## 4 Scaling Laws of Scheduling Gain--Imperfect CSI
Next, we determine the scaling laws when the near-ideal CSI is available for user scheduling. As mentioned earlier, there can be extra channel-related information available on top of the estimated channel at the BS, for user scheduling purposes. A typical example of obtaining additional information is to use sounding reference signals, as specified in 5G NR [34]; the additional uplink pilot used for SINR estimation, channel quality estimation, and beam direction estimation improves the user scheduling and is not directly related to uplink data demodulation. For simplicity, we assume that the exact SINR in (20) is available at the BS and used for the user selection metric in the case of near-ideal CSI availability. Then, it is observed from (15) that the achievable scheduling gain under near-ideal CSI availability can be obtained by \(\sqrt{2\sigma_{R}^{2}\log\left(\frac{N}{K}\right)}\), where \(\tilde{R}\) is derived from the exact SINR in (20). In this case, two different SINRs \(\gamma_{1}\) and \(\gamma_{2}\) in (16) are not independent of each other, even for the ZF receiver, because, unlike perfect CSI, there is residual multiuser interference under imperfect CSI. For tractable analysis, we consider the lower bounds for the variance of the sum rate for both MF and ZF receivers. By substituting (24) and (25) into the first term in (16), we obtain  
\[r_{k} =\sqrt{p_{u}}\hat{\mathbf{g}}_{k}^{H}\mathbf{h}_{k}x_{k}+\sqrt{p_{ u}}\sum\
olimits_{j\
eq k}\hat{\mathbf{g}}_{k}^{H}\mathbf{h}_{j}x_{j}+\hat{ \mathbf{g}}_{k}^{H}\mathbf{n}\]3  
and  
\[r_{k} =\sqrt{p_{u}}\hat{\mathbf{g}}_{k}^{H}\mathbf{h}_{k}x_{k}+\sqrt{p_{ u}}\sum\
olimits_{j\
eq k}\hat{\mathbf{g}}_{k}^{H}\mathbf{h}_{j}x_{j}+\hat{ \mathbf{g}}_{k}^{H}\mathbf{n}\]4  
From (26) and (27), it is observed that the scaling law for the variance of the sum rate is the same as that of the individual rate. Finally, we can summarize the scaling laws of the scheduling gain with imperfect CSI as follows:  
* If non-ideal CSI is available for user scheduling at the BS, the scaling law of the scheduling gain with imperfect CSI is similar to that with perfect CSI. That is, under imperfect CSI for data demodulation with non-ideal CSI for user scheduling, the user selection is still beneficial for the MF receiver, whereas this benefit is negligible for the ZF receiver. * If near-ideal CSI is available for user scheduling at the BS, i.e., under imperfect CSI for data demodulation with near-ideal CSI for user scheduling, the scaling law of the scheduling gain for the MF receiver with imperfect CSI is similar to that with perfect CSI. However, the scaling law of the scheduling gain for the ZF receiver under imperfect CSI is different from that under perfect CSI and depends on the channel estimation error \(\tau\). In the low \(\tau\) regime, the scheduling gain decreases. Meanwhile, in the high \(\tau\) regime, the scheduling increases as \(M\) increases and eventually converges to a constant value, i.e., scaled by \(\mathcal{O}\left(1\right)\) as \(M\rightarrow\infty\). Therefore, under imperfect CSI for data demodulation, with near-ideal CSI for user scheduling, the user selection is still beneficial for the MF receiver, whereas it can be different for the ZF receiver depending on the imperfectness of CSI.