## 3 Scaling Laws of Scheduling Gain--Perfect CSI
### Sum Rate Analysis  
The scheduling gain is related to the fluctuations in the mutual information; thus, more scheduling gain can be realized as the fluctuations in the sum rate become larger. To explicitly understand the relationship between the variance of sum rate and the scheduling gain, we introduce the result in [21] with a slight modification. We assume that the BS selects \(K\) active users among \(N\) total users through a user scheduling algorithm. Let \(R\) denote the sum rate, that is, \(R\overset{\Delta}{=}\sum_{k=1}^{K}R_{k}\) and \(\mathcal{R}\) denote the achievable rate, defined as the maximum sum rate after the optimal multiuser scheduling, that is  
\[\mathcal{R}\overset{\Delta}{=}\max_{\mathcal{S}\subset\{1,\ldots,N\},| \mathcal{S}|=K}R\left(\mathcal{S}\right),\]  
where \(\mathcal{S}\) is a set of users and \(R\left(\mathcal{S}\right)\overset{\Delta}{=}\sum_{k\in\mathcal{S}}R_{k}\). Combining the Gaussianity of the sum rate of the linear receiver [25] and the results in [21], we can approximate the achievable rate \(\mathcal{R}\) as  
\[\mathcal{R}\approx\mu_{R}+\sqrt{2\sigma_{R}^{2}\log\left(\frac{N}{K}\right)}, \tag{15}\]  
where \(\mu_{R}\) and \(\sigma_{R}^{2}\) are the mean and variance of \(R\), respectively. According to (15), the maximum scheduling gain by the optimum algorithm is approximately \(\sqrt{2\sigma_{R}^{2}\log\left(\frac{N}{K}\right)}\). Hereafter, we investigate the variance of the sum rate \(\sigma_{R}^{2}\) to understand the achievable scheduling gain with respect to the number of antennas. Using the results in [25], the second-order joint cumulant moment of \(R\) can be expressed as  
\[\kappa_{2}^{sum} =\sum_{m=1}^{K}\sum_{n=1}^{K}\mathbb{E}_{c}\left[R_{m},R_{n}\right]\] \[=\sum_{m=1}^{K}\mathbb{E}_{c}\left[R_{m},R_{m}\right]+\sum_{m=1} ^{K}\sum_{n=m}^{K}\mathbb{E}_{c}\left[R_{m},R_{n}\right]\] \[\overset{(a)}{=}K\mathbb{E}_{c}\left[R_{1},R_{1}\right]+K\left( K-1\right)\mathbb{E}_{c}\left[R_{1},R_{2}\right]\] \[\overset{(b)}{\approx}(\log_{2}\!e)^{2}K\frac{\sigma_{\gamma}^{ 2}}{\left(1+\mu_{\gamma}\right)^{2}}+(\log_{2}\!e)^{2}K\left(K-1\right)\frac{ \mathbb{E}\left[\gamma_{1}\gamma_{2}\right]-\mu_{\gamma}^{2}}{\left(1+\mu_{ \gamma}\right)^{2}}. \tag{16}\]  
Here, (a) follows the symmetry of the joint cumulant matrix, [25] and (b) follows the same approximation used in (7). Although the first term of (16), the variance of the individual rate, was already derived, the second term of (16), the covariance of the SINR, should be derived and is defined as \(\text{Cov}\left[\gamma_{1},\gamma_{2}\right]=\mathbb{E}\left[\gamma_{1}\gamma _{2}\right]-\mathbb{E}\left[\gamma_{1}\right]\mathbb{E}\left[\gamma_{2}\right]\). To calculate the exact covariance, the joint PDF of \(Z\overset{\Delta}{=}\gamma_{1}\gamma_{2}\) is required. However, the joint PDF is intractable because \(\gamma_{1}\) and \(\gamma_{2}\) are not independent of each other. Therefore, as an alternative approach for the MF receiver, we consider a lower bound of the variance of the sum rate by considering only the first term in (16). Substituting (12) into the first term in (16), we obtain the lower bound for the variance of the sum rate for the MF receiver at the high \(p_{u}\) regime as follows:  
\[\kappa_{2,MF}^{sum}>\frac{\left(\log_{2}\!e\right)^{2}MK}{\left(M+K-2\right) \left(K-3\right)}. \tag{17}\]  
Meanwhile, for the ZF receiver, the approximated covariance of the SINR can be derived as \(\text{Cov}\left[\gamma_{1},\gamma_{2}\right]\approx p_{u}^{2}\) in [25] by the joint distribution of the eigenvalues and Noviokv's theorem. By combining (14) and (16), the approximated variance of the sum rate for the ZF receiver can be obtained as  
\[\kappa_{2,ZF}^{sum}\approx\frac{p_{u}^{2}(\log_{2}\!e)^{2}MK}{\left(1+p_{u} \left(M-K+1\right)\right)^{2}}. \tag{18}\]  
From (17) and (18), it is observed that the scaling laws of the sum rate are equivalent to those of the individual rate.