IDEAS home Printed from https://ideas.repec.org/a/spr/metcap/v13y2011i2d10.1007_s11009-009-9140-8.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11009-009-9140-8
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s11009-009-9140-8?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Cohen, J. W., 1974. "Superimposed renewal processes and storage with gradual input," Stochastic Processes and their Applications, Elsevier, vol. 2(1), pages 31-57, January.
    2. ARJAS, Elja & SPEED, T.P., 1973. "Symmetric Wiener-Hopf factorisations in Markov additive processes," LIDAM Reprints CORE 134, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. Yiqiang Q. Zhao & Wei Li & W. John Braun, 2003. "Censoring, Factorizations, and Spectral Analysis for Transition Matrices with Block-Repeating Entries," Methodology and Computing in Applied Probability, Springer, vol. 5(1), pages 35-58, March.
    4. Asmussen, Søren & Avram, Florin & Pistorius, Martijn R., 2004. "Russian and American put options under exponential phase-type Lévy models," Stochastic Processes and their Applications, Elsevier, vol. 109(1), pages 79-111, January.
    5. Søren Asmussen, 1994. "Busy period analysis, rare events and transient behavior in fluid flow models," International Journal of Stochastic Analysis, Hindawi, vol. 7, pages 1-31, January.
    6. K. Dębicki & A. B. Dieker & T. Rolski, 2007. "Quasi-Product Forms for Lévy-Driven Fluid Networks," Mathematics of Operations Research, INFORMS, vol. 32(3), pages 629-647, August.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Ivanovs, Jevgenijs, 2017. "Splitting and time reversal for Markov additive processes," Stochastic Processes and their Applications, Elsevier, vol. 127(8), pages 2699-2724.
    2. 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.
    3. 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.
    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Çağlar, M. & Kyprianou, A. & Vardar-Acar, C., 2022. "An optimal stopping problem for spectrally negative Markov additive processes," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 1109-1138.
    2. Hongzhong Zhang, 2018. "Stochastic Drawdowns," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 10078, August.
    3. Doney, Ron & Maller, Ross & Savov, Mladen, 2009. "Renewal theorems and stability for the reflected process," Stochastic Processes and their Applications, Elsevier, vol. 119(4), pages 1270-1297, April.
    4. Kimura, Toshikazu, 2008. "Valuing finite-lived Russian options," European Journal of Operational Research, Elsevier, vol. 189(2), pages 363-374, September.
    5. Kazutoshi Yamazaki, 2017. "Inventory Control for Spectrally Positive Lévy Demand Processes," Mathematics of Operations Research, INFORMS, vol. 42(1), pages 212-237, January.
    6. Zbigniew Palmowski & Jos'e Luis P'erez & Kazutoshi Yamazaki, 2020. "Double continuation regions for American options under Poisson exercise opportunities," Papers 2004.03330, arXiv.org.
    7. Chi, Yichun, 2010. "Analysis of the expected discounted penalty function for a general jump-diffusion risk model and applications in finance," Insurance: Mathematics and Economics, Elsevier, vol. 46(2), pages 385-396, April.
    8. Michael C. Fu & Bingqing Li & Guozhen Li & Rongwen Wu, 2017. "Option Pricing for a Jump-Diffusion Model with General Discrete Jump-Size Distributions," Management Science, INFORMS, vol. 63(11), pages 3961-3977, November.
    9. Irmina Czarna & Zbigniew Palmowski, 2014. "Dividend Problem with Parisian Delay for a Spectrally Negative Lévy Risk Process," Journal of Optimization Theory and Applications, Springer, vol. 161(1), pages 239-256, April.
    10. Bakshi, Gurdip & Panayotov, George, 2010. "First-passage probability, jump models, and intra-horizon risk," Journal of Financial Economics, Elsevier, vol. 95(1), pages 20-40, January.
    11. Kim, Jerim & Kim, Jeongsim & Joo Yoo, Hyun & Kim, Bara, 2015. "Pricing external barrier options in a regime-switching model," Journal of Economic Dynamics and Control, Elsevier, vol. 53(C), pages 123-143.
    12. Benjamin Avanzi & Jos'e-Luis P'erez & Bernard Wong & Kazutoshi Yamazaki, 2016. "On optimal joint reflective and refractive dividend strategies in spectrally positive L\'evy models," Papers 1607.01902, arXiv.org, revised Nov 2016.
    13. Ning Cai & Wei Zhang, 2020. "Regime Classification and Stock Loan Valuation," Operations Research, INFORMS, vol. 68(4), pages 965-983, July.
    14. 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.
    15. Bayraktar, Erhan & Kyprianou, Andreas E. & Yamazaki, Kazutoshi, 2014. "Optimal dividends in the dual model under transaction costs," Insurance: Mathematics and Economics, Elsevier, vol. 54(C), pages 133-143.
    16. Kazutoshi Yamazaki, 2017. "Phase-type Approximation of the Gerber-Shiu Function," Papers 1701.02798, arXiv.org.
    17. Christian Paroissin & Landy Rabehasaina, 2015. "First and Last Passage Times of Spectrally Positive Lévy Processes with Application to Reliability," Methodology and Computing in Applied Probability, Springer, vol. 17(2), pages 351-372, June.
    18. He, Yue & Kawai, Reiichiro & Shimizu, Yasutaka & Yamazaki, Kazutoshi, 2023. "The Gerber-Shiu discounted penalty function: A review from practical perspectives," Insurance: Mathematics and Economics, Elsevier, vol. 109(C), pages 1-28.
    19. Marzia De Donno & Zbigniew Palmowski & Joanna Tumilewicz, 2020. "Double continuation regions for American and Swing options with negative discount rate in Lévy models," Mathematical Finance, Wiley Blackwell, vol. 30(1), pages 196-227, January.
    20. Olivier Le Courtois & François Quittard-Pinon & Xiaoshan Su, 2020. "Pricing and hedging defaultable participating contracts with regime switching and jump risk," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 43(1), pages 303-339, June.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:metcap:v:13:y:2011:i:2:d:10.1007_s11009-009-9140-8. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.