A Marginal Productivity Index Rule for Scheduling Multiclass Queues with Setups

Abstract

This paper addresses the problem of designing a tractable scheduling rule for a multiclass M /G /1 queue incurring class-dependent linear holding costs and setup costs, as well as class-dependent generally distributed setup times, which performs well relative to the discounted or average cost objective. We introduce a new dynamic scheduling rule based on priority indices which emerges from deployment of a systematic methodology for obtaining marginal productivity index policies in the framework of restless bandit models, introduced by Whittle (1988) and developed by the author over the last decade. For each class, two indices are defined: an active and a passive index, depending on whether the class is or is not set up, which are functions of the class state (number in system). The index rule prescribes to engage at each time a class of highest index: it thus dynamically indicates both when to leave the class being currently served, and which class to serve next. The paper (i) formulates the problem as a semi-Markov multiarmed restless bandit problem; (ii) introduces the required extensions to previous indexation theory; and (iii) gives closed index formulae for the average criterion.