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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
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    References listed on IDEAS

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    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.
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    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.

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