## 5 Numerical Evaluation  
In this section, we perform computer simulations to validate our analysis in Section 4. Numerical examples are also provided to evaluate the sum-rate performance of the proposed hybrid opportunistic scheduling method for finite parameters \(N\) and SNR. In our simulations, each channel coefficient in Equations (1) and (2) is generated \(10^{4}\) times for each system parameter. Unless otherwise stated, it is assumed that the large-scale path-loss component (i.e., \(\beta_{i}^{(d)}\), \(\beta_{i}^{(u)}\), and \(\beta_{ij}^{(M)}\) for all \(i\) and \(j\)) is given by \(1\) in our simulations. The average interference decaying rate is first evaluated numerically according to the total number of MSs, \(N\). Even if it seems unrealistic to have a large number of MSs in a cell, the range of parameter \(N\) is taken into account to precisely see some trends of curves varying with \(N\). In Figure 2, the log-log plot of the average interference decaying rate versus \(N\) is shown as \(N\) increases for system parameter \(M\in\{2,3\}\), indicating the number of transmit or receive antennas at the BS. This numerical result reveals that the interference decaying rate tends to decrease almost linearly with \(N\), but the slopes of the curves vary according to \(M\). The dotted lines are obtained from Theorem 2 (theoretical results) with proper biases, and, thus, only the slopes of the dotted lines are relevant. It is shown that the bound in Theorem 2 is indeed tight since the average interference decaying rates shown in Figure 2 are consistent with the user scaling law derived in Theorem 1. Moreover, it is shown that the average interference decaying rate gets increased as \(M\) increases since the user scaling law in Theorems 1 and 2 is expressed as an increasing function of \(M\). As shown in Figure 3, when \(M\in\{2,3\}\), the achievable sum-rates of the proposed hybrid opportunistic scheduling method are now evaluated according to the received SNR (in dB scale) and are compared with the conventional scheduling method in [26] where downlink MSs are opportunistically selected while uplink MSs are arbitrarily selected. Note that \(N\) is set to a different scalable value  
Figure 2: The average interference decaying rate versus \(N\) when \(M\in\{2,3\}\). according to SNR, i.e., \(N=\text{SNR}^{M}\), to see whether the slope of a curve follows the DoF in Theorems 1. It is obvious to see that the proposed method outperforms the conventional one in terms of sum-rates for all SNR regimes. This is because the DoF achieved by the method in [26] is surely lower than \(2M=4\) due to the fact that its user scaling law \(N=\omega(\text{SNR}^{2M-1})\) is not fulfilled and thus there exists more residual interference at each receiver side. It indicates that the performance gap between the two methods becomes large in the high SNR regime. In addition, the effect of the path-loss attenuation factor on the sum-rates is examined. For convenience of an illustration, suppose that \(\beta\triangleq\beta_{i}^{(d)}=\beta_{i}^{(u)}\) and \(\beta_{ij}^{(M)}=1\). That is, we consider the case where both downlink and uplink channels experience the same degree of path-loss attenuation. Figure 3: The achievable sum-rates versus SNR. (**a**) \(M=2\); (**b**) \(M=3\). Here, \(0<\beta<1\) corresponds to the case where an MS and the BS are relatively far away from each other while most MSs are co-located. On the other hand, \(\beta>1\) corresponds to the case where the distance between an MS and the BS is relatively close and MSs are separated by an obstacle (e.g., a wall). In Figure 4, the achievable sum-rates of the proposed hybrid opportunistic scheduling method versus the received SNR (in dB scale) are evaluated for \(M=2\) and \(\beta\in\{0.2,0.5,0.8,1.2,1.5\}\). As \(\beta\) decreases, both the desired signal power at MS \(\pi_{i}\) and the downlink interference power at MS \(\pi_{i}\) get reduced due to a more severe path-loss attenuation between an MS and the BS.