IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v11y2023i9p2088-d1134990.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/11/9/2088/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/11/9/2088/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Pang, Guodong & Zhou, Yuhang, 2018. "Functional limit theorems for a new class of non-stationary shot noise processes," Stochastic Processes and their Applications, Elsevier, vol. 128(2), pages 505-544.
    2. Sen, Ankita & Selvaraju, N., 2023. "Diffusion approximation of an infinite-server queue under Markovian environment with rapid switching," Statistics & Probability Letters, Elsevier, vol. 195(C).
    3. Kaj, Ingemar & Tahir, Daniah, 2019. "Stochastic equations and limit results for some two-type branching models," Statistics & Probability Letters, Elsevier, vol. 150(C), pages 35-46.
    4. Hongyuan Lu & Guodong Pang & Michel Mandjes, 2016. "A functional central limit theorem for Markov additive arrival processes and its applications to queueing systems," Queueing Systems: Theory and Applications, Springer, vol. 84(3), pages 381-406, December.
    5. Michel Mandjes & Birgit Sollie, 2022. "A Numerical Approach for Evaluating the Time-Dependent Distribution of a Quasi Birth-Death Process," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 1693-1715, September.
    6. D. T. Koops & O. J. Boxma & M. R. H. Mandjes, 2017. "Networks of $$\cdot /G/\infty $$ · / G / ∞ queues with shot-noise-driven arrival intensities," Queueing Systems: Theory and Applications, Springer, vol. 86(3), pages 301-325, August.
    7. Yiran Liu & Harsha Honnappa & Samy Tindel & Nung Kwan Yip, 2021. "Infinite server queues in a random fast oscillatory environment," Queueing Systems: Theory and Applications, Springer, vol. 98(1), pages 145-179, June.
    8. G. A. Delsing & M. R. H. Mandjes & P. J. C. Spreij & E. M. M. Winands, 2020. "Asymptotics and Approximations of Ruin Probabilities for Multivariate Risk Processes in a Markovian Environment," Methodology and Computing in Applied Probability, Springer, vol. 22(3), pages 927-948, September.
    9. Ioannis Dimitriou, 2022. "Stationary analysis of certain Markov-modulated reflected random walks in the quarter plane," Annals of Operations Research, Springer, vol. 310(2), pages 355-387, March.
    10. Wenwen Li & Alexander Goldenshluger, 2024. "Adaptive minimax estimation of service time distribution in the $$M_t/G/\infty $$ M t / G / ∞ queue from departure data," Queueing Systems: Theory and Applications, Springer, vol. 108(1), pages 81-123, October.
    11. Dieter Fiems, 2022. "Retrial queues with generally distributed retrial times," Queueing Systems: Theory and Applications, Springer, vol. 100(3), pages 189-191, April.
    12. Pang, Guodong & Zheng, Yi, 2017. "On the functional and local limit theorems for Markov modulated compound Poisson processes," Statistics & Probability Letters, Elsevier, vol. 129(C), pages 131-140.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:11:y:2023:i:9:p:2088-:d:1134990. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.