On the uniqueness of solutions to the Poisson equations for average cost Markov chains with unbounded cost functions

S. Bhulai, F.M. Spieksma

Research output: Contribution to JournalArticleAcademicpeer-review

Abstract

We consider the Poisson equations for denumerable Markov chains with unbounded cost functions. Solutions to the Poisson equations exist in the Banach space of bounded real-valued functions with respect to a weighted supremum norm such that the Markov chain is geometrically ergodic. Under minor additional assumptions the solution is also unique. We give a novel probabilistic proof of this fact using relations between ergodicity and recurrence. The expressions involved in the Poisson equations have many solutions in general. However, the solution that has a finite norm with respect to the weighted supremum norm is the unique solution to the Poisson equations. We illustrate how to determine this solution by considering three queueing examples: a multi-server queue, two independent single server queues, and a priority queue with dependence between the queues. © Springer-Verlag 2003.
Original languageEnglish
Pages (from-to)221-236
JournalMathematical Methods of Operations Research
Volume58
DOIs
Publication statusPublished - 2003

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