## 4 Scaling Laws of Scheduling Gain--Imperfect CSI  
In this section, we extend the scaling laws of the scheduling gain of optimal user scheduling in Section 3 to the case of imperfect CSI at the BS. The imperfectness of CSI affects two major operations at the BS: data demodulation and user scheduling. For data demodulation, the imperfect CSI typically implies a CSI with a channel estimation error that occurs due to the use of a practical channel estimator at the BS. The channel estimation error causes additional multiuser interference; thus, the channel hardening effect for the cases of imperfect CSI will be different compared to that for the perfect CSI case. Conversely, from the user scheduling perspective, the CSI imperfectness corresponds to the degree of CSI availability for calculating proper user selection metrics to determine a set of users to be scheduled. The CSI availability for user scheduling can be affected by not only the accuracy of the estimated channel, but also the extra channel-related information, which can contribute to better user selection. Thus, we consider two scenarios of CSI availability for user scheduling: non-ideal CSI availability as the worst case and near-ideal CSI availability as the best case. Based on these scenarios, which will be explained in detail later in this section, the effect of imperfect CSI on the scaling laws of scheduling gain will be investigated. We begin by analyzing the effect of imperfect CSI on the fluctuation of individual rate. Without loss of generality, the estimated channel \(\mathbf{\hat{h}}_{k}\) of user \(k\) can be modeled as [18]:  
\[\mathbf{h}_{k}=\sqrt{1-\tau^{2}}\mathbf{\hat{h}}_{k}+\tau\mathbf{e}_{k}, \tag{19}\]  
where \(\mathbf{e}_{k}\sim\mathcal{CN}\left(\mathbf{0},\mathbf{I}_{M}\right)\) is the Gaussian noise vector uncorrelated with \(\mathbf{\hat{h}}_{k}\) and \(\tau\in[0,1]\), which represents the imperfectness of \(\mathbf{\hat{h}}_{k}\). Typically, \(\tau\) is determined by the pilot sequence length \(t_{p}\) and pilot power \(p_{p}\), such as \(\tau=\sqrt{\frac{1}{t_{p}t_{p}+1}}\) when the estimated channel is obtained by the MMSE channel estimator, assuming that the orthogonal uplink pilot sequences are used across the users, as indicatedin [32]. Based on the estimated channel, the receiver matrix \(\hat{\mathbf{G}}\) is calculated as (1) at the BS, and the received \(k\)th stream and corresponding received SINR for user \(k\) are expressed as:  
\[r_{k} =\sqrt{p_{u}}\hat{\mathbf{g}}_{k}^{H}\mathbf{h}_{k}x_{k}+\sqrt{p_{ u}}\sum\
olimits_{j\
eq k}\hat{\mathbf{g}}_{k}^{H}\mathbf{h}_{j}x_{j}+\hat{ \mathbf{g}}_{k}^{H}\mathbf{n}\] \[=\sqrt{p_{u}\left(1-\tau^{2}\right)}\hat{\mathbf{g}}_{k}^{H} \hat{\mathbf{h}}_{k}x_{k}+\sqrt{p_{u}}\tau^{2}\hat{\mathbf{g}}_{k}^{H} \mathbf{e}_{k}x_{k}+\sqrt{p_{u}}\sum\
olimits_{j\
eq k}\hat{\mathbf{g}}_{k}^{H }\mathbf{h}_{j}x_{j}+\hat{\mathbf{g}}_{k}^{H}\mathbf{n},\]  
and  
\[\gamma_{k}^{ip}=\frac{p_{u}\left(1-\tau^{2}\right)\left|\hat{\mathbf{w}}_{k}^{ H}\hat{\mathbf{h}}_{k}\right|^{2}}{p_{u}\tau^{2}\left|\hat{\mathbf{w}}_{k}^{H} \mathbf{e}_{k}\right|^{2}+p_{u}\sum\
olimits_{j\
eq k}\left|\hat{\mathbf{w}}_ {k}^{H}\mathbf{h}_{j}\right|^{2}+1}, \tag{20}\]  
where \(\hat{\mathbf{w}}_{k}=\hat{\mathbf{g}}_{k}/\left|\hat{\mathbf{g}}_{k}\right|\) is the normalized \(k\)th column vector of \(\hat{\mathbf{G}}\). Next, for the MF receiver, the PDF and MGF of the SINR are derived. Considering imperfect CSI, we introduce Lemma 3. **Lemma 3**.