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Queueing Systems

, Volume 52, Issue 3, pp 193–198 | Cite as

Tail asymptotics for the queue length in an M/G/1 retrial queue

  • Weixin Shang
  • Liming Liu
  • Quan-Lin Li
Article

Abstract

In this paper, we study the tail behavior of the stationary queue length of an M/G/1 retrial queue. We show that the subexponential tail of the stationary queue length of an M/G/1 retrial queue is determined by that of the corresponding M/G/1 queue, and hence the stationary queue length in an M/G/1 retrial queue is subexponential if the stationary queue length in the corresponding M/G/1 queue is subexponential. Our results for subexponential tails also apply to regularly varying tails, and we provide the regularly varying tail asymptotics for the stationary queue length of the M/G/1 retrial queue.

Keywords

M/G/1 retrial queue Queue length Subexponentiality Regular variation Tail asymptotics 

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References

  1. 1.
    J.R. Artalejo, A classified bibliography of research on retrial queues: Progress in 1990–1999, Top 7 (1999) 187–211.Google Scholar
  2. 2.
    S. Asmussen, C. Klüppelberg and K. Sigman, Sampling at subexponential times, with queueing applications, Stochastic Processes and Their Applications 79 (1999) 265–286.CrossRefGoogle Scholar
  3. 3.
    N.H. Bingham, C.M. Goldie and J.L. Teugels, Regular Variation (Cambridge University Press, 1987).Google Scholar
  4. 4.
    O.J. Boxma and J.W. Cohen, Heavy-traffic analysis for the GI/G/1 queue with heavy-tailed distributions, Queueing Systems 33 (1999) 177–204.CrossRefGoogle Scholar
  5. 5.
    O.J. Boxma and V. Dumas, The busy period in the fluid queue, Performance Evaluation Rev. 26 (1998) 100–110.Google Scholar
  6. 6.
    P. Embrechts, C. Klüppelberg and T. Mikosch, Modelling Extremal Events for Insurance and Finance (Springer, Heidelberg, 1997).Google Scholar
  7. 7.
    A. Erramilli, O. Narayan and W. Willinger, Experimental queueing analysis with long-range dependent packet traffic, IEEE/ACM Transactions on Networking 4 (1996) 209–223.Google Scholar
  8. 8.
    G.I. Falin, A survey of retrial queues, Queueing Systems 7 (1990) 127–167.CrossRefGoogle Scholar
  9. 9.
    G.I. Falin and J.G.C. Templeton, Retrial Queues (Chapman & Hall, London, 1997).Google Scholar
  10. 10.
    S. Foss and D. Korshunov, Sampling at a random time with a heavy-tailed distribution, Markov Processes Relat. Fields 6 (2000) 543–568.Google Scholar
  11. 11.
    C.M. Goldie and C. Klüppelberg, Subexponential distributions, in: A Practical Guide to Heavy Tails: Statistical Techniques for Analyzing Heavy Tailed Distributions, eds., R. Adler, R. Feldman, and M.S. Taqqu, (Birkhäuser, Boston, 1998) pp. 435–459.Google Scholar
  12. 12.
    P.R. Jelenković, Subexponential loss rates in a GI/G/1 queue with applications, Queueing Systems 33 (1999) 91–123.Google Scholar
  13. 13.
    C. Klüppelberg, Subexponential distributions and integrated tails, J. Appl. Prob. 25 (1988) 132–141.Google Scholar
  14. 14.
    V.G. Kulkarni and H.M. Liang, Retrial queues revisited, in: Frontiers in Queueing: Models and Applications in Science and Engineering, ed., J.H. Dshalalow (CRC Press, BocaRaton, FL, 1997) pp. 19–34.Google Scholar
  15. 15.
    W.E. Leland, M.S. Taqqu, W. Willinger and D.V. Wilson, On the self-similar nature of Ethernet traffic, in: ACM SIGCOMM93 (1993). An extended version of this paper was published in IEEE / ACM Trans. Networking 2 (1994) 1–15.Google Scholar
  16. 16.
    Q.L. Li, Y. Ying and Y.Q. Zhao, A BMAP/G/1 retrial queue with a server subject to breakdowns and repairs, Ann. Oper. Res. (2005) accepted.Google Scholar
  17. 17.
    Q.L. Li and Y.Q. Zhao, Heavy-tailed asymptotics of stationary probability vectors of Markov chains of GI/G/1 type, Adv. in Appl. Probab. 37 (2005) 482–509.Google Scholar
  18. 18.
    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, J. Appl. Prob. 17 (1980) 802–813.Google Scholar
  19. 19.
    S. Resnick, Performance decay in a single server exponential queueing model with long range dependence, Oper. Res. 45 (1997) 235–243.Google Scholar
  20. 20.
    M. Roughan, D. Veitch and M.P. Rumsewicz, Computing queue-length distributions for power-law queues, in: Proceedings, Inforcom’98 (1998).Google Scholar
  21. 21.
    T. Takine, Subexponential asymptotics of the waiting time distribution in a single-server queue with multiple Markovian arrival streams, Stochastic Models 17 (2001) 429–448.CrossRefGoogle Scholar
  22. 22.
    W. Whitt, The impact of a heavy-tailed service-time distribution upon the M/G/s waiting-time distribution, Queueing Systems 36 (2000) 71–87.Google Scholar
  23. 23.
    T. Yang and J.G.C. Templeton, A survey on retrial queues, Queueing Systems 2 (1987) 201–233.CrossRefGoogle Scholar
  24. 24.
    A.P. Zwart, Tail asymptotics for the busy period in the GI/G/1 queue, Math. Oper. Res. 26 (2001) 485–493.CrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media, LLC 2006

Authors and Affiliations

  1. 1.Department of Industrial Engineering and Logistics ManagementHong Kong University of Science and TechnologyKowloonHong Kong
  2. 2.Department of Industrial EngineeringTsinghua UniversityBeijingP.R. China

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