## 4 Scaling Laws of Scheduling Gain--Imperfect CSI
: _For a random variable \(Z\overset{\Delta}{=}\frac{aX}{bY_{1}+cY_{2}+1}\), where \(X\sim\mathcal{X}_{2d_{1}}^{2}\), \(Y_{1}\sim\mathcal{X}_{2d_{2}}^{2}\), and \(Y_{2}\sim\mathrm{Exp}\left(1\right)\), the PDF of \(Z\) is obtained by_  
\[f_{Z}\left(z\right) =\frac{d_{1}\left(\frac{c-b}{c}\right)^{-d_{2}}\left(\frac{z}{a} \right)^{d_{1}-1}e^{-\frac{z}{a}}}{ac}\times\left(\sum\
olimits_{l=0}^{d_{1} }\mathcal{A}_{1}\left(\frac{z}{a}+\frac{1}{c}\right)^{-\left(l+1\right)}-\sum \
olimits_{i=0}^{d_{2}-1}\sum\
olimits_{p=0}^{d_{1}}\mathcal{A}_{2}\left( \frac{z}{a}+\frac{1}{b}\right)^{-\left(p+i+1\right)}\right). \tag{21}\]  
_Here, \(\mathcal{A}_{1}=\frac{1}{\left(d_{1}-l\right)! }\) and \(\mathcal{A}_{2}=\frac{\left(p+i\right)!\left(\frac{1}{b}-\frac{1}{c}\right)^{ i}}{\left(d_{1}-p\right)!p!\left(i\right)}\). Moreover, the MGF of \(Z\) is obtained by_  
\[\mathcal{M}_{Z}\left(s\right) =\frac{d_{1}\left(1-\frac{b}{c}\right)^{-d_{2}}}{c}\sum\limits_{ l=0}^{d_{1}}\mathcal{A}_{1}\mathrm{Y}\left(d_{1},l,\frac{1}{c},\left(1-as \right)\right)\] \[\quad-\frac{d_{1}\left(1-\frac{b}{c}\right)^{-d_{2}}}{c}\sum \limits_{i=0}^{d_{2}-1}\sum\limits_{p=0}^{d_{1}}\mathcal{A}_{2}\mathrm{Y} \left(d_{1},p+i,\frac{1}{b},\left(1-as\right)\right), \tag{22}\]  
_where \(\mathrm{Y}\left(a,b,c,s\right)\) is defined as_  
\[\mathrm{Y}\left(a,b,c,s\right) =c^{a-b-1}\frac{\Gamma\left(a\right)\Gamma\left(1-a+b\right)}{ \Gamma\left(1+b\right)}{}_{1}F_{1}\left(a,a-b,cs\right)\] \[\quad+s^{1-a+b}\Gamma\left(a-b-1\right){}_{1}F_{1}\left(1+b,2-a+b,cs\right), \tag{23}\]  
_and \({}_{p}F_{q}\left(a,b,s\right)\) represents the generalized hypergeometric series [27]._  
Proof of Lemma 3.: See Appendix B. From Lemma 3, we can obtain the PDF and MGF of the SINR for the MF receiver under imperfect CSI. **Corollary 2**. : _Under imperfect CSI, the PDF and MGF of the SINR for the MF receiver are obtained from Lemma 3 after substituting \(d_{1}=M\), \(d_{2}=K-1\), \(a=p_{u}(1-\tau^{2})\), \(b=p_{u}\), and \(c=p_{u}\tau^{2}\)._  
Proof of Corollary 2.: Let \(v_{k}\overset{\Delta}{=}|\hat{\mathbf{w}}_{k}^{H}\hat{\mathbf{h}}_{k}|^{2}\), \(\varsigma_{j}\overset{\Delta}{=}|\hat{\mathbf{w}}_{k}^{H}\mathbf{h}_{j}|^{2}\)\(\forall j\) and \(\varepsilon_{k}\overset{\Delta}{=}\left|\hat{\mathbf{w}}_{k}^{H}\mathbf{e}_{k}\right|^{2}\). The SINR for the MF receiver under imperfect CSI conditions is obtained by  
\[\gamma_{k,MF}^{ip}=\frac{p_{u}\left(1-\tau^{2}\right)v_{k}}{p_{u}\tau^{2} \varepsilon_{k}+p_{u}\sum\
olimits_{j\
eq k}\xi_{j}+1}.\]As presented in the proof of Corollary 1, \(\
u_{k}\sim\mathcal{X}_{2M}^{2}\) and \(\sum_{j\
eq k}\xi_{j}\sim\mathcal{X}_{2(K-1)}^{2}\). \(\varepsilon_{k}\sim\text{Exp}\left(1\right)\) because \(\hat{\mathbf{w}}_{k}\) is a unit-norm random vector, independent of the Gaussian random vector \(\mathbf{e}_{k}\)[33]. In addition, \(\
u_{k}\), \(\sum_{j\
eq k}\xi_{j}\) and \(\varepsilon_{k}\) are independent of each other. Therefore, we obtain Corollary 2 from Lemma 3. To obtain a closed-form expression for the variance of the individual rate, we again use the F approximation. In the high \(\tau\) regime, the total interference term in the denominator of \(\gamma_{k,MF}^{ip}\) follows an approximate chi-square distribution with \(2K\) degrees of freedom, that is, \(\tau^{2}\big{|}\hat{\mathbf{w}}_{k}^{H}\mathbf{e}_{k}\big{|}^{2}+\sum_{j\
eq k} \big{|}\hat{\mathbf{w}}_{k}^{H}\mathbf{h}_{j}\big{|}^{2}\sim\mathcal{X}_{2K}^{2}\). Therefore, the SINR of the MF receiver can be approximated as a scaled F distribution as \(\left(\frac{K}{\left(1-\tau^{2}\right)M}\right)\gamma_{k,MF}^{ip}\sim F\left( 2M,2K\right)\) in the high \(p_{u}\) regime.