IDEAS home Printed from https://ideas.repec.org/a/spr/mathme/v81y2015i3p299-315.html
   My bibliography  Save this article

Loss rates in the single-server queue with complete rejection

Author

Listed:
  • Bert Zwart

Abstract

Consider the single-server queue in which customers are rejected if their total sojourn time would exceed a certain level $$K$$ K . A basic performance measure of this system is the probability $$P_K$$ P K that a customer gets rejected in steady state. This paper presents asymptotic expansions for $$P_K$$ P K as $$K\rightarrow \infty $$ K → ∞ . If the service time $$B$$ B is light-tailed and inter-arrival times are exponential, it is shown that the loss probability has an exponential tail. The proof of this result heavily relies on results on the two-sided exit problem for Lévy processes with no positive jumps. For heavy-tailed (subexponential) service times and generally distributed inter-arrival times, the loss probability is shown to be asymptotically equivalent to the trivial lower bound $$P(B>K)$$ P ( B > K ) . Copyright The Author(s) 2015

Suggested Citation

  • Bert Zwart, 2015. "Loss rates in the single-server queue with complete rejection," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 81(3), pages 299-315, June.
    • Handle: RePEc:spr:mathme:v:81:y:2015:i:3:p:299-315
    DOI: 10.1007/s00186-015-0497-x
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s00186-015-0497-x
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s00186-015-0497-x?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. Schmidli, Hanspeter, 1999. "On the Distribution of the Surplus Prior and at Ruin," ASTIN Bulletin, Cambridge University Press, vol. 29(2), pages 227-244, November.
    2. Bezalel Gavish & Paul J. Schweitzer, 1977. "The Markovian Queue with Bounded Waiting time," Management Science, INFORMS, vol. 23(12), pages 1349-1357, August.
    3. Lars Andersen, 2011. "Subexponential loss rate asymptotics for Lévy processes," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 73(1), pages 91-108, February.
    4. van Ommeren, J. C. W., 1987. "Exponential bounds for excess probabilities in systems with a finite capacity," Stochastic Processes and their Applications, Elsevier, vol. 24(1), pages 143-149, February.
    5. Veraverbeke, N., 1977. "Asymptotic behaviour of Wiener-Hopf factors of a random walk," Stochastic Processes and their Applications, Elsevier, vol. 5(1), pages 27-37, February.
    6. Nam Kyoo Boots & Henk Tijms, 1999. "A Multiserver Queueing System with Impatient Customers," Management Science, INFORMS, vol. 45(3), pages 444-448, March.
    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. Cheng, Yebin & Tang, Qihe & Yang, Hailiang, 2002. "Approximations for moments of deficit at ruin with exponential and subexponential claims," Statistics & Probability Letters, Elsevier, vol. 59(4), pages 367-378, October.
    2. Psarrakos, Georgios & Politis, Konstadinos, 2008. "Tail bounds for the joint distribution of the surplus prior to and at ruin," Insurance: Mathematics and Economics, Elsevier, vol. 42(1), pages 163-176, February.
    3. Mor Harchol-Balter, 2021. "Open problems in queueing theory inspired by datacenter computing," Queueing Systems: Theory and Applications, Springer, vol. 97(1), pages 3-37, February.
    4. Tang, Qihe & Wei, Li, 2010. "Asymptotic aspects of the Gerber-Shiu function in the renewal risk model using Wiener-Hopf factorization and convolution equivalence," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 19-31, February.
    5. Wang, Yuebao & Yang, Yang & Wang, Kaiyong & Cheng, Dongya, 2007. "Some new equivalent conditions on asymptotics and local asymptotics for random sums and their applications," Insurance: Mathematics and Economics, Elsevier, vol. 40(2), pages 256-266, March.
    6. Sgibnev, M. S., 2001. "Exact asymptotic behaviour of the distribution of the supremum," Statistics & Probability Letters, Elsevier, vol. 52(3), pages 301-311, April.
    7. Chiu, S. N. & Yin, C. C., 2003. "The time of ruin, the surplus prior to ruin and the deficit at ruin for the classical risk process perturbed by diffusion," Insurance: Mathematics and Economics, Elsevier, vol. 33(1), pages 59-66, August.
    8. Emilio Gómez-Déniz & José María Sarabia & Enrique Calderín-Ojeda, 2019. "Ruin Probability Functions and Severity of Ruin as a Statistical Decision Problem," Risks, MDPI, vol. 7(2), pages 1-16, June.
    9. Søren Asmussen & Serguei Foss & Dmitry Korshunov, 2003. "Asymptotics for Sums of Random Variables with Local Subexponential Behaviour," Journal of Theoretical Probability, Springer, vol. 16(2), pages 489-518, April.
    10. Gajek, Leslaw, 2005. "On the deficit distribution when ruin occurs--discrete time model," Insurance: Mathematics and Economics, Elsevier, vol. 36(1), pages 13-24, February.
    11. Xindan Li & Dan Tang & Yongjin Wang & Xuewei Yang, 2014. "Optimal processing rate and buffer size of a jump-diffusion processing system," Annals of Operations Research, Springer, vol. 217(1), pages 319-335, June.
    12. Gao, Qingwu & Wang, Yuebao, 2009. "Ruin probability and local ruin probability in the random multi-delayed renewal risk model," Statistics & Probability Letters, Elsevier, vol. 79(5), pages 588-596, March.
    13. Sheldon Lin, X. & E. Willmot, Gordon & Drekic, Steve, 2003. "The classical risk model with a constant dividend barrier: analysis of the Gerber-Shiu discounted penalty function," Insurance: Mathematics and Economics, Elsevier, vol. 33(3), pages 551-566, December.
    14. Mihalis G. Markakis & Eytan Modiano & John N. Tsitsiklis, 2018. "Delay Analysis of the Max-Weight Policy Under Heavy-Tailed Traffic via Fluid Approximations," Mathematics of Operations Research, INFORMS, vol. 43(2), pages 460-493, May.
    15. Psarrakos, Georgios, 2009. "Asymptotic results for heavy-tailed distributions using defective renewal equations," Statistics & Probability Letters, Elsevier, vol. 79(6), pages 774-779, March.
    16. Tang, Qihe, 2007. "The overshoot of a random walk with negative drift," Statistics & Probability Letters, Elsevier, vol. 77(2), pages 158-165, January.
    17. Nam Boots & Henk Tijms, 1999. "AnM/M/c queue with impatient customers," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 7(2), pages 213-220, December.
    18. Wang, Qinan, 2004. "Modeling and analysis of high risk patient queues," European Journal of Operational Research, Elsevier, vol. 155(2), pages 502-515, June.
    19. Youri Raaijmakers & Sem Borst & Onno Boxma, 2023. "Fork–join and redundancy systems with heavy-tailed job sizes," Queueing Systems: Theory and Applications, Springer, vol. 103(1), pages 131-159, February.
    20. Serguei Foss & Takis Konstantopoulos & Stan Zachary, 2007. "Discrete and Continuous Time Modulated Random Walks with Heavy-Tailed Increments," Journal of Theoretical Probability, Springer, vol. 20(3), pages 581-612, September.

    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:mathme:v:81:y:2015:i:3:p:299-315. 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.