On the uniqueness of solutions to the Poisson equations for average cost Markov chains with unbounded cost functions |
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Authors: | Email author" target="_blank">Sandjai?BhulaiEmail author Flora M?Spieksma |
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Institution: | (1) Faculty of Sciences, Vrije universiteit Amsterdam, De Boelelaan 1081a, 1081, HV, Amsterdam, The Netherlands;(2) Mathematical Institute, University of Leiden, P.O. Box 9512, 2300, RA, Leiden, The Netherlands |
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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. |
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Keywords: | Geometric ergodicity Poisson equations Lyapunov functions Multi-server queue Priority queue |
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