approximate sum rate $\tilde{\tilde{\mathcal{R}}}$ for $K^*$-RUS is then given by $$\label{eq_s4_1}\begin{split}
&\tilde{\tilde{\mathcal{R}}}^*_{\text{RUS}}=\left(1-\frac{K^*_{\text{RUS}}}{T}\right)K^*_{\text{RUS}}\\
&\times\log_2\left(1+\frac{Pc(M-K^*_{\text{RUS}})(\alpha+2)(R_{\max}^2-R_{\min}^2)}{2\sigma_n^2K(R_{\max}^{\alpha+2}-R_{\min}^{\alpha+2})}\right).
\end{split}$$For $K^*$-LUS, the sum rate is approximated as $$\label{eq_s4_2}\begin{split}&\tilde{\tilde{\mathcal{R}}}^*_{\text{LUS}}=\left(1-\frac{K^*_{\text{LUS}}}{T}\right)K^*_{\text{LUS}}\\&\times\log_2\!\!\left(\!\!1\!+\!\frac{Pc(M-K^*_{\text{LUS}})}{\sigma_n^2\sum_{k=1}^{K^*_{\text{LUS}}}R_{\min}^{\alpha}{}_2F_1\left(k,-\frac{\alpha}{2};N\!\!+\!\!1;1-\frac{R_{\max}^2}{R_{\min}^2}\!\!\right)}\!\!\right).\end{split}$$Compared to and , there is no integral calculation of logarithmic functions in and , which simplifies the system sum rate expression. Furthermore, as shown in Fig. \[fig3\], and are capable of providing tight approximations in the case of $\rho=0,\delta=0$.

According to and , it is clear that both of the sum rates $\tilde{\tilde{\mathcal{R}}}^*_{\text{RUS}}$ and $\tilde{\tilde{\mathcal{R}}}^*_{\text{LUS}}$ increase when $P$ and $M$ become larger. Additionally, when $N$ increases, the sum rate of $K^*$-LUS increases, while the sum rate of $K^*$-RUS remains unchanged.

As far as SUS is concerned in the TDD scenario, as proved in [@yoo_optimality_2006], the ergodic sum rate is upper bounded by $\bar{\mathcal{R}}_{\text{SUS}}$, which is given by $$\label{eq_s4_3}\bar{\mathcal{R}}_{\text{SUS}}=\left[1-\frac{N}{T}\right]^+M\Theta(\log_2\log_2N),$$ where $[\cdot]^+$ is defined as $[x]^+=\max\{x,0\}$. Due to the pilot overhead imposed by the channel estimation, the system sum rate is scaled by the factor $[1-N/T]^+$. More specifically, in this case the instantaneous CSI of all the $N$ candidate-UEs are required for select ingactive-UEs at each coherence slot.

Comparing the pre-log parts $[1-N/T]^+$ in , $(1-K^*_{\text{RUS}}/T)$ in and $(1-K^*_{\text{LUS}}/T)$ in , we can see that the advantages of $K^*$-RUS and 