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An alternative approach to heavy-traffic limits for finite-pool queues

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  • G. Bet

    (Università degli Studi di Firenze)

Abstract

We consider a model for transitory queues in which only a finite number of customers can join. The queue thus operates over a finite time horizon. In this system, also known as the $$\Delta _{(i)}/G/1$$Δ(i)/G/1 queue, the customers decide independently when to join the queue by sampling their arrival time from a common distribution. We prove that, when the queue satisfies a certain heavy-traffic condition and under the additional assumption that the second moment of the service time is finite, the rescaled queue length process converges to a reflected Brownian motion with parabolic drift. Our result holds for general arrival times, thus improving on an earlier result Bet et al. (Math Oper Res 2019, https://doi.org/10.1287/moor.2018.0947) which assumes exponential arrival times.

Suggested Citation

  • G. Bet, 2020. "An alternative approach to heavy-traffic limits for finite-pool queues," Queueing Systems: Theory and Applications, Springer, vol. 95(1), pages 121-144, June.
    • Handle: RePEc:spr:queues:v:95:y:2020:i:1:d:10.1007_s11134-020-09653-z
    DOI: 10.1007/s11134-020-09653-z
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    References listed on IDEAS

    as
    1. William A. Massey, 1985. "Asymptotic Analysis of the Time Dependent M/M/1 Queue," Mathematics of Operations Research, INFORMS, vol. 10(2), pages 305-327, May.
    2. 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.
    3. Song‐Hee Kim & Ward Whitt, 2014. "Choosing arrival process models for service systems: Tests of a nonhomogeneous Poisson process," Naval Research Logistics (NRL), John Wiley & Sons, vol. 61(1), pages 66-90, February.
    4. Ward Whitt, 1980. "Some Useful Functions for Functional Limit Theorems," Mathematics of Operations Research, INFORMS, vol. 5(1), pages 67-85, February.
    5. Song-Hee Kim & Ward Whitt, 2014. "Are Call Center and Hospital Arrivals Well Modeled by Nonhomogeneous Poisson Processes?," Manufacturing & Service Operations Management, INFORMS, vol. 16(3), pages 464-480, July.
    6. Lawrence Brown & Noah Gans & Avishai Mandelbaum & Anat Sakov & Haipeng Shen & Sergey Zeltyn & Linda Zhao, 2005. "Statistical Analysis of a Telephone Call Center: A Queueing-Science Perspective," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 36-50, March.
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    Cited by:

    1. Moshe Haviv & Liron Ravner, 2021. "A survey of queueing systems with strategic timing of arrivals," Queueing Systems: Theory and Applications, Springer, vol. 99(1), pages 163-198, October.

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