamma}(\mathcal{K}_{k})}\right)\frac{\prod\limits_{l\in\mathcal{ L}_{k}}\Gamma(\mathcal{A}\mathcal{K}_{k})}{\Gamma\left(\mathcal{A}\sum\limits_{l\in \mathcal{L}_{k}}\mathcal{K}_{k}\right)}\text{ and }\]  
\[\mathbb{E}(\xi_{k}^{n})=\sum\limits_{n_{1}=0}^{n}\sum\limits_{n_{2}=0}^{n_{1}} \ldots\sum\limits_{n_{L-1}=0}^{n_{L-2}}\binom{n}{n_{1}}\left( \begin{array}{ c}n_{1}\\ n_{2}\end{array} \right)\ldots\left( \begin{array}{c}n_{L-2}\\ n_{L-1}\end{array} \right)\mathbb{E}\left(X_{1}^{n-n_{1}}\right)\mathbb{E} \left(X_{2}^{n_{1}-n_{2}}\right)\ldots\mathbb{E}\left(X_{L}^{n_{L-1}}\right).\]  
_The CDF of \(\zeta_{k}\) is given by_  
\[F_{\zeta_{k}}\big{(}z;\mathcal{A}_{k},\mathcal{K}_{k},\hat{\Omega}_{k}\big{)} \cong\sum\limits_{v=1}^{V}c_{v}\frac{\gamma\Big{(}\mathcal{K}_{k^{\prime}} \left(\mathcal{D}_{k}\frac{z}{\omega_{v}\hat{\Omega}_{k}}\right)^{\mathcal{A} _{k}}\Big{)}}{\Gamma(\mathcal{K}_{k})}, \tag{10}\]  
_where \(\gamma(\cdot,\cdot)\) is the lower incomplete Gamma function and is expressed by_  
\[\gamma(s,x)=\int_{0}^{x}t^{s-1}\exp(-t)dt. \tag{11}\]  
_The MGF of \(\zeta_{k}\) is_  
\[ \begin{array}{l}\mathcal{M}_{\zeta_{k}}(s)\\ =\frac{\mathcal{A}_{k}\mathcal{D}_{k}^{\mathcal{A}_{k}\mathcal{K}_{k}}}{\Omega _{k}^{\mathcal{A}_{k}\mathcal{K}_{k}}\Gamma(\mathcal{K}_{k})}\sum\limits_{v=1 }^{V}\frac{c_{v}}{\omega_{v}^{\mathcal{A}_{k}\mathcal{K}_{k}}}\frac{q^{1/2}p^ {4_{k}\mathcal{K}_{k}-1/2}}{(2\pi)^{(p+q)/2-1}(-s)^{\mathcal{A}_{k}\mathcal{K }_{k}}}\mathrm{G}_{P,q}^{q,p}\bigg{(}\left(\frac{\mathcal{D}_{k}}{\omega_{v} \hat{\Omega}_{k}}\right)^{\mathcal{A}_{k}q}\frac{p^{p}}{(-s)^{q}q^{p}}\bigg{|} \begin{array}{c}\Delta(p,1-\mathcal{A}_{k}\mathcal{K}_{k})\\ \Delta(q,0)\end{array}\right)\\ \end{array}  \tag{12}\]  
_where \(\mathrm{G}_{P,q}^{q,p}(\cdot)\) is the Meijer \(G\)-function and \(\frac{p}{q}=\mathcal{A}_{k}=\mathcal{A}_{n,k}=1\).