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Exponential Approximations for Tail Probabilities in Queues, I: Waiting Times

Author

Listed:
  • Joseph Abate

    (Ridgewood, New Jersey)

  • Gagan L. Choudhury

    (AT&T Bell Laboratories, Holmdel, New Jersey)

  • Ward Whitt

    (AT&T Bell Laboratories, Murray Hill, New Jersey)

Abstract

This paper focuses on simple exponential approximations for tail probabilities of the steady-state waiting time in infinite-capacity multiserver queues based on small-tail asymptotics. For the GI/GI/s model, we develop a heavy-traffic asymptotic expansion in powers of one minus the traffic intensity for the waiting-time asymptotic decay rate. We propose a two-term approximation for the asymptotic decay rate based on the first three moments of the interarrival-time and service-time distributions. We also suggest approximating the asymptotic constant by the product of the mean and the asymptotic decay rate. We evaluate the exponential approximations based on the exact asymptotic parameters and their approximations by making comparisons with exact results obtained numerically for the BMAP/GI/1 queue, which has a batch Markovian arrival process, and the GI/GI/s queue. Numerical examples show that the exponential approximations are remarkably accurate, especially for higher percentiles, such as the 90th percentile and beyond.

Suggested Citation

  • Joseph Abate & Gagan L. Choudhury & Ward Whitt, 1995. "Exponential Approximations for Tail Probabilities in Queues, I: Waiting Times," Operations Research, INFORMS, vol. 43(5), pages 885-901, October.
    • Handle: RePEc:inm:oropre:v:43:y:1995:i:5:p:885-901
    DOI: 10.1287/opre.43.5.885
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    Citations

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    Cited by:

    1. Dimitris Bertsimas & Ioannis Ch. Paschalidis, 2001. "Probabilistic Service Level Guarantees in Make-to-Stock Manufacturing Systems," Operations Research, INFORMS, vol. 49(1), pages 119-133, February.
    2. Søren Asmussen & Colm O'cinneide, 2002. "On the Tail of the Waiting Time in a Markov-Modulated M/G/1 Queue," Operations Research, INFORMS, vol. 50(3), pages 559-565, June.
    3. Gad Allon & Awi Federgruen, 2008. "Service Competition with General Queueing Facilities," Operations Research, INFORMS, vol. 56(4), pages 827-849, August.
    4. Opher Baron, 2008. "Regulated Random Walks and the LCFS Backlog Probability: Analysis and Application," Operations Research, INFORMS, vol. 56(2), pages 471-486, April.
    5. Amr Rizk & Felix Poloczek & Florin Ciucu, 2016. "Stochastic bounds in Fork–Join queueing systems under full and partial mapping," Queueing Systems: Theory and Applications, Springer, vol. 83(3), pages 261-291, August.
    6. Asmussen, Søren & Klüppelberg, Claudia & Sigman, Karl, 1999. "Sampling at subexponential times, with queueing applications," Stochastic Processes and their Applications, Elsevier, vol. 79(2), pages 265-286, February.
    7. Slotnick, Susan A. & Sobel, Matthew J., 2005. "Manufacturing lead-time rules: Customer retention versus tardiness costs," European Journal of Operational Research, Elsevier, vol. 163(3), pages 825-856, June.
    8. Jalili Marand, Ata & Li, Hongyan & Thorstenson, Anders, 2019. "Joint inventory control and pricing in a service-inventory system," International Journal of Production Economics, Elsevier, vol. 209(C), pages 78-91.
    9. Sergio Chayet & Panos Kouvelis & Dennis Z. Yu, 2011. "Product Variety and Capacity Investments in Congested Production Systems," Manufacturing & Service Operations Management, INFORMS, vol. 13(3), pages 390-403, July.
    10. So, Kut C. & Song, Jing-Sheng, 1998. "Price, delivery time guarantees and capacity selection," European Journal of Operational Research, Elsevier, vol. 111(1), pages 28-49, November.
    11. 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.
    12. Kumar, Sameer & Yang, Muer & Goldschmidt, Kyle H., 2018. "Will aging voting machines cause more voters to experience long waits?," International Journal of Production Economics, Elsevier, vol. 198(C), pages 1-10.
    13. Soroush Saghafian & Mark P. Van Oyen, 2016. "Compensating for Dynamic Supply Disruptions: Backup Flexibility Design," Operations Research, INFORMS, vol. 64(2), pages 390-405, April.

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