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A note on bias optimality in controlled queueing systems

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Mark E. Lewis  and
Martin L. Puterman
Mark E. Lewis*
Affiliation:
University of Michigan
Martin L. Puterman*
Affiliation:
University of British Columbia
*
∗Postal address: Department of Industrial and Operations Engineering, University of Michigan, 1205 Beal Avenue, Ann Arbor, MI 48109-2117, USA. Email address: melewis@engin.umich.edu
∗∗Postal address: Faculty of Commerce and Business Administration, University of British Columbia, 2053 Main Mall, Vancouver, BC, Canada V6T 1Z2
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Abstract

The use of bias optimality to distinguish among gain optimal policies was recently studied by Haviv and Puterman [1] and extended in Lewis et al. [2]. In [1], upon arrival to an M/M/1 queue, customers offer the gatekeeper a reward R. If accepted, the gatekeeper immediately receives the reward, but is charged a holding cost, c(s), depending on the number of customers in the system. The gatekeeper, whose objective is to ‘maximize’ rewards, must decide whether to admit the customer. If the customer is accepted, the customer joins the queue and awaits service. Haviv and Puterman [1] showed there can be only two Markovian, stationary, deterministic gain optimal policies and that only the policy which uses the larger control limit is bias optimal. This showed the usefulness of bias optimality to distinguish between gain optimal policies. In the same paper, they conjectured that if the gatekeeper receives the reward upon completion of a job instead of upon entry, the bias optimal policy will be the lower control limit. This note confirms that conjecture.

Keywords

Markov processesqueueing theory

MSC classification

Primary: 90C40: Markov and semi-Markov decision processes
Secondary: 60K25: Queueing theory
Type
Short Communications
Information
Journal of Applied Probability , Volume 37 , Issue 1 , March 2000 , pp. 300 - 305
Copyright
Copyright © 2000 by The Applied Probability Trust 

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Footnotes

This work was completed while MEL was a postdoctoral fellow at the University of British Columbia

References

Haviv, M., and Puterman, M. L. (1998). Bias optimality in controlled queueing systems. J. Appl. Prob. 35, 136–150.Google Scholar
Lewis, M. E., Ayhan, H., and Foley, R. D. (1999). Bias optimality in a queue with admission control. Prob. Eng. Inf. Sci. 13, 309–327.Google Scholar
Lippman, S. A. (1975). Applying a new device in the optimization of exponential queueing systems. Operat. Res. 23, 687–712.Google Scholar
Stidham, S. Jr, (1978). Socially and individually optimal control of arrivals to a GI/M/1 queue. Management Sci. 24, 1598–1610.Google Scholar