Comparing Markov Chains: Aggregation and Precedence Relations Applied to Sets of States, with Applications to Assemble-to-Order Systems

Authors:
Ana Bušić;Ingrid Vliegen;Alan Scheller-Wolf
Affiliations:
INRIA and École normale supérieure, 75013 Paris, France;Center for Telematics and Information Technology, University of Twente, 7500 AE Enschede, The Netherlands;Tepper School of Business, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
Venue:
Mathematics of Operations Research
Year:
2012

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Abstract

Solving Markov chains is, in general, difficult if the state space of the chain is very large (or infinite) and lacking a simple repeating structure. One alternative to solving such chains is to construct models that are simple to analyze and provide bounds for a reward function of interest. We present a new bounding method for Markov chains inspired by Markov reward theory: Our method constructs bounds by redirecting selected sets of transitions, facilitating an intuitive interpretation of the modifications of the original system. We show that our method is compatible with strong aggregation of Markov chains; thus we can obtain bounds for an initial chain by analyzing a much smaller chain. We illustrate our method by using it to prove monotonicity results and bounds for assemble-to-order systems.