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Large Deviations for the Single-Server Queue and the Reneging Paradox

Author

Listed:
  • Rami Atar

    (Viterbi Faculty of Electrical Engineering, Technion, Haifa 32000, Israel)

  • Amarjit Budhiraja

    (Department of Statistics and Operations Research, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599)

  • Paul Dupuis

    (Division of Applied Mathematics, Brown University, Providence, Rhode Island 02906)

  • Ruoyu Wu

    (Department of Mathematics, Iowa State University, Ames, Iowa 50011)

Abstract

For the M/M/1+M model at the law-of-large-numbers scale, the long-run reneging count per unit time does not depend on the individual (i.e., per customer) reneging rate. This paradoxical statement has a simple proof. Less obvious is a large deviations analogue of this fact, stated as follows: the decay rate of the probability that the long-run reneging count per unit time is atypically large or atypically small does not depend on the individual reneging rate. In this paper, the sample path large deviations principle for the model is proved and the rate function is computed. Next, large time asymptotics for the reneging rate are studied for the case when the arrival rate exceeds the service rate. The key ingredient is a calculus of variations analysis of the variational problem associated with atypical reneging. A characterization of the aforementioned decay rate, given explicitly in terms of the arrival and service rate parameters of the model, is provided yielding a precise mathematical description of this paradoxical behavior.

Suggested Citation

  • Rami Atar & Amarjit Budhiraja & Paul Dupuis & Ruoyu Wu, 2022. "Large Deviations for the Single-Server Queue and the Reneging Paradox," Mathematics of Operations Research, INFORMS, vol. 47(1), pages 232-258, February.
    • Handle: RePEc:inm:ormoor:v:47:y:2022:i:1:p:232-258
    DOI: 10.1287/moor.2021.1127
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