Journal of Applied Probability

Open bandit processes with uncountable states and time-backward effects

Xianyi Wu and Xian Zhou

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Abstract

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.

Article information

Source
J. Appl. Probab., Volume 50, Number 2 (2013), 388-402.

Dates
First available in Project Euclid: 19 June 2013

Permanent link to this document
https://projecteuclid.org/euclid.jap/1371648948

Digital Object Identifier
doi:10.1239/jap/1371648948

Mathematical Reviews number (MathSciNet)
MR3102487

Zentralblatt MATH identifier
1266.90112

Subjects
Primary: 90B36: Scheduling theory, stochastic [See also 68M20] 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 90C40: Markov and semi-Markov decision processes

Keywords
Open bandit process generalized bandit process Gittins index priority scheduling

Citation

Wu, Xianyi; Zhou, Xian. Open bandit processes with uncountable states and time-backward effects. J. Appl. Probab. 50 (2013), no. 2, 388--402. doi:10.1239/jap/1371648948. https://projecteuclid.org/euclid.jap/1371648948


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