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Average optimal policies in Markov decision drift processes with applications to a queueing and a replacement model

Published online by Cambridge University Press:  01 July 2016

Arie Hordijk  and
Frank A. Van Der Duyn Schouten
Arie Hordijk*
Affiliation:
University of Leiden
Frank A. Van Der Duyn Schouten*
Affiliation:
Free University, Amsterdam
*
∗Postal address: Department of Mathematics, University of Leiden, Wassenaarseweg 80, Postbus 9512, 2300 RA Leiden, The Netherlands.
∗∗Postal address: Department of Actuarial Sciences and Econometrics, Free University, Postbus 7161, 1007 MC Amsterdam, The Netherlands.
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Abstract

Recently the authors introduced the concept of Markov decision drift processes. A Markov decision drift process can be seen as a straightforward generalization of a Markov decision process with continuous time parameter. In this paper we investigate the existence of stationary average optimal policies for Markov decision drift processes. Using a well-known Abelian theorem we derive sufficient conditions, which guarantee that a ‘limit point' of a sequence of discounted optimal policies with the discounting factor approaching 1 is an average optimal policy. An alternative set of sufficient conditions is obtained for the case in which the discounted optimal policies generate regenerative stochastic processes. The latter set of conditions is easier to verify in several applications. The results of this paper are also applicable to Markov decision processes with discrete or continuous time parameter and to semi-Markov decision processes. In this sense they generalize some well-known results for Markov decision processes with finite or compact action space. Applications to an M/M/1 queueing model and a maintenance replacement model are given. It is shown that under certain conditions on the model parameters the average optimal policy for the M/M/1 queueing model is monotone non-decreasing (as a function of the number of waiting customers) with respect to the service intensity and monotone non-increasing with respect to the arrival intensity. For the maintenance replacement model we prove the average optimality of a bang-bang type policy. Special attention is paid to the computation of the optimal control parameters.

Keywords

MARKOV DECISION DRIFT PROCESSAVERAGE COST CRITERIONREGENERATIVE STOCHASTIC PROCESSLIMIT POINT OF DISCOUNTED OPTIMAL POLICIESM/M/1 QUEUEING MODELMAINTENANCE-REPLACEMENT MODEL
Type
Research Article
Information
Advances in Applied Probability , Volume 15 , Issue 2 , June 1983 , pp. 274 - 303
Copyright
Copyright © Applied Probability Trust 1983 

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