IDEAS home Printed from https://ideas.repec.org/a/spr/annopr/v300y2021i1d10.1007_s10479-020-03864-6.html
   My bibliography  Save this article

A functional law of the iterated logarithm for multi-class queues with batch arrivals

Author

Listed:
  • Yongjiang Guo

    (Beijing University of Posts and Telecommunications)

  • Xiyang Hou

    (Huazhong University of Science and Technology)

  • Yunan Liu

    (North Carolina State University)

Abstract

A functional law of the iterated logarithm (LIL) and its corresponding LIL are established for a multiclass single-server queue with first come first served (FCFS) service discipline. The functional LIL and its LIL quantify the magnitude of asymptotic stochastic fluctuations of the stochastic processes compensated by their deterministic fluid limits. The functional LIL and LIL are established in three cases: underloaded, critically loaded and overloaded, for performance measures including the total workload, idle time, queue length, workload, busy time, departure and sojourn time processes. The proofs of the functional LIL and LIL are based on a strong approximation approach, which approximates discrete performance processes with reflected Brownian motions. Numerical examples are considered to provide insights on these limit results.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:annopr:v:300:y:2021:i:1:d:10.1007_s10479-020-03864-6
    DOI: 10.1007/s10479-020-03864-6
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10479-020-03864-6
    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/s10479-020-03864-6?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. Hong Chen & Hanqin Zhang, 2000. "Diffusion Approximations for Some Multiclass Queueing Networks with FIFO Service Disciplines," Mathematics of Operations Research, INFORMS, vol. 25(4), pages 679-707, November.
    2. Hong Chen & Hanqin Zhang, 1997. "Stability of Multiclass Queueing Networks Under FIFO Service Discipline," Mathematics of Operations Research, INFORMS, vol. 22(3), pages 691-725, August.
    3. Sofian De Clercq & Koenraad Laevens & Bart Steyaert & Herwig Bruneel, 2013. "A multi-class discrete-time queueing system under the FCFS service discipline," Annals of Operations Research, Springer, vol. 202(1), pages 59-73, January.
    4. Tsai, Tsung-Hsi, 2000. "Empirical law of the iterated logarithm for Markov chains with a countable state space," Stochastic Processes and their Applications, Elsevier, vol. 89(2), pages 175-191, October.
    5. Peter W. Glynn & Ward Whitt, 1988. "An LIL Version of L = (lambda) W," Mathematics of Operations Research, INFORMS, vol. 13(4), pages 693-710, November.
    6. Avi Mandelbaum & William A. Massey, 1995. "Strong Approximations for Time-Dependent Queues," Mathematics of Operations Research, INFORMS, vol. 20(1), pages 33-64, February.
    7. Sakalauskas, L. L. & Minkevicius, S., 2000. "On the law of the iterated logarithm in open queueing networks," European Journal of Operational Research, Elsevier, vol. 120(3), pages 632-640, February.
    8. 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.
    9. Rishi Talreja & Ward Whitt, 2008. "Fluid Models for Overloaded Multiclass Many-Server Queueing Systems with First-Come, First-Served Routing," Management Science, INFORMS, vol. 54(8), pages 1513-1527, August.
    10. Caramellino, Lucia, 1998. "Strassen's law of the iterated logarithm for diffusion processes for small time," Stochastic Processes and their Applications, Elsevier, vol. 74(1), pages 1-19, May.
    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. 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.
    2. Noa Zychlinski, 2023. "Applications of fluid models in service operations management," Queueing Systems: Theory and Applications, Springer, vol. 103(1), pages 161-185, February.
    3. Chang Cao & J. G. Dai & Xiangyu Zhang, 2022. "State space collapse for multi-class queueing networks under SBP service policies," Queueing Systems: Theory and Applications, Springer, vol. 102(1), pages 87-122, October.
    4. Francisco Castro & Hamid Nazerzadeh & Chiwei Yan, 2020. "Matching queues with reneging: a product form solution," Queueing Systems: Theory and Applications, Springer, vol. 96(3), pages 359-385, December.
    5. Carfagnini, Marco & Földes, Juraj & Herzog, David P., 2022. "A functional law of the iterated logarithm for weakly hypoelliptic diffusions at time zero," Stochastic Processes and their Applications, Elsevier, vol. 149(C), pages 188-223.
    6. Saulius Minkevičius & Edvinas Greičius, 2019. "Heavy Traffic Limits for the Extreme Waiting Time in Multi-phase Queueing Systems," Methodology and Computing in Applied Probability, Springer, vol. 21(1), pages 109-124, March.
    7. 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.
    8. 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.
    9. Avi Mandelbaum & Kavita Ramanan, 2010. "Directional Derivatives of Oblique Reflection Maps," Mathematics of Operations Research, INFORMS, vol. 35(3), pages 527-558, August.
    10. Ivo Adan & Brett Hathaway & Vidyadhar G. Kulkarni, 2019. "On first-come, first-served queues with two classes of impatient customers," Queueing Systems: Theory and Applications, Springer, vol. 91(1), pages 113-142, February.
    11. Saulius MinkeviÄ ius, 2020. "On the law of the iterated logarithm in hybrid multiphase queueing systems," Operations Research and Decisions, Wroclaw University of Science Technology, Faculty of Management, vol. 30(4), pages 57-64.
    12. Otis B. Jennings & Josh E. Reed, 2012. "An Overloaded Multiclass FIFO Queue with Abandonments," Operations Research, INFORMS, vol. 60(5), pages 1282-1295, October.
    13. Kouki, Chaaben & Arts, Joachim & Babai, M. Zied, 2024. "Performance evaluation of a two-echelon inventory system with network lost sales," European Journal of Operational Research, Elsevier, vol. 314(2), pages 647-664.
    14. Edvinas Greičius & Saulius Minkevičius, 2017. "Diffusion limits for the queue length of jobs in multi-server open queueing networks," Operations Research and Decisions, Wroclaw University of Science and Technology, Faculty of Management, vol. 27(4), pages 71-84.
    15. Ramandeep S. Randhawa & Sunil Kumar, 2009. "Multiserver Loss Systems with Subscribers," Mathematics of Operations Research, INFORMS, vol. 34(1), pages 142-179, February.
    16. 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.
    17. 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.
    18. Hong Chen & Hanqin Zhang, 2000. "Stability of Multiclass Queueing Networks Under Priority Service Disciplines," Operations Research, INFORMS, vol. 48(1), pages 26-37, February.
    19. 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.
    20. 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.

    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:annopr:v:300:y:2021:i:1:d:10.1007_s10479-020-03864-6. 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.