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A Numerical Approach for Evaluating the Time-Dependent Distribution of a Quasi Birth-Death Process

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  • Michel Mandjes

    (University of Amsterdam
    Eurandom, Eindhoven University of Technology
    University of Amsterdam)

  • Birgit Sollie

    (Vrije Universiteit Amsterdam)

Abstract

This paper considers a continuous-time quasi birth-death (qbd) process, which informally can be seen as a birth-death process of which the parameters are modulated by an external continuous-time Markov chain. The aim is to numerically approximate the time-dependent distribution of the resulting bivariate Markov process in an accurate and efficient way. An approach based on the Erlangization principle is proposed and formally justified. Its performance is investigated and compared with two existing approaches: one based on numerical evaluation of the matrix exponential underlying the qbd process, and one based on the uniformization technique. It is shown that in many settings the approach based on Erlangization is faster than the other approaches, while still being highly accurate. In the last part of the paper, we demonstrate the use of the developed technique in the context of the evaluation of the likelihood pertaining to a time series, which can then be optimized over its parameters to obtain the maximum likelihood estimator. More specifically, through a series of examples with simulated and real-life data, we show how it can be deployed in model selection problems that involve the choice between a qbd and its non-modulated counterpart.

Suggested Citation

  • Michel Mandjes & Birgit Sollie, 2022. "A Numerical Approach for Evaluating the Time-Dependent Distribution of a Quasi Birth-Death Process," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 1693-1715, September.
    • Handle: RePEc:spr:metcap:v:24:y:2022:i:3:d:10.1007_s11009-021-09882-6
    DOI: 10.1007/s11009-021-09882-6
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    References listed on IDEAS

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    1. Benjamin Melamed & Micha Yadin, 1984. "Randomization Procedures in the Computation of Cumulative-Time Distributions over Discrete State Markov Processes," Operations Research, INFORMS, vol. 32(4), pages 926-944, August.
    2. Jason Xu & Peter Guttorp & Midori Kato-Maeda & Vladimir N. Minin, 2015. "Likelihood-based inference for discretely observed birth–death-shift processes, with applications to evolution of mobile genetic elements," Biometrics, The International Biometric Society, vol. 71(4), pages 1009-1021, December.
    3. Blom, Joke & De Turck, Koen & Mandjes, Michel, 2017. "Refined large deviations asymptotics for Markov-modulated infinite-server systems," European Journal of Operational Research, Elsevier, vol. 259(3), pages 1036-1044.
    4. Donald Gross & Douglas R. Miller, 1984. "The Randomization Technique as a Modeling Tool and Solution Procedure for Transient Markov Processes," Operations Research, INFORMS, vol. 32(2), pages 343-361, April.
    5. D. Anderson & J. Blom & M. Mandjes & H. Thorsdottir & K. Turck, 2016. "A Functional Central Limit Theorem for a Markov-Modulated Infinite-Server Queue," Methodology and Computing in Applied Probability, Springer, vol. 18(1), pages 153-168, March.
    6. V. Ramaswami & Douglas Woolford & David Stanford, 2008. "The erlangization method for Markovian fluid flows," Annals of Operations Research, Springer, vol. 160(1), pages 215-225, April.
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