## 3 Scaling Laws of Scheduling Gain--Perfect CSI
### Individual Rate Analysis  
According to the Shannon capacity formula [24], the individual rate for user \(k\) is defined as  
\[R_{k}=\log_{2}\left(1+\gamma_{k}\right). \tag{3}\]  
To understand the fluctuations of the individual rate, it is necessary to analyze the second moment of mutual information. The direct derivation of the exact distribution of the mutual information is unfeasible [25]; thus, the approximated second moment is obtained in this subsection. For (3), the Taylor series is obtained as follows:  
\[\log_{2}\left(1+\gamma_{k}\right)=\log_{2}e\left(\log\left(1+\mu_{ \gamma}\right)+\sum_{n=1}^{\infty}\frac{\left(-1\right)^{n-1}}{n}\frac{\left( \gamma_{k}-\mu_{\gamma}\right)^{n}}{\left(1+\mu_{\gamma}\right)^{n}}\right), \tag{4}\]  
where \(\mu_{\gamma}=\mathbb{E}\{\gamma_{k}\}\). From the definition of the joint cumulant moment of \(n\) random variables \(X_{1},..,X_{n}\)[26], we can obtain  
\[\kappa_{n} =\mathbb{E}_{c}\left\{X_{1},X_{2},...,X_{n}\right\}\] \[\triangleq\sum_{\psi}\left(-1\right)^{\left|\psi\right|-1}\left( \left|\psi\right|-1\right)!\prod_{\beta\in\psi}\mathbb{E}\left\{\prod_{i\in \beta}X_{i}\right\}, \tag{5}\]  
where \(\psi\) runs through all partitions of \(\{1,...,n\}\), \(\left|\psi\right|\) denotes the number of blocks in \(\psi\), and \(\beta\) runs through the list of all blocks of \(\psi\). From (4) and (5), the second cumulant moment, i.e., the variance of individual rate, can be expressed as  
\[\kappa_{2} =\mathbb{E}_{c}\left\{\log_{2}\left(1+\gamma_{k}\right),\log_{2} \left(1+\gamma_{k}\right)\right\}\] \[=\left(\log_{2}e\right)^{2}\sum_{m=1}^{\infty}\sum_{n=1}^{ \infty}\frac{\left(-1\right)^{m+n}}{mn}\frac{\mathbb{E}_{c}\left\{\left(\gamma _{k}-\mu_{\gamma}\right)^{m},\left(\gamma_{k}-\mu_{\gamma}\right)^{n}\right\} }{\left(1+\mu_{\gamma}\right)^{m+n}} \tag{6}\] \[\overset{(a)}{\approx}\frac{\left(\log_{2}e\right)^{2}\sigma_{ \gamma}^{2}}{\left(1+\mu_{\gamma}\right)^{2}}. \tag{7}\]  
Here, the approximation (a) is obtained by neglecting the higher-order terms in the summation in (6), and \(\sigma_{\gamma}^{2}=\text{Var}\left\{\gamma_{k}\right\}\). Therefore, as shown in (7), the moments of the SINR, \(\mu_{\gamma}\) and \(\sigma_{\gamma}^{2}\), should be derived to obtain the variance of mutual information. Next, the variance of the mutual information is analyzed for the case of the MF receiver. Thus, we first derive the following lemma:  
**Lemma 1**. : _For a random variable \(Z\overset{\Delta}{=}\frac{\theta X}{BY+1}\), where \(X\sim\mathcal{X}_{2d_{1}}^{2}\), \(Y\sim\mathcal{X}_{2d_{2}}^{2}\), and \(X\) and \(Y\) are independent, the PDF of \(Z\) is expressed as_  
\[f_{Z}\left(z\right)=\frac{d_{1}\left(\frac{z}{a}\right)^{d_{1}-1}e^{-\frac{z}{ a}}}{a\Gamma\left(d_{2}\right)}\sum_{i=0}^{d_{1}}\frac{\left(i+d_{2}-1\right)!b^{i} }{\left(d_{1}-i\right)!i!}\left(\frac{b}{a}z+1\right)^{-\left(i+d_{2}\right)}. \tag{8}\]  
_Furthermore, the MGF of \(Z\) is expressed as_  
\[\mathcal{M}_{Z}\left(s\right)=\frac{d_{1}\Gamma\left(d_{1}\right)}{\Gamma \left(d_{2}\right)}\sum_{i=0}^{d_{1}}\frac{\left(i+d_{2}-1\right)!b^{i-d_{1}} }{\left(d_{1}-i\right)!i!}\times\Psi\left(d_{1},1+d_{1}-i-d_{2},\frac{1}{b}- \frac{a}{b}s\right), \tag{9}\]  
_where \(\Gamma\left(x\right)=\int_{0}^{\infty}t^{x-1}e^{-t}dt\) is the Gamma function; \(\Psi(a,b,s)\) is the confluent hypergeometric function of the second kind, which is defined by \(\Psi\left(a,b,s\right)=\frac{1}{\Gamma\left(a\right)}\int_{0}^{\infty}e^{-st}t ^{a-1}(1+t)^{b-a-1}dt\)[27]._  
Proof of Lemma 1.: See Appendix A. Using Lemma 1, we can obtain the PDF and MGF of the SINR for the MF receiver. **Corollary 1**. : _The PDF and MGF of the SINR for the MF receiver are obtained from Lemma 1 after substituting \(d_{1}=M\), \(d_{2}=K-1\), and \(a=b=p_{u}\)._Proof of Corollary 1.: Let \(\
u_{j}\overset{\Delta}{=}\left|\mathbf{w}_{k}^{H}\mathbf{h}_{j}\right|^{2}=\left| \mathbf{h}_{k}^{H}\mathbf{h}_{j}\right|^{2}/\left\|\mathbf{h}_{k}\right\|^{2}\)\(\forall j\).