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Asymptotic behaviour of the tandem queueing system with identical service times at both queues

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  • O. J. Boxma
  • Q. Deng

Abstract

Consider a tandem queue consisting of two single-server queues in series, with a Poisson arrival process at the first queue and arbitrarily distributed service times, which for any customer are identical in both queues. For this tandem queue, we relate the tail behaviour of the sojourn time distribution and the workload distribution at the second queue to that of the (residual) service time distribution. As a by-result, we prove that both the sojourn time distribution and the workload distribution at the second queue are regularly varying at infinity of index 1−ν, if the service time distribution is regularly varying at infinity of index −ν (ν>1). Furthermore, in the latter case we derive a heavy-traffic limit theorem for the sojourn time S (2) at the second queue when the traffic load ρ↑ 1. It states that, for a particular contraction factor Δ (ρ), the contracted sojourn time Δ (ρ) S (2) converges in distribution to the limit distribution H(·) as ρ↑ 1 where . Copyright Springer-Verlag Berlin Heidelberg 2000

Suggested Citation

  • O. J. Boxma & Q. Deng, 2000. "Asymptotic behaviour of the tandem queueing system with identical service times at both queues," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 52(2), pages 307-323, November.
    • Handle: RePEc:spr:mathme:v:52:y:2000:i:2:p:307-323
    DOI: 10.1007/s186-000-8317-z
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    Cited by:

    1. Saulius Minkevičius & Vladimiras Dolgopolovas & Leonidas L. Sakalauskas, 2016. "A Law of the Iterated Logarithm for the Sojourn Time Process in Queues in Series," Methodology and Computing in Applied Probability, Springer, vol. 18(1), pages 37-57, March.
    2. H. Christian Gromoll & Bryce Terwilliger & Bert Zwart, 2018. "Heavy traffic limit for a tandem queue with identical service times," Queueing Systems: Theory and Applications, Springer, vol. 89(3), pages 213-241, August.
    3. S. Minkevičius & S. Steišūnas, 2006. "About the Sojourn Time Process in Multiphase Queueing Systems," Methodology and Computing in Applied Probability, Springer, vol. 8(2), pages 293-302, June.

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