IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v119y2009i2p428-452.html
   My bibliography  Save this article

The Skorokhod problem in a time-dependent interval

Author

Listed:
  • Burdzy, Krzysztof
  • Kang, Weining
  • Ramanan, Kavita

Abstract

We consider the Skorokhod problem in a time-varying interval. We prove existence and uniqueness of the solution. We also express the solution in terms of an explicit formula. Moving boundaries may generate singularities when they touch. Under the assumption that the first time [tau] when the moving boundaries touch after time zero is strictly positive, we derive two sets of conditions on the moving boundaries. We show that the variation of the local time of the associated reflected Brownian motion on [0,[tau]] is finite under the first set of conditions and infinite under the second set of conditions. We also apply these results to study the semimartingale property of a class of two-dimensional reflected Brownian motions.

Suggested Citation

  • Burdzy, Krzysztof & Kang, Weining & Ramanan, Kavita, 2009. "The Skorokhod problem in a time-dependent interval," Stochastic Processes and their Applications, Elsevier, vol. 119(2), pages 428-452, February.
    • Handle: RePEc:eee:spapps:v:119:y:2009:i:2:p:428-452
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304-4149(08)00046-X
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    As the access to this document is restricted, you may want to

    for a different version of it.

    References listed on IDEAS

    as
    1. Avi Mandelbaum & William A. Massey, 1995. "Strong Approximations for Time-Dependent Queues," Mathematics of Operations Research, INFORMS, vol. 20(1), pages 33-64, February.
    Full references (including those not matched with items on IDEAS)

    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. Onno Boxma & Jamol Pender, 2025. "Overlap times in the G/G/1 queue via Laplace transforms," Queueing Systems: Theory and Applications, Springer, vol. 109(1), pages 1-20, March.
    2. K. Srinivasa Rao & J. Durga Aparajitha, 2019. "On two node tandem queueing model with time dependent service rates," International Journal of System Assurance Engineering and Management, Springer;The Society for Reliability, Engineering Quality and Operations Management (SREQOM),India, and Division of Operation and Maintenance, Lulea University of Technology, Sweden, vol. 10(1), pages 19-34, February.
    3. Allan, Andrew L. & Liu, Chong & Prömel, David J., 2021. "Càdlàg rough differential equations with reflecting barriers," Stochastic Processes and their Applications, Elsevier, vol. 142(C), pages 79-104.
    4. Avi Mandelbaum & Kavita Ramanan, 2010. "Directional Derivatives of Oblique Reflection Maps," Mathematics of Operations Research, INFORMS, vol. 35(3), pages 527-558, August.
    5. Yongjiang Guo & Yunan Liu & Renhu Pei, 2018. "Functional law of the iterated logarithm for multi-server queues with batch arrivals and customer feedback," Annals of Operations Research, Springer, vol. 264(1), pages 157-191, May.
    6. Defraeye, Mieke & Van Nieuwenhuyse, Inneke, 2016. "Staffing and scheduling under nonstationary demand for service: A literature review," Omega, Elsevier, vol. 58(C), pages 4-25.
    7. Jamol Pender, 2017. "Sampling the Functional Kolmogorov Forward Equations for Nonstationary Queueing Networks," INFORMS Journal on Computing, INFORMS, vol. 29(1), pages 1-17, February.
    8. Jabari, Saif Eddin & Liu, Henry X., 2013. "A stochastic model of traffic flow: Gaussian approximation and estimation," Transportation Research Part B: Methodological, Elsevier, vol. 47(C), pages 15-41.
    9. Ad Ridder, 2022. "Rare-event analysis and simulation of queues with time-varying rates," Queueing Systems: Theory and Applications, Springer, vol. 100(3), pages 545-547, April.
    10. Schwarz, Justus Arne & Selinka, Gregor & Stolletz, Raik, 2016. "Performance analysis of time-dependent queueing systems: Survey and classification," Omega, Elsevier, vol. 63(C), pages 170-189.
    11. Avishai Mandelbaum & Petar Momčilović, 2008. "Queues with Many Servers: The Virtual Waiting-Time Process in the QED Regime," Mathematics of Operations Research, INFORMS, vol. 33(3), pages 561-586, August.
    12. Yongjiang Guo & Xiyang Hou & Yunan Liu, 2021. "A functional law of the iterated logarithm for multi-class queues with batch arrivals," Annals of Operations Research, Springer, vol. 300(1), pages 51-77, May.
    13. Noa Zychlinski, 2023. "Applications of fluid models in service operations management," Queueing Systems: Theory and Applications, Springer, vol. 103(1), pages 161-185, February.
    14. Wanyang Dai, 2024. "Stochastic Differential Games and a Unified Forward–Backward Coupled Stochastic Partial Differential Equation with Lévy Jumps," Mathematics, MDPI, vol. 12(18), pages 1-46, September.
    15. Elvin Coban & Aliza Heching & Alan Scheller‐Wolf, 2019. "Service Center Staffing with Cross‐Trained Agents and Heterogeneous Customers," Production and Operations Management, Production and Operations Management Society, vol. 28(4), pages 788-809, April.
    16. Tan, Xiaoqian & Knessl, Charles & Yang, Yongzhi (Peter), 2013. "On finite capacity queues with time dependent arrival rates," Stochastic Processes and their Applications, Elsevier, vol. 123(6), pages 2175-2227.
    17. Kerry Fendick & Ward Whitt, 2022. "Heavy traffic limits for queues with non-stationary path-dependent arrival processes," Queueing Systems: Theory and Applications, Springer, vol. 101(1), pages 113-135, June.
    18. Gianmarco Bet & Remco van der Hofstad & Johan S. H. van Leeuwaarden, 2019. "Heavy-Traffic Analysis Through Uniform Acceleration of Queues with Diminishing Populations," Mathematics of Operations Research, INFORMS, vol. 44(3), pages 821-864, August.
    19. Ward Whitt, 2001. "The Reflection Map with Discontinuities," Mathematics of Operations Research, INFORMS, vol. 26(3), pages 447-484, August.
    20. Addabbo, Tommaso & Kocarev, Ljupco, 2009. "Periodic dynamics in queuing networks," Chaos, Solitons & Fractals, Elsevier, vol. 41(4), pages 2178-2192.

    More about this item

    Keywords

    ;

    Statistics

    Access and download statistics

    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:eee:spapps:v:119:y:2009:i:2:p:428-452. 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: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    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.