## 3 Performance Evaluation  
By substituting (4) into (6) and applying some arithmetic manipulations, the effective capacity of the \(m\)-user can be rewritten as \[C_{m,e}=-\frac{1}{n\theta_{m}}\ln\Big{\{}\mathbb{E}\Big{[}\varepsilon+(1- \varepsilon)(1+\gamma_{m})^{\kappa}e^{\beta_{m}\delta_{m}}\Big{]}\Big{\}}, \tag{8}\]  
where \(\delta_{m}=\frac{1}{\ln 2}\sqrt{1-\left(1+\gamma_{m}\right)^{-2}}\). In light of the power series expansion and taking the first three items, i.e., \(e^{x}\approx 1\)\(+x+\frac{x^{2}}{2}\), an approximated expression can be obtained as follows:  
\[C_{m,e}\approx-\frac{1}{n\theta_{m}}\ln\Bigg{\{}\mathbb{E}\big{[}\varepsilon+ (1-\varepsilon)(1+\gamma_{m})^{\kappa_{m}}\times\left(1+\beta_{m}\delta_{m}+ \frac{(\beta_{m}\delta_{m})^{2}}{2}\right)\right]\Bigg{\}} \tag{9}\]  
\[+\underbrace{\mathbb{E}\Bigg{\{}\frac{\beta_{m}}{\ln 2}(1- \varepsilon)(1+\gamma_{m})^{\kappa_{m}}\sqrt{\left[1-\left(1+\gamma_{m} \right)^{-2}\right]}\Bigg{\}}}_{I_{2}}\] \[+\underbrace{\mathbb{E}\Bigg{\{}(1-\varepsilon)(1+\gamma_{m})^{ \kappa_{m}}\frac{\beta_{m}^{2}\Big{[}1-(1+\gamma_{m})^{-2}\Big{]}(\log_{2} \!e)^{2}}{2}\Bigg{\}}}_{I_{3}}\Bigg{\}}. \tag{10}\]  
Upon substituting (2) into (10), we obtain \(I_{1}\) as follows:  
\[I_{1}=(1-\varepsilon)\mathbb{E}\Bigg{[}\Bigg{(}1+\frac{\rho|h_{m}|^{2}a_{m}}{ \rho|h_{m}|^{2}\tilde{a}_{m}+1}\Bigg{)}^{a_{m}}\Bigg{]}=(1-\varepsilon)\int_{ 0}^{\infty}\left(1+\frac{\rho a_{m}x}{\rho\tilde{a}_{m}x+1}\right)^{a_{m}}f_{ |h_{m}|^{2}}(x)dx. \tag{11}\]  
With the help of order statistics in [25], the channel gain PDF of user \(m\)\(|h_{m}|^{2}\) is  
\[f_{|h_{m}|^{2}}(x)=\frac{M!}{(m-1)!(M-m)!}f_{|\tilde{h}_{m}|^{2}}(x)\quad \times\bigg{[}F_{|\tilde{h}_{m}|^{2}}(x)\bigg{]}^{m-1}\bigg{[}1-F_{|\tilde{h} _{m}|^{2}}(x)\bigg{]}^{M-m}, \tag{12}\]  
where \(|\tilde{h}_{m}|^{2}\) is the unsorted channel gain between the BS and user \(m\). The corresponding cumulative distribution function (CDF) is expressed as \(F_{|\tilde{h}_{m}|^{2}}(x)=1-e^{-x}\). By substituting (12) into (11) and then applying a binomial theorem, \(I_{1}\) can be calculated as follows:  
\[I_{1}= (1-\varepsilon)\phi_{m}\sum_{r=0}^{m-1}\binom{m-1}{r}(-1)^{r} \times\int_{0}^{\infty}\left(1+\frac{\rho a_{m}x}{\rho\tilde{a}_{m}x+1}\right) ^{a_{m}}e^{-x\varphi}dx\] \[= (1-\varepsilon)\phi_{m}\sum_{r=0}^{m-1}\binom{m-1}{r}\frac{(-1)^{ r}}{\varphi}\times\int_{0}^{\infty}\left(1+\frac{\rho a_{m}y}{\rho\tilde{a}_{m}y+ \varphi}\right)^{a_{m}}e^{-y}dy, \tag{13}\]  
where \(\phi_{m}=\frac{M!}{(m-1)!(M-m)! }\) and \(\varphi=M-m+r+1\). The expression of (13) can be further approximated as integral expressions using the virtue of the Gauss-Laguerre quadrature [26],  
\[I_{1}\approx(1-\varepsilon)\phi_{m}\sum_{r=0}^{m-1}\binom{m-1}{r}\frac{(-1)^{ r}}{\varphi}\times\sum_{p=1}^{p}H_{p}\bigg{(}1+\frac{\rho a_{m}x_{p}}{\rho \tilde{a}_{m}x_{p}+\varphi}\bigg{)}^{a_{m}}, \tag{14}\]where \(x_{p}\) is the abscissas, and \(H_{p}\) is the weight of the Gauss-Laguerre quadrature. \(x_{p}\) is the \(p\)-th zero of \(L_{p}\big{(}x_{p}\big{)}\), \(H_{p}\) can be formulated as \(H_{p}=\frac{(P! )^{2}x_{p}}{\big{[}L_{p+1}(x_{p})\big{]}^{2}}\), and \(P\) is a complexity-vs-accuracy trade-off parameter, which can ensure a trade-off between complexity and accuracy.