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Tail Behaviour of the Area Under the Queue Length Process of the Single Server Queue with Regularly Varying Service Times

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Abstract

This paper considers a stable GI∨GI∨1 queue with a regularly varying service time distribution. We derive the tail behaviour of the integral of the queue length process Q(t) over one busy period. We show that the occurrence of a large integral is related to the occurrence of a large maximum of the queueing process over the busy period and we exploit asymptotic results for this variable. We also prove a central limit theorem for ∫0t Q(s) ds.

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References

  1. J. Abate and W. Whitt, Asymptotics for M/G/1 low-priority waiting-time tail probabilities, Queueing Systems (1997) 173–233.

    Article  Google Scholar 

  2. S. Asmussen, Subexponential asymptotics for stochastic processes: Extremal behaviour, stationary distributions and first passage times, Annals of Applied Probability 8 (1998) 354–374.

    Article  Google Scholar 

  3. S. Asmussen, Extreme value theory for queues via cycle maxima, Extremes 1 (1998) 137–168.

    Article  Google Scholar 

  4. S. Asmussen and C. Klüppelberg, Large deviations results for subexponential tails, with applications to insurance risk, Stochastic Processes and their Applications 64 (1996) 103–125.

    Article  Google Scholar 

  5. S. Asmussen, C. Klüppelberg and K. Sigman, Sampling at subexponential times, with queueing applications, Stochastic Processes and their Applications 79 (1999) 265–286.

    Article  Google Scholar 

  6. S. Asmussen and J. Møller, Tail asymptotics for M/G/1 type queueing processes with subexponential increments, Queueing Systems 33 (1999) 153–176.

    Article  Google Scholar 

  7. S. Asmussen and J.L. Teugels, Convergence rates for M/G/1 queues and ruin problems with heavy tails, Journal of Applied Probability 33 (1996) 1181–1190.

    Google Scholar 

  8. A. Baltrunas, D.J. Daley and C. Klüppelberg, Tail behaviour of the busy period of a GI/GI/1 queue with subexponential service times, Stochastic Processes and their Applications 111 (2004) 237–258.

    Article  Google Scholar 

  9. N.H. Bingham, C.M. Goldie and J.L. Teugels, Regular Variation (Cambridge University Press, Cambridge, 1987).

    Google Scholar 

  10. A.A. Borovkov, O.J. Boxma and Z. Palmowski, On the integral of the workload process of the single server queue, Journal of Applied Probability 40 (2003) 200–225.

    Article  Google Scholar 

  11. D.J. Daley, The total waiting time in a busy period of a stable single-server queue. I, Journal of Applied Probability 6 (1969) 550–564.

    Google Scholar 

  12. D.J. Daley and D.R. Jacobs Jr., The total waiting time in a busy period of a stable single-server queue. II, Journal of Applied Probability 6 (1969) 565–572.

    Google Scholar 

  13. S. Foss and S. Zachary, The maximum on a random time interval of a random walk with long-tailed increments and negative drift, Annals of Applied Probability 13 (2003) 37–53.

    Article  Google Scholar 

  14. S. Foss, Z. Palmowski and S. Zachary, The probability of exceeding a high boundary on a random time interval for a heavy-tailed random walk, to appear in Annals of Applied Probability (2005).

  15. D.K. Fuk and S.V. Nagaev, Probability inequalities for sums of independent random variables, Th. Prob. Appl. 16 (1971) 643–660.

    Article  Google Scholar 

  16. F. Guillemin and D. Pinchon, On the area swept under the occupation process of an M/M/1 queue in a busy period, Queueing Systems 29 (1998) 383–398.

    Article  Google Scholar 

  17. F. Guillemin and A. Simonian, Transient characteristics of an M/M/∞ system, Advances in Applied Probability 27 (1997) 862–888.

    Google Scholar 

  18. A. Gut, Stopped two-dimensional perturbed random walks and Lévy processes, Statistics and Probability Letters 35 (1997) 317–325.

    Article  Google Scholar 

  19. D. Heath, S. Resnick and G. Samorodnitsky, Patterns of buffer overflow in a class of queues with long memory in the input stream, Annals of Applied Probability 7 (1997) 1021–1057.

    Article  Google Scholar 

  20. C.C. Heyde, On large deviation probabilities in the case of attraction to a non-normal law, Sankhya Ser. A 30 (1968) 253–258.

    Google Scholar 

  21. D.L. Iglehart, Functional limit theorems for the queue GI/G/1 in light traffic, Advances in Applied Probability 3 (1971) 269–281.

    Google Scholar 

  22. P. Jelenkovic and P. Momcilovic, Large deviations for square root insensitive random sums, Mathematics of Operations Research 29(2) (2004), 398–406.

    Article  Google Scholar 

  23. C. Klüppelberg, Subexponential distributions and integrated tails, Journal of Applied Probability 35 (1988) 325–347.

    Google Scholar 

  24. A.De Meyer and J.L. Teugels, On the asymptotic behaviour of the distributions of the busy period and service time in M/G/1, Journal of Applied Probability (1980), 802–813.

  25. A.V. Nagaev, Integral limit theorems taking large deviations into account when Cramér’s condition does not hold. I, Th. Prob. Appl. 14 (1969) 51–64.

    Article  Google Scholar 

  26. A.V. Nagaev, Probabilities of large deviations of sums of independent random variables, Doctor of Science Thesis (1970).

  27. I.F. Pinelis, Asymptotics equivalence of the probabilities of large deviations for sums and maximum of independent random variables. Limit theorems of probability theory, Trudy Inst. Mat. 5 (1985), “Nauka” Sibirsk. Otdel., Novosybirsk, pp. 144–173.

  28. A.P. Zwart, Tail asymptotics for the busy period in the GI/G/1 queue, Mathematics of Operations Research 26 (2001) 485–493.

    Article  Google Scholar 

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Authors and Affiliations

  1. Wrocław University, Mathematical Institute, Pl. Grunwaldzki 2/4, 50-384, Wrocław, Poland

    Rafał Kulik

  2. Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Av., K1N 6N5, Ottawa, ON

    Rafał Kulik

  3. Wrocław University, Mathematical Institute, Pl. Grunwaldzki 2/4, 50-384, Wrocław, Poland

    Zbigniew Palmowski

  4. Utrecht University, Mathematical Institute, P.O. Box 80.010, 3508, TA, Utrecht, The Netherlands

    Zbigniew Palmowski

Authors
  1. Rafał Kulik
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  2. Zbigniew Palmowski
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Correspondence to Rafał Kulik.

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AMS subject classification: 60K25, 90B22.

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Kulik, R., Palmowski, Z. Tail Behaviour of the Area Under the Queue Length Process of the Single Server Queue with Regularly Varying Service Times. Queueing Syst 50, 299–323 (2005). https://doi.org/10.1007/s11134-005-0926-2

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  • DOI: https://doi.org/10.1007/s11134-005-0926-2

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