## 4 Scaling Laws of Scheduling Gain--Imperfect CSI
Therefore, assuming (7) and (10), the variance of the individual rate for the MF receiver under imperfect CSI can be approximated as  
\[r_{k} =\sqrt{p_{u}}\hat{\mathbf{g}}_{k}^{H}\mathbf{h}_{k}x_{k}+\sqrt{p_{ u}}\sum\
olimits_{j\
eq k}\hat{\mathbf{g}}_{k}^{H}\mathbf{h}_{j}x_{j}+\hat{ \mathbf{g}}_{k}^{H}\mathbf{n}\]0  
Meanwhile, for the ZF receiver, the PDF and MGF of the SINR under the imperfect CSI can be obtained from Lemma 1. Under imperfect CSI, the PDF and MGF of the SINR for the ZF receiver are obtained from Lemma 1 after substituting \(d_{1}=M-K+1\), \(d_{2}=K\), \(a=p_{u}(1-\tau^{2})\) and \(b=p_{u}\tau^{2}\). Proof of Corollary 3.: Let \(\
u_{k}\overset{\Delta}{=}|\hat{\mathbf{w}}_{k}^{H}\hat{\mathbf{h}}_{k}|^{2}\) and \(\varepsilon_{j}\overset{\Delta}{=}|\hat{\mathbf{w}}_{k}^{H}\mathbf{e}_{j}|^{2 }\ \forall j\). Then, the SINR for the ZF receiver under imperfect CSI is given by  
\[r_{k} =\sqrt{p_{u}}\hat{\mathbf{g}}_{k}^{H}\mathbf{h}_{k}x_{k}+\sqrt{p_{ u}}\sum\
olimits_{j\
eq k}\hat{\mathbf{g}}_{k}^{H}\mathbf{h}_{j}x_{j}+\hat{ \mathbf{g}}_{k}^{H}\mathbf{n}\]1  
because \(\hat{\mathbf{w}}_{k}^{H}\hat{\mathbf{h}}_{j}=0\) for \(\forall j\
eq k\). \(\
u_{k}\sim\mathcal{X}_{2(M-K+1)}^{2}\) and \(\sum_{j}\varepsilon_{j}\sim\mathcal{X}_{2K}^{2}\). Moreover, \(\
u_{k}\) and \(\sum_{j}\varepsilon_{j}\) are independent of each other [33]. Therefore, we obtain Corollary 3 from Lemma 1. Similar to the case of the MF receiver, the SINR of the ZF receiver in the high \(p_{u}\) regime can be approximated as a scaled F distribution as \(\left(\left(\frac{\tau^{2}}{1-\tau^{2}}\right)\frac{K}{M-K+1}\right)\gamma_{k,ZF }^{ip}\sim F\left(2\left(M-K+1\right),2K\right)\). Therefore, using (7) and (10), the variance of the individual rate for the ZF receiver under imperfect CSI is expressed as  
\[r_{k} =\sqrt{p_{u}}\hat{\mathbf{g}}_{k}^{H}\mathbf{h}_{k}x_{k}+\sqrt{p_{ u}}\sum\
olimits_{j\
eq k}\hat{\mathbf{g}}_{k}^{H}\mathbf{h}_{j}x_{j}+\hat{ \mathbf{g}}_{k}^{H}\mathbf{n}\]2  
From the analysis, the following scaling laws of the individual rate with imperfect CSI can be observed. For the MF receiver, the variance of the individual rate \(\kappa_{2,MF}^{(ip)}\) is a monotonically increasing function of \(M\) (Appendix C) and scales with \(\mathcal{O}(1)\) as \(M\rightarrow\infty\). That is, the fluctuation characteristic of the individual rate for the MF receiver does not change according to the imperfectness of CSI. However, the channel hardening of the ZF receiver depends on \(\tau\) (Appendix D). As \(M\) increases, the variance of the individual rate \(\kappa_{2,ZF}^{(ip)}\) tends to monotonically increase for a large \(\tau\), whereas \(\kappa_{2,ZF}^{(ip)}\) tends to decrease monotonically for a small \(\tau\). Thus, \(\kappa_{2,ZF}^{(ip)}\) is scaled with \(\mathcal{O}(1)\) as \(M\rightarrow\infty\) for a large \(\tau\), and the channel hardening effect for the ZF receiver occurs only in the low \(\tau\) regime This is because the multiuser interference for the ZF receiver increases with \(\tau\). Hereafter, the scaling laws of the scheduling gain under imperfect CSI are investigated. We first consider the case in which the user scheduling relies on non-ideal CSI availability, assuming that the estimated channel is directly used for the user selection algorithm, as in [13; 17]. In this scenario, the BS will regard the estimated channel as the actual channel as the BS cannot estimate the channel estimation error by itself. Thus, a metric for user selection can be calculated by the estimated channel, and the BS can select several users to be served based on the calculated metrics. Accordingly, it can be observed from (15) that the achievable scheduling gain follows \(\sqrt{2\sigma_{R}^{2}\log\left(\frac{N}{K}\right)}\), where \(\hat{R}=\sum_{k}\hat{R}_{k}\) corresponds to the estimated sum rate derived from the estimated SINR in (2), after replacing \(\mathbf{h}\) with \(\mathbf{\hat{h}}\). Therefore, we can expect that there is no difference between the scaling laws of scheduling gain with imperfect and perfect CSI.