LookAhead: the optimal non-decreasing index policy for a time-varying holding cost problem

2026-01-13

Queueing systems

AI Summary

This work addresses the problem of minimizing time-varying holding costs (TVHC) in a two-class M/M/1 queueing system, where one class incurs a non-decreasing per-unit holding cost as a function of waiting time while the other exhibits a constant cost. The paper proposes the LookAhead scheduling policy, which introduces a “lookahead horizon” \( X \) to evaluate the expected holding cost \( X \) time units into the future rather than the instantaneous cost at decision epochs. This approach yields an optimal non-decreasing index rule. Notably, the study provides the first rigorous characterization of a globally optimal policy for the TVHC problem outside asymptotic regimes, overcoming prior limitations confined to asymptotic analysis. Leveraging Markov decision processes and dynamic optimization techniques, the authors derive a closed-form expression for the optimal index, achieving global optimality in the time-averaged total holding cost.

Technology Category

• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Search and Optimization: Learning to Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Web Mining and Content Analysis: Models for Web evolution
• Security and Privacy: Large-scale security measurements

Abstract

In practice, the cost of delaying a job can grow as the job waits. Such behavior is modeled by the time-varying holding cost (TVHC) problem, where each job’s instantaneous holding cost increases with its current age (a job’s age is the time since it arrived). The goal of the TVHC problem is to find a scheduling policy that minimizes the time-average total holding cost across all jobs. However, no optimality results are known for the TVHC problem outside of the asymptotic regime. In this paper, we study a simple yet still challenging special case: A two-class M/M/1 queue in which class 1 jobs incur a non-decreasing, time-varying holding cost and class 2 jobs incur a constant holding cost. Our main contribution is deriving the first optimal (non-decreasing) index policy for this special case of the TVHC problem. Our optimal policy, called LookAhead, stems from the following idea: Rather than considering each job’s current holding cost when making scheduling decisions, we should look at their cost some X time into the future, where this X is intuitively called the “lookahead amount." This paper derives that optimal lookahead amount.

Problem

• Time-Varying Holding Cost
• Optimal Scheduling
• Non-decreasing Index Policy
• M/M/1 Queue
• Holding Cost

Innovation

• LookAhead policy
• time-varying holding cost
• optimal index policy
• M/M/1 queue
• scheduling