## 5 Simulation Results
In addition to the ergodic sum rate shown in Figure 4, we plot the corresponding scheduling gain as a function of \(M\) under perfect CSI in Figure 5, where the simulation environment is equivalent to that in Figure 4. In Figure 5, the scheduling gain is defined as the performance gain of a specific user selection algorithm compared to RR in terms of the ergodic sum rate. Figure 3: Variance of sum rate under perfect CSI as a function of \(M\), where \(K=20\) and \(p_{u}=20\) dB. As shown in Figures 4 and 5, the performance gap between the SUS and RR of the ZF receiver decreases as \(M\) increases. This is because the variance of the sum rate decreases for the ZF receiver for a large \(M\). For the MF receiver, the gain of SIRUS increases and maintains a constant positive performance gap compared to the RR scheduler as \(M\) increases. However, the performance gap between the SUS and RR of the MF receiver shrinks as \(M\) increases, although the variance of the MF receiver increases. That is, for massive MIMO systems with SUS for the MF receiver, multiuser diversity gain is not sufficiently obtained compared to SIRUS. This implies that the user scheduling algorithm should be carefully chosen according to the types of receivers to fully utilize the multiuser diversity. From the simulation results, we can conclude that user scheduling is more important for the MF receiver than for the ZF receiver in massive MIMO systems under perfect CSI, as analyzed in Section 3. Hereafter, the simulation results under imperfect CSI are presented from Figure 6-9. In Figure 6, the variance of the individual rate under imperfect CSI is presented as a function of \(M\), where \(N=100\), \(K=20\), and \(p_{u}=0\) dB. For simplicity, the analytical results of the F approximation are omitted in Figure 6. As demonstrated by the analysis for the MF receiver, the variance of the individual rate always increases, regardless of \(\tau\). However, for the ZF receiver, the variance of the individual rate  
Figure 4: Ergodic sum rate under perfect CSI as a function of \(M\), where \(N=100\), \(K=20\), and \(p_{u}=0\) dB. Figure 5: Scheduling gain under perfect CSI as a function of \(M\), where \(N=100\), \(K=20\), and \(p_{u}=0\) dB. decreases when \(\tau^{2}=0.1\) and increases when \(\tau^{2}=0.4\), according to \(M\). Therefore, it can be observed from Figure 6 that the imperfectness of CSI affects the channel hardening effect for the ZF receiver. Furthermore, under imperfect CSI, all variances of the individual rate converge to a constant value as \(M\) increases. Figure 7 shows the variance of the sum rate under imperfect CSI as a function of \(M\), where \(N=100\), \(K=20\), and \(p_{u}=0\) dB. For the analytical results, the lower bounds (26) and (27) are shown for the MF and ZF receivers, respectively. Similar to the results in Figure 6, the scaling law on the variance of the sum rate with \(M\) for the MF receiver does not change, regardless of \(\tau\), while that for the ZF receiver tends to change depending on \(\tau\). Figure 8 represents the ergodic sum rate as a function of \(M\) with the user selection algorithm at the BS, where \(N=100\), \(K=20\), \(p_{u}=0\) dB and \(\tau=0.4\). Figure 9 depicts the corresponding scheduling gain. The cases of a small \(\tau\) are not presented for simplicity. As explained in Section 4, we consider two scenarios for CSI availability for user scheduling at the BS, namely non-ideal CSI and near-ideal  
Figure 6: Variance of individual rate under imperfect CSI as a function of \(M\), where \(N=100\), \(K=20\), and \(p_{u}=20\) dB. Figure 7: Variance of sum rate under imperfect CSI as a function of \(M\), where \(N=100\), \(K=20\), and \(p_{u}=20\) dB. CSI. The exhaustive search for the optimal user selection algorithm uses the exact SINR formula in (4) as the selection metric; thus, the CSI availability for user scheduling only affects the scheduling gain for the optimal user selection algorithm.