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Some indexable families of restless bandit problems

Part of: Hamilton-Jacobi theories, including dynamic programming Numerical methods in calculus of variations and optimal control

Published online by Cambridge University Press:  01 July 2016

K. D. Glazebrook ,
D. Ruiz-Hernandez  and
C. Kirkbride
K. D. Glazebrook*
Affiliation:
Lancaster University
D. Ruiz-Hernandez*
Affiliation:
Universitat Pompeu Fabra
C. Kirkbride*
Affiliation:
Lancaster University
*
∗ Postal address: Department of Mathematics and Statistics, Lancaster University, Lancaster, LA1 4YF, UK. Email address: k.glazebrook@lancaster.ac.uk
∗∗ Postal address: Department of Economics and Business, Universitat Pompeu Fabra, E-08005 Barcelona, Spain.
∗∗∗ Postal address: Department of Management Science, Lancaster University, Lancaster, LA1 4YX, UK.
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Abstract

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In 1988 Whittle introduced an important but intractable class of restless bandit problems which generalise the multiarmed bandit problems of Gittins by allowing state evolution for passive projects. Whittle's account deployed a Lagrangian relaxation of the optimisation problem to develop an index heuristic. Despite a developing body of evidence (both theoretical and empirical) which underscores the strong performance of Whittle's index policy, a continuing challenge to implementation is the need to establish that the competing projects all pass an indexability test. In this paper we employ Gittins' index theory to establish the indexability of (inter alia) general families of restless bandits which arise in problems of machine maintenance and stochastic scheduling problems with switching penalties. We also give formulae for the resulting Whittle indices. Numerical investigations testify to the outstandingly strong performance of the index heuristics concerned.

Keywords

Bandit problemdynamic programmingGittins indexmachine maintenancerestless banditstochastic schedulingswitching cost

MSC classification

Primary: 90C40: Markov and semi-Markov decision processes
Secondary: 49L20: Dynamic programming method 90C39: Dynamic programming 49M20: Methods of relaxation type
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 38 , Issue 3 , September 2006 , pp. 643 - 672
Copyright
Copyright © Applied Probability Trust 2006 

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