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

Author

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  • Sandjai Bhulai
  • Flora M. Spieksma

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. Copyright Springer-Verlag 2003

Suggested Citation

  • Sandjai Bhulai & Flora M. Spieksma, 2003. "On the uniqueness of solutions to the Poisson equations for average cost Markov chains with unbounded cost functions," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 58(2), pages 221-236, November.
    • Handle: RePEc:spr:mathme:v:58:y:2003:i:2:p:221-236
    DOI: 10.1007/s001860300292
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    Cited by:

    1. Jiang, Shuxia & Liu, Yuanyuan & Yao, Shuai, 2014. "Poisson’s equation for discrete-time single-birth processes," Statistics & Probability Letters, Elsevier, vol. 85(C), pages 78-83.
    2. Olivier Bilenne, 2021. "Dispatching to parallel servers," Queueing Systems: Theory and Applications, Springer, vol. 99(3), pages 199-230, December.

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