# Scaling Laws of Scheduling Gain for Uplink Massive MIMO Systems: Is User Scheduling Still Beneficial for Massive MIMO?

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## Abstract

**:**

## 1. Introduction

## 2. System Model

## 3. Scaling Laws of Scheduling Gain—Perfect CSI

#### 3.1. Individual Rate Analysis

**Lemma**

**1.**

**Corollary**

**1.**

**Proof**

**of Corollary 1.**

**Lemma**

**2.**

**Proof**

**of Lemma 2.**

#### 3.2. Sum Rate Analysis

- For the MF receiver, the scheduling gain first increases and then scales with $\mathcal{O}\left(1\right)$ as $M\to \infty $ under perfect CSI. This implies that the user scheduling to maximize the sum rate is still beneficial for massive MIMO systems with the MF receiver.
- For the ZF receiver, the scheduling gain decreases with $\mathcal{O}\left(\sqrt{\frac{1}{M}}\right)$ under the perfect CSI. Therefore, if the ZF receiver is used at the BS, only a limited scheduling gain can be achievable for large M. This implies that the benefit of user scheduling tends to disappear for massive MIMO systems using the ZF receiver.

## 4. Scaling Laws of Scheduling Gain—Imperfect CSI

**Lemma**

**3.**

**Corollary**

**2.**

**Proof**

**of Corollary 2.**

**Corollary**

**3.**

**Proof**

**of Corollary 3.**

- If non-ideal CSI is available for user scheduling at the BS, the scaling law of the scheduling gain with imperfect CSI is similar to that with perfect CSI. That is, under imperfect CSI for data demodulation with non-ideal CSI for user scheduling, the user selection is still beneficial for the MF receiver, whereas this benefit is negligible for the ZF receiver.
- If near-ideal CSI is available for user scheduling at the BS, i.e., under imperfect CSI for data demodulation with near-ideal CSI for user scheduling, the scaling law of the scheduling gain for the MF receiver with imperfect CSI is similar to that with perfect CSI. However, the scaling law of the scheduling gain for the ZF receiver under imperfect CSI is different from that under perfect CSI and depends on the channel estimation error $\tau $. In the low $\tau $ regime, the scheduling gain decreases. Meanwhile, in the high $\tau $ regime, the scheduling increases as M increases and eventually converges to a constant value, i.e., scaled by $\mathcal{O}\left(1\right)$ as $M\to \infty $. Therefore, under imperfect CSI for data demodulation, with near-ideal CSI for user scheduling, the user selection is still beneficial for the MF receiver, whereas it can be different for the ZF receiver depending on the imperfectness of CSI.

## 5. Simulation Results

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Proof of Lemma 1

## Appendix B. Proof of Lemma 3

## Appendix C. Proof of Monotonicity of (24)

## Appendix D. Proof of Monotonicity of (25)

- Case (i) When $a=2$, $f\left(M\right)=2cM-{c}^{2}$ is an increasing function of M. Since the root of $f\left(M\right)$ is $\omega =\frac{c}{2}=\frac{K-1}{2}$, $f\left(M\right)$ is always positive where $M\ge K$.
- Case (ii) When $0\le a<2$, $f\left(M\right)$ is a convex function and the two roots are given by$${\omega}_{1}=\frac{c\left(a-1\right)}{a}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{and}\phantom{\rule{0.277778em}{0ex}}{\omega}_{2}=\frac{c\left(a-1\right)}{a-2}.$$When $1\le a<2$, ${\omega}_{1}\ge {\omega}_{2}$ and $K>{\omega}_{1}$. When $0\le a<1$, ${\omega}_{2}>{\omega}_{1}$ and $K>{\omega}_{2}$. Therefore, $f\left(M\right)$ is always positive where $M\ge K$.
- Case (iii) When $a>2$, $f\left(M\right)$ is a concave function and ${\omega}_{2}>{\omega}_{1}$ and $K>{\omega}_{2}$. Therefore, $f\left(M\right)$ is always negative where $M\ge K$,

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**Figure 1.**Comparison between the Monte Carlo simulations and analytical probability density functions (PDFs) of signal-to-interference-plus-noise ratio (SINR), where $M=64$, $K=20$, ${p}_{u}=20$ dB, and ${\tau}^{2}=0.2$.

**Figure 2.**Variance of individual rate under perfect channel state information (CSI) as a function of M, where $K=20$ and ${p}_{u}=20$ dB.

**Figure 3.**Variance of sum rate under perfect CSI as a function of M, where $K=20$ and ${p}_{u}=20$ dB.

**Figure 4.**Ergodic sum rate under perfect CSI as a function of M, where $N=100$, $K=20$, and ${p}_{u}=0$ dB.

**Figure 5.**Scheduling gain under perfect CSI as a function of M, where $N=100$, $K=20$, and ${p}_{u}=0$ dB.

**Figure 6.**Variance of individual rate under imperfect CSI as a function of M, where $N=100$, $K=20$, and ${p}_{u}=20$ dB.

**Figure 7.**Variance of sum rate under imperfect CSI as a function of M, where $N=100$, $K=20$, and ${p}_{u}=20$ dB.

**Figure 8.**Ergodic sum rate under imperfect CSI as a function of M, where $N=100$, $K=20$, ${p}_{u}=0$ dB, and ${\tau}^{2}=0.4$.

**Figure 9.**Scheduling gain under imperfect CSI as a function of M, where $N=100$, $K=20$ and ${p}_{u}=0$ dB, and ${\tau}^{2}=0.4$.

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**MDPI and ACS Style**

Kim, T.; Park, S.
Scaling Laws of Scheduling Gain for Uplink Massive MIMO Systems: Is User Scheduling Still Beneficial for Massive MIMO? *Electronics* **2020**, *9*, 1650.
https://doi.org/10.3390/electronics9101650

**AMA Style**

Kim T, Park S.
Scaling Laws of Scheduling Gain for Uplink Massive MIMO Systems: Is User Scheduling Still Beneficial for Massive MIMO? *Electronics*. 2020; 9(10):1650.
https://doi.org/10.3390/electronics9101650

**Chicago/Turabian Style**

Kim, Taehyoung, and Sangjoon Park.
2020. "Scaling Laws of Scheduling Gain for Uplink Massive MIMO Systems: Is User Scheduling Still Beneficial for Massive MIMO?" *Electronics* 9, no. 10: 1650.
https://doi.org/10.3390/electronics9101650