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Utility Optimal Scheduling with a Slow Time-Scale Index-Bias for Achieving Rate Guarantees in Cellular Networks

Anurag Kumar, Rajesh Sundaresan

TL;DR

The paper tackles rate-guaranteed, utility-maximizing downlink scheduling for 5G network slicing by introducing a three-time-scale algorithm PF-RG-LM that combines fast slot-level scheduling with EWMA throughput tracking and a slow stochastic-approximation update of index-biases, interpreted as Lagrange multipliers. It proves convergence to the unique primal-dual optimum via a pair of coupled o.d.e.s and demonstrates, through extensive simulations, that slower updates to the index-bias yield throughput on the rate-region boundary with smoothly varying multipliers, outperforming a two-time-scale baseline. The approach offers tunable time-scales to balance accuracy of rate guarantees against tracking dynamics, enabling practical control for operators and robust performance across multiple UEs and channel states. Overall, the method provides a principled, scalable path to enforce minimum rates while maintaining high network utility in sliced cellular networks.

Abstract

One of the requirements of network slicing in 5G networks is RAN (radio access network) scheduling with rate guarantees. We study a three-time-scale algorithm for maximum sum utility scheduling, with minimum rate constraints. As usual, the scheduler computes an index for each UE in each slot, and schedules the UE with the maximum index. This is at the fastest, natural time-scale of channel fading. The next time-scale is of the exponentially weighted moving average (EWMA) rate update. The slowest time scale in our algorithm is an "index-bias" update by a stochastic approximation algorithm, with a step-size smaller than the EWMA. The index-biases are related to Lagrange multipliers, and bias the slot indices of the UEs with rate guarantees, promoting their more frequent scheduling. We obtain a pair of coupled ordinary differential equations (o.d.e.) such that the unique stable points of the two o.d.e.s are the primal and dual solutions of the constrained utility optimization problem. The UE rate and index-bias iterations track the asymptotic behaviour of the o.d.e. system for small step-sizes of the two slower time-scale iterations. Simulations show that, by running the index-bias iteration at a slower time-scale than the EWMA iteration and using the EWMA throughput itself in the index-bias update, the UE rates stabilize close to the optimum operating point on the rate region boundary, and the index-biases have small fluctuations around the optimum Lagrange multipliers. We compare our results with a prior two-time-scale algorithm and show improved performance.

Utility Optimal Scheduling with a Slow Time-Scale Index-Bias for Achieving Rate Guarantees in Cellular Networks

TL;DR

The paper tackles rate-guaranteed, utility-maximizing downlink scheduling for 5G network slicing by introducing a three-time-scale algorithm PF-RG-LM that combines fast slot-level scheduling with EWMA throughput tracking and a slow stochastic-approximation update of index-biases, interpreted as Lagrange multipliers. It proves convergence to the unique primal-dual optimum via a pair of coupled o.d.e.s and demonstrates, through extensive simulations, that slower updates to the index-bias yield throughput on the rate-region boundary with smoothly varying multipliers, outperforming a two-time-scale baseline. The approach offers tunable time-scales to balance accuracy of rate guarantees against tracking dynamics, enabling practical control for operators and robust performance across multiple UEs and channel states. Overall, the method provides a principled, scalable path to enforce minimum rates while maintaining high network utility in sliced cellular networks.

Abstract

One of the requirements of network slicing in 5G networks is RAN (radio access network) scheduling with rate guarantees. We study a three-time-scale algorithm for maximum sum utility scheduling, with minimum rate constraints. As usual, the scheduler computes an index for each UE in each slot, and schedules the UE with the maximum index. This is at the fastest, natural time-scale of channel fading. The next time-scale is of the exponentially weighted moving average (EWMA) rate update. The slowest time scale in our algorithm is an "index-bias" update by a stochastic approximation algorithm, with a step-size smaller than the EWMA. The index-biases are related to Lagrange multipliers, and bias the slot indices of the UEs with rate guarantees, promoting their more frequent scheduling. We obtain a pair of coupled ordinary differential equations (o.d.e.) such that the unique stable points of the two o.d.e.s are the primal and dual solutions of the constrained utility optimization problem. The UE rate and index-bias iterations track the asymptotic behaviour of the o.d.e. system for small step-sizes of the two slower time-scale iterations. Simulations show that, by running the index-bias iteration at a slower time-scale than the EWMA iteration and using the EWMA throughput itself in the index-bias update, the UE rates stabilize close to the optimum operating point on the rate region boundary, and the index-biases have small fluctuations around the optimum Lagrange multipliers. We compare our results with a prior two-time-scale algorithm and show improved performance.
Paper Structure (19 sections, 1 theorem, 54 equations, 7 figures)

This paper contains 19 sections, 1 theorem, 54 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Related Literature
  3. Contributions and Outline
  4. System Model, Notation, & Problem Definition
  5. Scheduling with Rate Guarantees
  6. Motivation for the algorithm
  7. Utility optimal scheduling with rate guarantees
  8. Convergence Analysis
  9. Outline of the convergence analysis: a two UE example
  10. Main result on convergence with rate guarantees
  11. Stable rest points and the optimum solution
  12. Simulation Experiments
  13. The PF-RG-TC algorithm
  14. Wireless communication model
  15. Two UEs: Rate regions
  16. ...and 4 more sections

Key Result

Theorem 1

Let $q_a$ be any sequence of integers such that $a q_a \rightarrow \infty$ as $a \rightarrow 0$. For the model considered in this paper, under Conditions cond:compact-channel-state-space, cond:strict-convexity-rate-regions, cond:continuity-arg-max-over-rate-regions, and cond:nu-max-existence, there

Figures (7)

  • Figure 1: Demonstration of the optimality of the o.d.e. limit. Two UEs at distances of $100$ m and $200$ m, system bandwidth $40$ MHz, slot time $1$ ms, gNB transmit power $100$ mW.
  • Figure 2: Left plot: A slot rate region $\mathcal{R}_s$. Right plot: The rate region $\overline{\mathcal{R}}$.
  • Figure 3: PF-RG-TC, PF-RG-LM with $a=b$, and PF-RG-LM: Left plots $a = 0.0005, b = 0.000005$; Right plots $a = 0.00005, b = 0.0000005$. $\text{UE}_1$ rate guarantee $60$ Mbps. The final rate pairs provided by the three algorithms superimposed on $\overline{\mathcal{R}}$.
  • Figure 4: PF-RG-TC, PF-RG-LM with $a=b$, and PF-RG-LM: Left plots $a = 0.0005, b = 0.000005$; Right plots $a = 0.00005, b = 0.0000005$. $\text{UE}_1$ rate guarantee $60$ Mbps. Time series of EWMA throughputs (plots are in color).
  • Figure 5: PF-RG-TC, PF-RG-LM with $a=b$, and PF-RG-LM: Left plots $a = 0.0005, b = 0.000005$; Right plots $a = 0.00005, b = 0.0000005$. $\text{UE}_1$ rate guarantee $60$ Mbps. Time series of index-biases (plots are in color).
  • ...and 2 more figures

Theorems & Definitions (1)

  • Theorem 1