IDEAS home Printed from https://ideas.repec.org/a/spr/queues/v98y2021i3d10.1007_s11134-021-09689-9.html
   My bibliography  Save this article

Large deviations for acyclic networks of queues with correlated Gaussian inputs

Author

Listed:
  • Martin Zubeldia

    (Eindhoven University of Technology
    University of Amsterdam)

  • Michel Mandjes

    (University of Amsterdam
    Eurandom, Eindhoven University of Technology
    University of Amsterdam)

Abstract

We consider an acyclic network of single-server queues with heterogeneous processing rates. It is assumed that each queue is fed by the superposition of a large number of i.i.d. Gaussian processes with stationary increments and positive drifts, which can be correlated across different queues. The flow of work departing from each server is split deterministically and routed to its neighbors according to a fixed routing matrix, with a fraction of it leaving the network altogether. We study the exponential decay rate of the probability that the steady-state queue length at any given node in the network is above any fixed threshold, also referred to as the ‘overflow probability’. In particular, we first leverage Schilder’s sample-path large deviations theorem to obtain a general lower bound for the limit of this exponential decay rate, as the number of Gaussian processes goes to infinity. Then, we show that this lower bound is tight under additional technical conditions. Finally, we show that if the input processes to the different queues are nonnegatively correlated, non-short-range dependent fractional Brownian motions, and if the processing rates are large enough, then the asymptotic exponential decay rates of the queues coincide with the ones of isolated queues with appropriate Gaussian inputs.

Suggested Citation

  • Martin Zubeldia & Michel Mandjes, 2021. "Large deviations for acyclic networks of queues with correlated Gaussian inputs," Queueing Systems: Theory and Applications, Springer, vol. 98(3), pages 333-371, August.
  • Handle: RePEc:spr:queues:v:98:y:2021:i:3:d:10.1007_s11134-021-09689-9
    DOI: 10.1007/s11134-021-09689-9
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11134-021-09689-9
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.

    File URL: https://libkey.io/10.1007/s11134-021-09689-9?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Mandjes, Michel & Mannersalo, Petteri & Norros, Ilkka & van Uitert, Miranda, 2006. "Large deviations of infinite intersections of events in Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 116(9), pages 1269-1293, September.
    2. Lavancier, Frédéric & Philippe, Anne & Surgailis, Donatas, 2009. "Covariance function of vector self-similar processes," Statistics & Probability Letters, Elsevier, vol. 79(23), pages 2415-2421, December.
    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. Krzysztof Dȩbicki, 2022. "Exact asymptotics of Gaussian-driven tandem queues," Queueing Systems: Theory and Applications, Springer, vol. 100(3), pages 285-287, April.

    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. Braunsteins, Peter & Mandjes, Michel, 2023. "The Cramér-Lundberg model with a fluctuating number of clients," Insurance: Mathematics and Economics, Elsevier, vol. 112(C), pages 1-22.
    2. Ranieri Dugo & Giacomo Giorgio & Paolo Pigato, 2024. "The Multivariate Fractional Ornstein-Uhlenbeck Process," CEIS Research Paper 581, Tor Vergata University, CEIS, revised 28 Aug 2024.
    3. Li, Bao-Gen & Ling, Dian-Yi & Yu, Zu-Guo, 2021. "Multifractal temporally weighted detrended partial cross-correlation analysis of two non-stationary time series affected by common external factors," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 573(C).
    4. Gustavo Didier & Vladas Pipiras, 2012. "Exponents, Symmetry Groups and Classification of Operator Fractional Brownian Motions," Journal of Theoretical Probability, Springer, vol. 25(2), pages 353-395, June.
    5. Tim Leung & Theodore Zhao, 2024. "A Noisy Fractional Brownian Motion Model for Multiscale Correlation Analysis of High-Frequency Prices," Mathematics, MDPI, vol. 12(6), pages 1-21, March.
    6. Lavancier, Frédéric & Philippe, Anne & Surgailis, Donatas, 2010. "A two-sample test for comparison of long memory parameters," Journal of Multivariate Analysis, Elsevier, vol. 101(9), pages 2118-2136, October.
    7. Bao-Gen Li & Dian-Yi Ling & Zu-Guo Yu, 2020. "Multifractal temporally weighted detrended partial cross-correlation analysis to quantify intrinsic power-law cross-correlation of two non-stationary time series affected by common external factors," Papers 2006.09154, arXiv.org.
    8. Norros, Ilkka & Saksman, Eero, 2009. "Local independence of fractional Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3155-3172, October.
    9. Characiejus, Vaidotas & Račkauskas, Alfredas, 2014. "Operator self-similar processes and functional central limit theorems," Stochastic Processes and their Applications, Elsevier, vol. 124(8), pages 2605-2627.
    10. Lucia Caramellino & Barbara Pacchiarotti & Simone Salvadei, 2015. "Large Deviation Approaches for the Numerical Computation of the Hitting Probability for Gaussian Processes," Methodology and Computing in Applied Probability, Springer, vol. 17(2), pages 383-401, June.
    11. Düker, Marie-Christine, 2020. "Limit theorems in the context of multivariate long-range dependence," Stochastic Processes and their Applications, Elsevier, vol. 130(9), pages 5394-5425.
    12. Davydov, Yu., 2012. "On convex hull of d-dimensional fractional Brownian motion," Statistics & Probability Letters, Elsevier, vol. 82(1), pages 37-39.

    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:spr:queues:v:98:y:2021:i:3:d:10.1007_s11134-021-09689-9. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.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.