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Open Bandit Processes with Uncountable States and Time-Backward Effects

Part of: Stochastic processes Operations research and management science

Published online by Cambridge University Press:  30 January 2018

Xianyi Wu  and
Xian Zhou
Xianyi Wu*
Affiliation:
East China Normal University and Macquarie University
Xian Zhou*
Affiliation:
Macquarie University
*
∗ Postal address: Department of Statistics and Actuarial Science, East China Normal University, Shanghai, China.
∗∗ Postal address: Department of Applied Finance and Actuarial Studies, Macquarie University, Sydney, Australia. Email address: [email protected]
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Abstract

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Bandit processes and the Gittins index have provided powerful and elegant theory and tools for the optimization of allocating limited resources to competitive demands. In this paper we extend the Gittins theory to more general branching bandit processes, also referred to as open bandit processes, that allow uncountable states and backward times. We establish the optimality of the Gittins index policy with uncountably many states, which is useful in such problems as dynamic scheduling with continuous random processing times. We also allow negative time durations for discounting a reward to account for the present value of the reward that was received before the present time, which we refer to as time-backward effects. This could model the situation of offering bonus rewards for completing jobs above expectation. Moreover, we discover that a common belief on the optimality of the Gittins index in the generalized bandit problem is not always true without additional conditions, and provide a counterexample. We further apply our theory of open bandit processes with time-backward effects to prove the optimality of the Gittins index in the generalized bandit problem under a sufficient condition.

Keywords

Open bandit processgeneralized bandit processGittins indexpriority scheduling

MSC classification

Primary: 90B36: Scheduling theory, stochastic 60G40: Stopping times; optimal stopping problems; gambling theory 90C40: Markov and semi-Markov decision processes
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 2 , June 2013 , pp. 388 - 402
Copyright
© Applied Probability Trust 

Footnotes

This research was partially supported by the Natural Science Foundation of China, under grant number 71071056, and the Australian Research Council Discovery Project, under grant number DP1094153.

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