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Some indexable families of restless bandit problems

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

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 

References

Agrawal, R., Hedge, M. and Teneketzis, D. (1988). Asymptotically efficient adaptive allocation rules for the multiarmed bandit problem with switching cost. IEEE Trans. Automatic Control 33, 899–906.Google Scholar
Ansell, P. S., Glazebrook, K. D., Niño-Mora, J. and O'Keeffe, M. (2003). Whittle's index policy for a multi-class queueing system with convex holding costs. Math. Meth. Operat. Res. 57, 21–39.Google Scholar
Asawa, M. and Teneketzis, D. (1996). Multi-armed bandits with switching penalties. IEEE Trans. Automatic Control 41, 328–348.CrossRefGoogle Scholar
Banks, J. S. and Sundaram, R. (1994). Switching costs and the Gittins index. Econometrica 62, 687–694.CrossRefGoogle Scholar
Gittins, J. C. (1979). Bandit processes and dynamic allocation indices (with discussion). J. R. Statist. Soc. B 41, 148–177.Google Scholar
Gittins, J. C. (1989). Multi-Armed Bandit Allocation Indices. John Wiley, Chichester.Google Scholar
Glazebrook, K. D. (1980). On stochastic scheduling with precedence relations and switching costs. J. Appl. Prob. 17, 1016–1024.Google Scholar
Glazebrook, K. D., Mitchell, H. M. and Ansell, P. S. (2005). Index policies for the maintenance of a collection of machines by a set of repairmen. Europ. J. Operat. Res. 165, 267–284.Google Scholar
Glazebrook, K. D., Niño-Mora, J. and Ansell, P. S. (2002). Index policies for a class of discounted restless bandits. Adv. Appl. Prob. 34, 754–774.Google Scholar
Nash, P. (1979). Optimal allocation of resources between research projects. , University of Cambridge.Google Scholar
Niño-Mora, J. (2001). Restless bandits, partial conservation laws and indexability. Adv. Appl. Prob. 33, 76–98.Google Scholar
Niño-Mora, J. (2002). Dynamic allocation indices for restless projects and queueing admission control: a polyhedral approach. Math. Program. 93, 361–413.CrossRefGoogle Scholar
Papadimitriou, C. H. and Tsitsiklis, J. N. (1999). The complexity of optimal queuing network control. Math. Operat. Res. 24, 293–305.CrossRefGoogle Scholar
Puterman, M. L. (1994). Markov Decision Processes: Discrete Stochastic Dynamic Programming. John Wiley, New York.Google Scholar
Reiman, M. I. and Wein, L. M. (1998). Dynamic scheduling of a two-class queue with setups. Operat. Res. 46, 532–547.Google Scholar
Van Oyen, M. P. and Teneketzis, D. (1994). Optimal stochastic scheduling of forest networks with switching penalties. Adv. Appl. Prob. 26, 474–479.Google Scholar
Weber, R. R. and Weiss, G. (1990). On an index policy for restless bandits. J. Appl. Prob. 27, 637–648. (Addendum: Adv. Appl. Prob. 23 (1991), 429-430.)Google Scholar
Whittle, P. (1980). Multi-armed bandits and the Gittins index. J. R. Statist. Soc. B 42, 143–149.Google Scholar
Whittle, P. (1988). Restless bandits: activity allocation in a changing world. In A Celebration of Applied Probability (J. Appl. Prob. Spec. Vol. 25A), ed. Gani, J., Applied Probability Trust, Sheffield, pp. 287–298.Google Scholar
Whittle, P. (1996). Optimal Control: Basics and Beyond. John Wiley, Chichester.Google Scholar