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A Moment Approach for a Conditional Central Limit Theorem of Infinite-Server Queue: A Case of M/M X / ∞ Queue

Author

Listed:
  • Ayane Nakamura

    (Graduate School of Science and Technology, University of Tsukuba, Tsukuba 305-8577, Japan
    These authors contributed equally to this work.)

  • Tuan Phung-Duc

    (Institute of Systems and Information Engineering, University of Tsukuba, Tsukuba 305-8577, Japan
    These authors contributed equally to this work.)

Abstract

Several studies have been conducted on scaling limits for Markov-modulated infinite-server queues. To the best of our knowledge, most of these studies adopt an approach to prove the convergence of the moment-generating function (or characteristic function) of the random variable that represents a scaled version of the number of busy servers and show the weak law of large numbers and the central limit theorem (CLT). In these studies, an essential assumption is the finiteness of the phase process and, in most of them, the CLT for the number of busy servers conditional on the phase (or the joint states) has not been considered. This paper proposes a new method called the moment approach to address these two limitations in an infinite-server batch service queue, which is called the M/M X / ∞ queue. We derive the conditional weak law of large numbers and a recursive formula that suggests the conditional CLT. We derive series expansion of the conditional raw moments, which are used to confirm the conditional CLT by a symbolic algorithm.

Suggested Citation

  • Ayane Nakamura & Tuan Phung-Duc, 2023. "A Moment Approach for a Conditional Central Limit Theorem of Infinite-Server Queue: A Case of M/M X / ∞ Queue," Mathematics, MDPI, vol. 11(9), pages 1-20, April.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:9:p:2088-:d:1134990
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    References listed on IDEAS

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    1. D. Anderson & J. Blom & M. Mandjes & H. Thorsdottir & K. Turck, 2016. "A Functional Central Limit Theorem for a Markov-Modulated Infinite-Server Queue," Methodology and Computing in Applied Probability, Springer, vol. 18(1), pages 153-168, March.
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    Cited by:

    1. Ayane Nakamura & Tuan Phung-Duc, 2023. "Equilibrium Analysis for Batch Service Queueing Systems with Strategic Choice of Batch Size," Mathematics, MDPI, vol. 11(18), pages 1-22, September.

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