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On the distributions of infinite server queues with batch arrivals

Author

Listed:
  • Andrew Daw

    (Cornell University)

  • Jamol Pender

    (Cornell University)

Abstract

Queues that feature multiple entities arriving simultaneously are among the oldest models in queueing theory, and are often referred to as “batch” (or, in some cases, “bulk”) arrival queueing systems. In this work, we study the effect of batch arrivals on infinite server queues. We assume that the arrival epochs occur according to a Poisson process, with treatment of both stationary and non-stationary arrival rates. We consider both exponentially and generally distributed service durations, and we analyze both fixed and random arrival batch sizes. In addition to deriving the transient mean, variance, and moment-generating function for time-varying arrival rates, we also find that the steady-state distribution of the queue is equivalent to the sum of scaled Poisson random variables with rates proportional to the order statistics of its service distribution. We do so through viewing the batch arrival system as a collection of correlated sub-queues. Furthermore, we investigate the limiting behavior of the process through a batch scaling of the queue and through fluid and diffusion limits of the arrival rate. In the course of our analysis, we make important connections between our model and the harmonic numbers, generalized Hermite distributions, and truncated polylogarithms.

Suggested Citation

  • Andrew Daw & Jamol Pender, 2019. "On the distributions of infinite server queues with batch arrivals," Queueing Systems: Theory and Applications, Springer, vol. 91(3), pages 367-401, April.
    • Handle: RePEc:spr:queues:v:91:y:2019:i:3:d:10.1007_s11134-019-09603-4
    DOI: 10.1007/s11134-019-09603-4
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    References listed on IDEAS

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    1. F. G. Foster, 1964. "Batched Queuing Processes," Operations Research, INFORMS, vol. 12(3), pages 441-449, June.
    2. R. Milne & M. Westcott, 1993. "Generalized multivariate Hermite distributions and related point processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 45(2), pages 367-381, June.
    3. Xuefeng Gao & Lingjiong Zhu, 2018. "Functional central limit theorems for stationary Hawkes processes and application to infinite-server queues," Queueing Systems: Theory and Applications, Springer, vol. 90(1), pages 161-206, October.
    4. Singha Chiamsiri & Michael S. Leonard, 1981. "A Diffusion Approximation for Bulk Queues," Management Science, INFORMS, vol. 27(10), pages 1188-1199, October.
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    Cited by:

    1. Ivan Ferretti & Matteo Camparada & Lucio Enrico Zavanella, 2022. "Queuing Theory-Based Design Methods for the Definition of Power Requirements in Manufacturing Systems," Energies, MDPI, vol. 15(20), pages 1-14, October.

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