sum rate improvements as $N$ increases because more multiuser diversity related to LSF can be exploited with larger $N$. For SUS, the sum rate decreases with $N$, because the pilot overhead becomes serious for large $N$, which overwhelms the increase of the multiuser diversity gain.

![Sum rate $\mathcal{R}$ vs. transmit power $P$. $M=32\text{ and }N=64.$[]{data-label="fig7"}](sim4){width="7.5cm"}

![Sum rate $\mathcal{R}$ vs. the number of candidate-UEs $N$. $P=30 \text{ dBm and }M=32.$[]{data-label="fig8"}](sim5){width="7.5cm"}

Fairness Performance
--------------------

In this subsection, we evaluate the system performance in terms of long-term fairness among UEs. The Jain’s Fairness Index (JFI)[@DEC_1984] $\mathcal{F}$, defined as $$\mathcal{F}=\frac{\left(\sum_{n=1}^N\omega_n\mathcal{R}_n\right)^2}{N\sum_{n=1}^N(\omega_n\mathcal{R}_n)^2},$$is employed in our investigation, where $\mathcal{R}_n$ is the rate of the $n$-th candidate-UE defined in and $\omega_n$ is the probability of the $n$-th candidate-UE being selected to be served at each coherence slot.

First, we briefly analyze the long-term fairness performance of $K^*$-RUS and $K^*$-LUS without considering the channel estimation error and transmit correlation. Then the simulation results are shown in Fig. \[fig9\] and Fig. \[fig10\].

In the case of $\rho=0,\delta=0$, for the $n$-th candidate-UE in $K^*$-RUS, we have $\omega_n=K^*_{\text{RUS}}/N$ and $\mathcal{R}_n=\mathcal{R}^*_{\text{RUS}}/K^*_{\text{RUS}}$. Therefore, the JFI of $K^*$-RUS is given by $$\label{eq_jfi1}\mathcal{F}_{\text{RUS}}=\frac{\left(\sum_{n=1}^N\frac{K^*_{\text{RUS}}}{N}\frac{\mathcal{R}_{\text{RUS}}^*}{K^*_{\text{RUS}}}\right)^2}{N\sum_{n=1}^N\left(\frac{K^*_{\text{RUS}}}{N}\frac{\mathcal{R}_{\text{RUS}}^*}{K^*_{\text{RUS}}}\right)^2}=1.$$It is clear that $K^*$-RUS is capable of offering the optimal fairness among candidate-UEs. 