IDEAS home Printed from https://ideas.repec.org/a/spr/queues/v107y2024i1d10.1007_s11134-024-09906-1.html
   My bibliography  Save this article

Exact results for the distribution of the partial busy period for a multi-server queue

Author

Listed:
  • Josef Zuk

    (Defence Science and Technology Group)

  • David Kirszenblat

    (Defence Science and Technology Group)

Abstract

Exact explicit results are derived for the distribution of the partial busy period of the M/M/c multi-server queue for a general number of servers. A rudimentary spectral method leads to a representation that is amenable to efficient numerical computation across the entire ergodic region. An alternative algebraic approach yields a representation as a finite sum of Marcum Q-functions depending on the roots of certain polynomials that are explicitly determined for an arbitrary number of servers. Asymptotic forms are derived in the limit of a large number of servers under two scaling regimes, and also for the large-time limit. Connections are made with previous work. The present work is the first to offer tangible exact results for the distribution when the number of servers is greater than two.

Suggested Citation

  • Josef Zuk & David Kirszenblat, 2024. "Exact results for the distribution of the partial busy period for a multi-server queue," Queueing Systems: Theory and Applications, Springer, vol. 107(1), pages 63-108, June.
    • Handle: RePEc:spr:queues:v:107:y:2024:i:1:d:10.1007_s11134-024-09906-1
    DOI: 10.1007/s11134-024-09906-1
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11134-024-09906-1
    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/s11134-024-09906-1?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. Shlomo Halfin & Ward Whitt, 1981. "Heavy-Traffic Limits for Queues with Many Exponential Servers," Operations Research, INFORMS, vol. 29(3), pages 567-588, June.
    2. Richard H. Davis, 1966. "Waiting-Time Distribution of a Multi-Server, Priority Queuing System," Operations Research, INFORMS, vol. 14(1), pages 133-136, February.
    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. Josef Zuk & David Kirszenblat, 2024. "Analytic approach to the non-pre-emptive Markovian priority queue," Queueing Systems: Theory and Applications, Springer, vol. 107(1), pages 159-198, June.

    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. Constantinos Maglaras & Assaf Zeevi, 2005. "Pricing and Design of Differentiated Services: Approximate Analysis and Structural Insights," Operations Research, INFORMS, vol. 53(2), pages 242-262, April.
    2. Jianfu Wang & Opher Baron & Alan Scheller-Wolf, 2015. "M/M/c Queue with Two Priority Classes," Operations Research, INFORMS, vol. 63(3), pages 733-749, June.
    3. Avishai Mandelbaum & Petar Momčilović, 2017. "Personalized queues: the customer view, via a fluid model of serving least-patient first," Queueing Systems: Theory and Applications, Springer, vol. 87(1), pages 23-53, October.
    4. Amy R. Ward & Mor Armony, 2013. "Blind Fair Routing in Large-Scale Service Systems with Heterogeneous Customers and Servers," Operations Research, INFORMS, vol. 61(1), pages 228-243, February.
    5. Achal Bassamboo & J. Michael Harrison & Assaf Zeevi, 2006. "Design and Control of a Large Call Center: Asymptotic Analysis of an LP-Based Method," Operations Research, INFORMS, vol. 54(3), pages 419-435, June.
    6. Kazuma Abe & Tuan Phung-Duc, 2023. "Diffusion limit of a modified Erlang-B system with sensing time of secondary users," Annals of Operations Research, Springer, vol. 331(2), pages 985-1006, December.
    7. Josh Reed & Yair Shaki, 2015. "A Fair Policy for the G / GI / N Queue with Multiple Server Pools," Mathematics of Operations Research, INFORMS, vol. 40(3), pages 558-595, March.
    8. Niyirora, Jerome & Zhuang, Jun, 2017. "Fluid approximations and control of queues in emergency departments," European Journal of Operational Research, Elsevier, vol. 261(3), pages 1110-1124.
    9. Achal Bassamboo & Assaf Zeevi, 2009. "On a Data-Driven Method for Staffing Large Call Centers," Operations Research, INFORMS, vol. 57(3), pages 714-726, June.
    10. Ari Arapostathis & Hassan Hmedi & Guodong Pang, 2021. "On Uniform Exponential Ergodicity of Markovian Multiclass Many-Server Queues in the Halfin–Whitt Regime," Mathematics of Operations Research, INFORMS, vol. 46(2), pages 772-796, May.
    11. van Leeuwaarden, Johan S.H. & Knessl, Charles, 2011. "Transient behavior of the Halfin-Whitt diffusion," Stochastic Processes and their Applications, Elsevier, vol. 121(7), pages 1524-1545, July.
    12. Ramandeep S. Randhawa & Sunil Kumar, 2008. "Usage Restriction and Subscription Services: Operational Benefits with Rational Users," Manufacturing & Service Operations Management, INFORMS, vol. 10(3), pages 429-447, December.
    13. Martino Banchio & Giacomo Mantegazza, 2022. "Artificial Intelligence and Spontaneous Collusion," Papers 2202.05946, arXiv.org, revised Sep 2023.
    14. Zhang, Ping & Serban, Nicoleta, 2007. "Discovery, visualization and performance analysis of enterprise workflow," Computational Statistics & Data Analysis, Elsevier, vol. 51(5), pages 2670-2687, February.
    15. Hsiao-Hui Lee & Edieal J. Pinker & Robert A. Shumsky, 2012. "Outsourcing a Two-Level Service Process," Management Science, INFORMS, vol. 58(8), pages 1569-1584, August.
    16. O. Garnet & A. Mandelbaum & M. Reiman, 2002. "Designing a Call Center with Impatient Customers," Manufacturing & Service Operations Management, INFORMS, vol. 4(3), pages 208-227, October.
    17. A. J. E. M. Janssen & J. S. H. van Leeuwaarden & Bert Zwart, 2011. "Refining Square-Root Safety Staffing by Expanding Erlang C," Operations Research, INFORMS, vol. 59(6), pages 1512-1522, December.
    18. Cui, Zhenyu & Kirkby, J. Lars & Nguyen, Duy, 2021. "Efficient simulation of generalized SABR and stochastic local volatility models based on Markov chain approximations," European Journal of Operational Research, Elsevier, vol. 290(3), pages 1046-1062.
    19. Mor Armony & Constantinos Maglaras, 2004. "On Customer Contact Centers with a Call-Back Option: Customer Decisions, Routing Rules, and System Design," Operations Research, INFORMS, vol. 52(2), pages 271-292, April.
    20. Bart Steyaert & Sabine Wittevrongel & Herwig Bruneel, 2017. "Characterisation of the output process of a discrete-time GI / D / 1 queue, and its application to network performance," Annals of Operations Research, Springer, vol. 252(1), pages 175-190, May.

    More about this item

    Keywords

    Partial busy period; Multi-server queues; Markovian queue; Spectral decomposition;
    All these keywords.

    JEL classification:

    • C44 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Operations Research; Statistical Decision Theory
    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques

    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:spr:queues:v:107:y:2024:i:1:d:10.1007_s11134-024-09906-1. 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.