## 5 Simulation Results  
In this section, simulation results are provided to verify our analyses. First, in Figure 1, the analytical PDFs of the SINR in the corollaries are compared with the simulation results, where \(M=64\), \(K=20\), \(p_{u}=20\) dB, and \(\tau^{2}=0.2\). The symbols represent the simulation results, and the lines depict the analytical results. From Figure 1, the derived PDFs match well with the simulation results. From Figure 2-9, simulation results for the ergodic sum rates and scheduling gains of user selection algorithms as well as the variances of individual rates and sum rates are shown. The margin of error considering the 95% confidence interval for each point is about \(2.73\cdot 10^{-2}\) at maximum. Figure 2 shows the variances of individual rate under perfect CSI according to \(M\) when \(K=20\) and \(p_{u}=20\) dB. For the MF receiver, two analytical results are considered: (7) after substituting the exact \(\mu_{\gamma}\) and \(\sigma_{\gamma}^{2}\) obtained by the MGF in Corollary 1, and (12) obtained by the F approximation. For the ZF receiver, the analytical result corresponds to (14). For the MF receiver, it is observed that the analytical results with the exact moments and F approximation are almost the same. The difference between the simulation and analytical results is due to neglecting the higher-order moments of SINR in the Taylor series expansion. Therefore, we can confirm that the variance of the MF receiver increases while that of the ZF receiver decreases as \(M\) increases. That is, under perfect CSI, the MF receiver does not exhibit the channel hardening phenomenon unlike the ZF receiver, as predicted. Figure 1: Comparison between the Monte Carlo simulations and analytical probability density functions (PDFs) of signal-to-interference-plus-noise ratio (SINR), where \(M=64\), \(K=20\), \(p_{u}=20\) dB, and \(\tau^{2}=0.2\). Figure 2: Variance of individual rate under perfect channel state information (CSI) as a function of \(M\), where \(K=20\) and \(p_{u}=20\) dB. Figure 3 shows the variances of the sum rate under perfect CSI as a function of \(M\) when \(K=20\) and \(p_{u}=20\) dB. For the MF receiver, the lower bound in (17) is represented; for the ZF receiver, the approximation in (18) is represented. Similar to the case of individual rate, the variance of the MF receiver increases, while that of the ZF receiver decreases as \(M\) increases. As demonstrated by the analysis, the sum rate of the ZF receiver still exhibits the channel hardening effect. Figure 4 illustrates the ergodic sum rate as a function of \(M\) according to the user selection algorithm at the BS under the assumption of perfect CSI, where \(N=100\), \(K=20\), and \(p_{u}=0\) dB. For the MF receiver, we consider SIRUS [14], which is proposed for MF-based massive MIMO systems. In SIRUS, a user who generates the maximum SIR among the remaining users is sequentially selected in a greedy manner until the number of selected users becomes equal to \(K\). For the ZF receiver, we consider SUS [13], which is designed for the ZF to maximize the sum rate. In SUS, a user set with near-orthogonal channel vectors is selected in the greedy manner. For comparison, the results of round robin (RR) scheduling and the achievable maximum sum rate in (15), i.e., optimal user scheduling, are presented. The RR scheduler selects the users randomly; hence, no scheduling gain occurs. Meanwhile, the maximum sum rate in (15) can be realized when an optimal user selection algorithm based on an exhaustive search is employed. Therefore, the simulation of the computational complexity of the optimal scheduler is infeasible; thus, the analytical results of (15) are shown, rather than the simulation results.