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Sampling at subexponential times, with queueing applications

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  • Asmussen, Søren
  • Klüppelberg, Claudia
  • Sigman, Karl

Abstract

We study the tail asymptotics of the r.v. X(T) where {X(t)} is a stochastic process with a linear drift and satisfying some regularity conditions like a central limit theorem and a large deviations principle, and T is an independent r.v. with a subexponential distribution. We find that the tail of X(T) is sensitive to whether or not T has a heavier or lighter tail than a Weibull distribution with tail . This leads to two distinct cases, heavy tailed and moderately heavy tailed, but also some results for the classical light-tailed case are given. The results are applied via distributional Little's law to establish tail asymptotics for steady-state queue length in GI/GI/1 queues with subexponential service times. Further applications are given for queues with vacations, and M/G/1 busy periods.

Suggested Citation

  • Asmussen, Søren & Klüppelberg, Claudia & Sigman, Karl, 1999. "Sampling at subexponential times, with queueing applications," Stochastic Processes and their Applications, Elsevier, vol. 79(2), pages 265-286, February.
    • Handle: RePEc:eee:spapps:v:79:y:1999:i:2:p:265-286
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    2. F. G. Badía & C. Sangüesa, 2017. "Log-Convexity of Counting Processes Evaluated at a Random end of Observation Time with Applications to Queueing Models," Methodology and Computing in Applied Probability, Springer, vol. 19(2), pages 647-664, June.
    3. Marc Lelarge, 2008. "Packet reordering in networks with heavy-tailed delays," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 67(2), pages 341-371, April.
    4. Hiroyuki Masuyama, 2016. "A sufficient condition for the subexponential asymptotics of GI/G/1-type Markov chains with queueing applications," Annals of Operations Research, Springer, vol. 247(1), pages 65-95, December.
    5. Predrag R. Jelenković & Petar Momčilović, 2004. "Large Deviations of Square Root Insensitive Random Sums," Mathematics of Operations Research, INFORMS, vol. 29(2), pages 398-406, May.
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    7. Bin Liu & Yiqiang Q. Zhao, 2020. "Tail asymptotics for the $$M_1,M_2/G_1,G_2/1$$ M 1 , M 2 / G 1 , G 2 / 1 retrial queue with non-preemptive priority," Queueing Systems: Theory and Applications, Springer, vol. 96(1), pages 169-199, October.
    8. A. P. Zwart, 2001. "Tail Asymptotics for the Busy Period in the GI/G/1 Queue," Mathematics of Operations Research, INFORMS, vol. 26(3), pages 485-493, August.
    9. Bin Liu & Yiqiang Q. Zhao, 2022. "Tail Asymptotics for a Retrial Queue with Bernoulli Schedule," Mathematics, MDPI, vol. 10(15), pages 1-13, August.
    10. Evsey V. Morozov & Irina V. Peshkova & Alexander S. Rumyantsev, 2023. "Bounds and Maxima for the Workload in a Multiclass Orbit Queue," Mathematics, MDPI, vol. 11(3), pages 1-15, January.
    11. Bin Liu & Jie Min & Yiqiang Q. Zhao, 2023. "Refined tail asymptotic properties for the $$M^X/G/1$$ M X / G / 1 retrial queue," Queueing Systems: Theory and Applications, Springer, vol. 104(1), pages 65-105, June.
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    13. Baltrunas, A. & Daley, D. J. & Klüppelberg, C., 2004. "Tail behaviour of the busy period of a GI/GI/1 queue with subexponential service times," Stochastic Processes and their Applications, Elsevier, vol. 111(2), pages 237-258, June.
    14. Debicki, Krzystof & Zwart, Bert & Borst, Sem, 2004. "The supremum of a Gaussian process over a random interval," Statistics & Probability Letters, Elsevier, vol. 68(3), pages 221-234, July.

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