## 3 Scaling Laws of Scheduling Gain--Perfect CSI
### Individual Rate Analysis
Then, the SINR for the MF receiver is expressed as  
\[\gamma_{k,MF}=\frac{p_{u}\
u_{k}}{p_{u}\sum_{j\
eq k}\
u_{j}+1}.\]  
\(\
u_{k}\) in the numerator of \(\gamma_{k,MF}\) follows the chi-square distribution with \(2M\) degrees of freedom. Furthermore, \(\sum_{j\
eq k}\
u_{j}\) in the denominator of \(\gamma_{k,MF}\) follows the chi-square distribution with \(2(K-1)\) degrees of freedom because each \(\
u_{j}\sim\text{Exp}\left(1\right)\) and the summation of independent \((K-1)\) Exp \(\left(1\right)\) random variables follow a chi-square distribution with \(2(K-1)\) degrees of freedom [28]. In [29], it was proven that \(\
u_{k}\) is independent of \(\
u_{j}\) for \(\forall j\
eq k\). Therefore, we can obtain the PDF and the MGF of the SINR of the MF receiver directly from Lemma 1. Although the MGF in Lemma 1 is a closed-form expression, it is still a challenge to explicitly understand the dependence of the variance on the number of BS antennas \(M\). Therefore, to obtain a simplified form of the variance, we use the following lemma:  
**Lemma 2**. : _Consider two chi-square random variables \(X\sim\mathcal{X}_{2d_{1}}^{2}\) and \(Y\sim\mathcal{X}_{2d_{2}}^{2}\). Then a random variable \(Z\overset{\Delta}{=}\frac{d_{2}}{d_{1}}\cdot\frac{X}{Y}\) follows \(\mathcal{F}\left(2d_{1},2d_{2}\right)\), i.e., the PDF of \(Z\) is expressed as:_  
\[\log_{2}\left(1+\gamma_{k}\right)=\log_{2}e\left(\log\left(1+\mu_{ \gamma}\right)+\sum_{n=1}^{\infty}\frac{\left(-1\right)^{n-1}}{n}\frac{\left( \gamma_{k}-\mu_{\gamma}\right)^{n}}{\left(1+\mu_{\gamma}\right)^{n}}\right), \tag{4}\]0  
_and the \(n\)th moment is given by:_  
\[\log_{2}\left(1+\gamma_{k}\right)=\log_{2}e\left(\log\left(1+\mu_{ \gamma}\right)+\sum_{n=1}^{\infty}\frac{\left(-1\right)^{n-1}}{n}\frac{\left( \gamma_{k}-\mu_{\gamma}\right)^{n}}{\left(1+\mu_{\gamma}\right)^{n}}\right), \tag{4}\]1  
Proof of Lemma 2.: From the definition of F distributions [30], this is straightforward and thus omitted here. At the high \(p_{u}\) regime, i.e., the interference-limited environment, the noise variance in the denominator of the SINR can be neglected. Therefore, the distribution of the SINR for the MF receiver can be approximated as a scaled F distribution, i.e., \(\frac{K-1}{M}\gamma_{k,MF}\sim F\left(2M,2\left(K-1\right)\right)\). Therefore, using (10) in Lemma 2, we obtain the first and second moments of the SINR for the MF receiver as follows:  
\[\log_{2}\left(1+\gamma_{k}\right)=\log_{2}e\left(\log\left(1+\mu_{ \gamma}\right)+\sum_{n=1}^{\infty}\frac{\left(-1\right)^{n-1}}{n}\frac{\left( \gamma_{k}-\mu_{\gamma}\right)^{n}}{\left(1+\mu_{\gamma}\right)^{n}}\right), \tag{4}\]2  
By substituting the mean and variance of the SINR of (11) into (7), the approximated variance of the individual rate for the MF receiver is expressed as  
\[\log_{2}\left(1+\gamma_{k}\right)=\log_{2}e\left(\log\left(1+\mu_{ \gamma}\right)+\sum_{n=1}^{\infty}\frac{\left(-1\right)^{n-1}}{n}\frac{\left( \gamma_{k}-\mu_{\gamma}\right)^{n}}{\left(1+\mu_{\gamma}\right)^{n}}\right), \tag{4}\]3  
Hereafter, the variance of mutual information is analyzed for the case of the ZF receiver.