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Extremes of Markov-additive Processes with One-sided Jumps, with Queueing Applications

Author

Listed:
  • A. B. Dieker

    (H. Milton Stewart School of Industrial and Systems Engineering)

  • M. Mandjes

    (University of Amsterdam)

Abstract

Through Laplace transforms, we study the extremes of a continuous-time Markov-additive process with one-sided jumps and a finite-state background Markovian state-space, jointly with the epoch at which the extreme is ‘attained’. For this, we investigate discrete-time Markov-additive processes and use an embedding to relate these to the continuous-time setting. The resulting Laplace transforms are given in terms of two matrices, which can be determined either through solving a nonlinear matrix equation or through a spectral method. Our results on extremes are first applied to determine the steady-state buffer-content distribution of several single-station queueing systems. We show that our framework comprises many models dealt with earlier, but, importantly, it also enables us to derive various new results. At the same time, our setup offers interesting insights into the connections between the approaches developed so far, including matrix-analytic techniques, martingale methods, the rate-conservation approach, and the occupation-measure method. We also study networks of fluid queues, and show how the results on single queues can be used to find the Laplace transform of the steady-state buffer-content vector; it has a matrix quasi-product form. Fluid-driven priority systems also have this property.

Suggested Citation

  • A. B. Dieker & M. Mandjes, 2011. "Extremes of Markov-additive Processes with One-sided Jumps, with Queueing Applications," Methodology and Computing in Applied Probability, Springer, vol. 13(2), pages 221-267, June.
    • Handle: RePEc:spr:metcap:v:13:y:2011:i:2:d:10.1007_s11009-009-9140-8
    DOI: 10.1007/s11009-009-9140-8
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    References listed on IDEAS

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    Cited by:

    1. Berkelmans, Wouter & Cichocka, Agata & Mandjes, Michel, 2020. "The correlation function of a queue with Lévy and Markov additive input," Stochastic Processes and their Applications, Elsevier, vol. 130(3), pages 1713-1734.
    2. Ivanovs, Jevgenijs, 2017. "Splitting and time reversal for Markov additive processes," Stochastic Processes and their Applications, Elsevier, vol. 127(8), pages 2699-2724.
    3. Søren Asmussen & Patrick J. Laub & Hailiang Yang, 2019. "Phase-Type Models in Life Insurance: Fitting and Valuation of Equity-Linked Benefits," Risks, MDPI, vol. 7(1), pages 1-22, February.
    4. Przemysław Klusik & Zbigniew Palmowski, 2014. "A Note on Wiener–Hopf Factorization for Markov Additive Processes," Journal of Theoretical Probability, Springer, vol. 27(1), pages 202-219, March.

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