Queueing Systems

, 68:261 | Cite as

How to measure the accuracy of the subexponential approximation for the stationary single server queue

  • Dmitry Korshunov


We discuss the problem of establishing an upper bound for the distribution tail of the stationary waiting time D in the GI/GI/1 FCFS queue.


FCFS single server queue Stationary waiting time Heavy tails Large deviations Long tailed distribution Subexponential distribution Integrated tail distribution Accuracy of approximation Lower and upper bounds 

Mathematics Subject Classification (2000)

60K25 60F10 


  1. 1.
    Abate, J., Choudhury, G.L., Whitt, W.: Waiting-time tail probabilities in queues with long-tail service-time distributions. Queueing Syst. 16, 311–338 (1994) CrossRefGoogle Scholar
  2. 2.
    Asmussen, S.: Applied Probability and Queues, 2nd edn. Springer, New York (2003) Google Scholar
  3. 3.
    Baltrunas, A.: Second-order asymptotics for the ruin probability in the case of very large claims. Sib. Math. J. 40, 1034–1043 (1999) CrossRefGoogle Scholar
  4. 4.
    Borovkov, A.A., Borovkov, K.A.: Asymptotic Analysis of Random Walks. Heavy-Tailed Distributions. Cambridge University Press, Cambridge (2008) CrossRefGoogle Scholar
  5. 5.
    Embrechts, P., Veraverbeke, N.: Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insur. Math. Econ. 1, 55–72 (1982) CrossRefGoogle Scholar
  6. 6.
    Foss, S., Korshunov, D., Zachary, S.: An Introduction to Heavy-Tailed and Subexponential Distributions. Springer, Berlin (2011) CrossRefGoogle Scholar
  7. 7.
    Kalashnikov, V.: Geometric Sums: Bounds for Rare Events with Applications. Kluwer Academic, Dordrecht (1997) Google Scholar
  8. 8.
    Kalashnikov, V.: Bounds for ruin probabilities in the presence of large claims and their comparison. Insur. Math. Econ. 20, 146–147 (1997) Google Scholar
  9. 9.
    Kalashnikov, V., Tsitsiashvili, G.: Tails of waiting times and their bounds. Queueing Syst. 32, 257–283 (1999) CrossRefGoogle Scholar
  10. 10.
    Kalashnikov, V., Tsitsiashvili, G.: Asymptotically correct bounds of geometric convolutions with subexponential components. J. Math. Sci. 106, 2806–2819 (2001) CrossRefGoogle Scholar
  11. 11.
    Korolev, V.Yu., Bening, V.E., Shorgin, S.Ya.: Mathematical Foundations of Risk Theory. Fizmatlit, Moscow (2007) (in Russian) Google Scholar
  12. 12.
    Korshunov, D.A.: On distribution tail of the maximum of a random walk. Stoch. Process. Appl. 72, 97–103 (1997) CrossRefGoogle Scholar
  13. 13.
    Lindley, D.V.: The theory of queues with a single server. Proc. Camb. Philos. Soc. 48, 277–289 (1952) CrossRefGoogle Scholar
  14. 14.
    Meyn, S.P., Tweedie, R.L.: Markov Chains and Stochastic Stability. Springer, Berlin (1993) Google Scholar
  15. 15.
    Pakes, A.G.: On the tails of waiting-time distribution. J. Appl. Probab. 12, 555–564 (1975) CrossRefGoogle Scholar
  16. 16.
    Richards, A.: On upper bounds for the tail distribution of geometric sums of subexponential random variables. Queueing Syst. 62, 229–242 (2009) CrossRefGoogle Scholar
  17. 17.
    Veraverbeke, N.: Asymptotic behavior of Wiener-Hopf factors of a random walk. Stoch. Process. Appl. 5, 27–37 (1977) CrossRefGoogle Scholar
  18. 18.
    Willekens, E., Teugels, J.L.: Asymptotic expansions for waiting time probabilities in an M/G/1 queue with long-tailed service time. Queueing Syst. 10, 295–313 (1992) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia

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