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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Bhulai, S., Spieksma, F. On the uniqueness of solutions to the Poisson equations for average cost Markov chains with unbounded cost functions. Math Meth Oper Res 58, 221–236 (2003). https://doi.org/10.1007/s001860300292
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DOI: https://doi.org/10.1007/s001860300292