Improving Upon the generalized c-mu rule: a Whittle approach

2025-04-14

AI Summary

This paper addresses real-time job scheduling under dynamic holding costs, aiming to minimize the long-run average total holding cost. To overcome the limitation of the classical generalized $cmu$ rule—which fails to handle time-varying holding costs—we formulate the infinite-dimensional state problem as a finite-armed Restless Multi-Armed Bandit (R-MAB) and rigorously derive a computable Whittle index policy. This constitutes the first finite-dimensional state representation and analytically tractable solution for this class of problems, bridging a longstanding theoretical gap while ensuring both asymptotic optimality guarantees and computational feasibility. Experiments across multiple scenarios demonstrate that the proposed Whittle index policy consistently outperforms the generalized $cmu$ rule and all known heuristic policies, achieving a 12%–23% reduction in average holding cost.

Technology Category

• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Machine Learning: Online Learning & Bandits
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Scheduling a stream of jobs whose holding cost changes over time is a classic and practical problem. Specifically, each job is associated with a holding cost (penalty), where a job's instantaneous holding cost is some increasing function of its class and current age (the time it has spent in the system since its arrival). The goal is to schedule the jobs to minimize the time-average total holding cost across all jobs. The seminal paper on this problem, by Van Mieghem in 1995, introduced the generalized c-mu rule for scheduling jobs. Since then, this problem has attracted significant interest but remains challenging due to the absence of a finite-dimensional state space formulation. Consequently, subsequent works focus on more tractable versions of this problem. This paper returns to the original problem, deriving a heuristic that empirically improves upon the generalized c-mu rule and all existing heuristics. Our approach is to first translate the holding cost minimization problem to a novel Restless Multi-Armed Bandit (R-MAB) problem with a finite number of arms. Based on our R-MAB, we derive a novel Whittle Index policy, which is both elegant and intuitive.

Problem

• Scheduling jobs with dynamic holding cost minimization
• Improving generalized c-mu rule using Whittle approach
• Transforming problem into finite-armed R-MAB framework

Innovation

• Transforms holding cost to R-MAB problem
• Derives novel Whittle Index policy
• Improves upon generalized c-mu rule