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Convergence of controlled models and finite-state approximation for discounted continuous-time Markov decision processes with constraints

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  • Guo, Xianping
  • Zhang, Wenzhao

Abstract

In this paper we consider the convergence of a sequence {Mn} of the models of discounted continuous-time constrained Markov decision processes (MDP) to the “limit” one, denoted by M∞. For the models with denumerable states and unbounded transition rates, under reasonably mild conditions we prove that the (constrained) optimal policies and the optimal values of {Mn} converge to those of M∞, respectively, using a technique of occupation measures. As an application of the convergence result developed here, we show that an optimal policy and the optimal value for countable-state continuous-time MDP can be approximated by those of finite-state continuous-time MDP. Finally, we further illustrate such finite-state approximation by solving numerically a controlled birth-and-death system and also give the corresponding error bound of the approximation.

Suggested Citation

  • Guo, Xianping & Zhang, Wenzhao, 2014. "Convergence of controlled models and finite-state approximation for discounted continuous-time Markov decision processes with constraints," European Journal of Operational Research, Elsevier, vol. 238(2), pages 486-496.
    • Handle: RePEc:eee:ejores:v:238:y:2014:i:2:p:486-496
    DOI: 10.1016/j.ejor.2014.03.037
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    References listed on IDEAS

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    1. Xianping Guo & Alexei Piunovskiy, 2011. "Discounted Continuous-Time Markov Decision Processes with Constraints: Unbounded Transition and Loss Rates," Mathematics of Operations Research, INFORMS, vol. 36(1), pages 105-132, February.
    2. Jorge Alvarez-Mena & Onésimo Hernández-Lerma, 2002. "Convergence of the optimal values of constrained Markov control processes," The Annals of Regional Science, Springer;Western Regional Science Association, vol. 55(3), pages 461-484, June.
    3. Cervellera, C. & Macciò, D., 2011. "A comparison of global and semi-local approximation in T-stage stochastic optimization," European Journal of Operational Research, Elsevier, vol. 208(2), pages 109-118, January.
    4. Jorge Alvarez-Mena & Onésimo Hernández-Lerma, 2006. "Existence of nash equilibria for constrained stochastic games," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 63(2), pages 261-285, May.
    5. Eugene A. Feinberg, 2000. "Constrained Discounted Markov Decision Processes and Hamiltonian Cycles," Mathematics of Operations Research, INFORMS, vol. 25(1), pages 130-140, February.
    6. Eugene A. Feinberg, 2004. "Continuous Time Discounted Jump Markov Decision Processes: A Discrete-Event Approach," Mathematics of Operations Research, INFORMS, vol. 29(3), pages 492-524, August.
    7. Jorge Alvarez-Mena & Onésimo Hernández-Lerma, 2002. "Convergence of the optimal values of constrained Markov control processes," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 55(3), pages 461-484, June.
    8. Alexey Piunovskiy & Yi Zhang, 2012. "The Transformation Method for Continuous-Time Markov Decision Processes," Journal of Optimization Theory and Applications, Springer, vol. 154(2), pages 691-712, August.
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    Cited by:

    1. Tomás Prieto-Rumeau & José Lorenzo, 2015. "Approximation of zero-sum continuous-time Markov games under the discounted payoff criterion," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(3), pages 799-836, October.
    2. Qingda Wei, 2016. "Continuous-time Markov decision processes with risk-sensitive finite-horizon cost criterion," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 84(3), pages 461-487, December.
    3. Ping Cao & Jingui Xie, 2016. "Optimal control of a multiclass queueing system when customers can change types," Queueing Systems: Theory and Applications, Springer, vol. 82(3), pages 285-313, April.
    4. Qingda Wei, 2017. "Finite approximation for finite-horizon continuous-time Markov decision processes," 4OR, Springer, vol. 15(1), pages 67-84, March.

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