On the Gittins index in the M/G/1 queue

Authors:
Samuli Aalto;Urtzi Ayesta;Rhonda Righter
Affiliations:
TKK Helsinki University of Technology, Espoo, Finland;LAAS-CNRS, Toulouse, France and Ikerbasque, BCAM--Basque Center for Applied Mathematics, Derio, Spain;University of California at Berkeley, Berkeley, USA
Venue:
Queueing Systems: Theory and Applications
Year:
2009

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Abstract

For an M/G/1 queue with the objective of minimizing the mean number of jobs in the system, the Gittins index rule is known to be optimal among the set of non-anticipating policies. We develop properties of the Gittins index. For a single-class queue it is known that when the service time distribution is of type Decreasing Hazard Rate (New Better than Used in Expectation), the Foreground---Background (First-Come-First-Served) discipline is optimal. By utilizing the Gittins index approach, we show that in fact, Foreground---Background and First-Come-First-Served are optimal if and only if the service time distribution is of type Decreasing Hazard Rate and New Better than Used in Expectation, respectively. For the multi-class case, where jobs of different classes have different service distributions, we obtain new results that characterize the optimal policy under various assumptions on the service time distributions. We also investigate distributions whose hazard rate and mean residual lifetime are not monotonic.