Skip to main content
SpringerLink
Log in

Extreme Value Theory for Queues Via Cycle Maxima

  • Published:
Extremes Aims and scope Submit manuscript

Abstract

The present state of extreme value theory for queues is surveyed. The exposition focuses on the regenerative properties of queueing systems, which reduces the problem to the study of the tail of the maximum \(\overline X \left( {\tau } \right)\) of the queueing process \(\left\{ {X\left( t \right)} \right\}\) during a regenerative cycle τ. For simple queues, methods for obtaining the distribution of \(\overline X \left( {\tau } \right)\) both explicitly and asymptotically are reviewed. In greater generality, the study leads into Wiener–Hopf problems. Extensions to queues in a Markov regime, for example governed by Markov-modulated Poisson arrivals, are also considered.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adler, R., An Introduction to Continuity, Extrema and Related Topics for General Gaussian Processes, IMS Monographs, 12, (1990).

  2. Aldous, D., Hofri, M., and Szpankowski, W., “Maximum size of a dynamic data structure: Hashing with lazy delay revisited,” Siam J. Comp. 21, 713–732, (1992).

    Google Scholar 

  3. Anantharam, V., “How large delays build up in a GI/GI/1 queue,” Queueing Systems 5, 345–368, (1988).

    Google Scholar 

  4. Andersson, C.W., “Extreme value theory for a class of discrete distributions with applications to some stochastic processes,” J. Appl. Probab. 7, 99–113, (1970).

    Google Scholar 

  5. Anick, D., Mitra, D., and Sondhi, M.M., “Stochastic theory of a data-handling system with multiple sources,” Bell System Tech. J. 61, 1871–1894, (1982).

    Google Scholar 

  6. Asmussen, S., “Conditioned limit theorems relating a random walk to its associate, with applications to risk reserve processes and the GI/G/1 queue,” Adv. Appl. Probab. 14, 143–170, (1982).

    Google Scholar 

  7. Asmussen, S., Applied Probability and Queues, John Wiley & Sons, Chichester New York, 1987.

    Google Scholar 

  8. Asmussen, S., “Phase-type representations in random walk and queueing problems,” Ann. Probab. 20, 772–789, (1992).

    Google Scholar 

  9. Asmussen, S., “Busy period analysis, rare events and transient behavior in fluid flow models,” Journal of Applied Mathematics and Stochastic Analysis 7, 269–299, (1994).

    Google Scholar 

  10. Asmussen, S., “Stationary distributions for fluid flow models with or without Brownian noise,” Stochastic Models 11, 21–49, (1995a).

    Google Scholar 

  11. Asmussen, S., “Stationary distributions via first passage times,” Advances in Queueing: Models, Methods & Problems, (J. Dshalalow ed.), 79–102, CRC Press, Boca Raton, Florida, (1995b).

    Google Scholar 

  12. Asmussen, S., “Rare events in the presence of heavy tails,” In Stochastic Networks: Rare Events and Stability, (P. Glasserman, K. Sigman, D. Yao eds.), pp. 197–214, Springer-Verlag, 1996.

  13. Asmussen, S., “A probabilistic look at the Wiener-Hopf equation,” SIAM Review 40, 189–201, (1998).

    Google Scholar 

  14. Asmussen, S., “Subexponential asymptotics for stochastic processes: extremal behavior, stationary distributions and first passage probabilities,” Ann. Appl. Probab. 8, 354–374, (1998).

    Google Scholar 

  15. Asmussen, S. and Nielsen, H.M., “Ruin probabilities via local adjustment coefficients,” J. Appl. Probab. 32, 736–755, (1995).

    Google Scholar 

  16. Asmussen, S. and Perry, D., “On cycle maxima, first passage problems and extreme value theory for queues,” Stochastic Models 8, 421–458, (1992).

    Google Scholar 

  17. Asmussen, S. and Rubinstein, R.Y., “Steady-state rare events simulation and its complexity properties,” Advances in Queueing: Models, Methods & Problems (J. Dshalalow ed.), 429–466, CRC Press, Boca Raton, Florida, 1995.

    Google Scholar 

  18. Barndorff-Nielsen, O., “On the limiting distribution of the maximum of a random number of random variables,” Acta Math. Acad. Sci. Hungar. 15, 399–403, (1964).

    Google Scholar 

  19. Berger, A.W. and Whitt, W., “Maximum values in queueing processes,” Prob. Eng. Inf. Sci. 9, 375–409, (1995).

    Google Scholar 

  20. Berman, S.M., “Limit distribution of the maximum term in a sequence of dependent random variables,” Ann. Math. Statist. 33, 894–908, (1962).

    Google Scholar 

  21. Bingham, N.H., Goldie, C.M., and Teugels, J.L., “Encyclopedia of Mathematics and its Applications,” Regular Variation 27, Cambridge University Press, (1987).

  22. Chang, K.-H., “Extreme and high-level sojourns of the single server queue in heavy traffic,” Queueing Systems 27, 17–35, (1997).

    Google Scholar 

  23. Chistyakov, V.P., “A theorem on sums of independent random variables and its application to branching random processes,” Th. Prob. Appl. 9, 640–648, (1964).

    Google Scholar 

  24. Cinlar, E., “Markov additive processes,” I. Z. Wahrscheinlichkeitsth. verw. Geb. 24, 85–95, (1972).

    Google Scholar 

  25. Cohen, J.W., “Extreme value distributions for the M/G/1 and GI/M/1 queueing systems,” Ann. Inst. Henri Poincaré Ser. B IV, 83–98, (1968).

    Google Scholar 

  26. Cohen, J.W., “On the tail of the stationary waiting time distribution and limit theorems for the M/G/1 queue,” Ann. Inst. Henri Poincaré VIII, 255–263, (1972a).

    Google Scholar 

  27. Cohen, J.W., “The supremum of the actual and virtual waiting times during a busy cycle of the K m/Kn/1 queueing system,” Adv. Appl. Probab. 4, 339–356, (1972b).

    Google Scholar 

  28. Cohen, J.W., “Lecture Notes in Economics and Mathematical Systems,” On Regenerative Processes in Queueing Theory 121, Springer, Berlin Heidelberg New York, 1982.

    Google Scholar 

  29. Cohen, J.W., The Single Server Queue, (2nd ed.), North-Holland, Amsterdam, 1982.

  30. Durrett, R., Probability: Theory and Examples, Duxbury Press, 1991.

  31. Embrechts, P., Klüppelberg, C., and Mikosch, T., Extremal Events in Finance and Insurance, Springer-Verlag, 1997.

  32. Embrechts, P. and Veraverbeke, N., “Estimates for the probability of ruin with special emphasis on the possibility of large claims,” Insurance: Mathematics and Economics 1, 55–72, (1982).

    Google Scholar 

  33. Feller, W., An Introduction to Probability Theory and its Applications, II (2nd ed.), Wiley, New York, 1971.

    Google Scholar 

  34. Glasserman, P. and Kou, S.-G., “Limits of first passage times to rare sets in regenerative processes” Ann. Appl. Probab. 5, 424–445, (1995).

    Google Scholar 

  35. Glynn, P.W., “Diffusion approximations,” Handbook of Operations Research (1993).

  36. Gnedenko, B.V. and Kovalenko, I.N., Introduction to Queueing Theory, (2nd ed.), Birkhäuser, Basel, 1989.

    Google Scholar 

  37. Goldie, C.M. and Klüppelberg, C., “Subexponential distributions,” In A Practical Guide to Heavy Tails: Statistical Techniques for Analysing Heavy Tails (R. Adler, R. Feldman and M.S. Taqqu, eds.), 435–459, Birkhäuser, Basel, 1998.

    Google Scholar 

  38. Goldie, C.M. and Resnick, S.I., Distributions that are both subexponential and in the domain of attraction of an extreme-value distribution,” J. Appl. Probab. 20, 706–718, (1988).

    Google Scholar 

  39. Heath, D., Resnick, S., and Samorodnitsky, G., “Patterns of buffer overflow in a class of queues with long memory in the input stream,” Ann. Appl. Probab. 7, 1021–1057, (1997).

    Google Scholar 

  40. Heathcote, C.R., “On the maximum of the queue GI/M/1,” J. Appl. Probab. 2, 206–214, (1965).

    Google Scholar 

  41. Heyde, C.C., “On the growth of the maximum queue length in a stable queue,” Oper. Res. 19, 447–452, (1970).

    Google Scholar 

  42. Hooghiemstra, G. and Meester, L.E., “The extremal index in 10 seconds,” J. Appl. Probab. 34, 818–822, (1997).

    Google Scholar 

  43. Iglehart, D.L., “Extreme values in the GI/G/1 queue,” Ann. Math. Statist. 43, 627–635, (1972).

    Google Scholar 

  44. Iglehart, D.L. and Stone, M.L., “Regenerative simulation for estimating extreme values,” Opns. Res. 31, 1145–1166, (1983).

    Google Scholar 

  45. Keilson, J., “A limit theorem for passage times in ergodic regenerative processes,” Ann. Math. Statist. 37, 866–870, (1966).

    Google Scholar 

  46. Keilson, J., Markov Chain Models - Rarity and Exponentiality, Springer-Verlag, 1979.

  47. Leadbetter, M.R., Lindgren, G., and Rootzén, H., Extremes and Related Properties of Random Sequences and Processes, Springer-Verlag, 1983.

  48. Louchard, G., Kenynon, C., and Schott, R., “Data structures maxima,” SIAM J. Comp. 26, 1006–1019, (1997).

    Google Scholar 

  49. McCormick, W.P. and Park, Y.S., “Approximating the distribution of the maximum queue length for M/M/s queues. In Queueing and Related Models (V.N. Bhat and I.W. Basawa, eds.), 240–261, Oxford University Press, 1992.

  50. Møller, J.R., PhD Dissertation in preparation, Department of Mathematical Statistics, University of Lund, 1998.

  51. Neuts, M.F., “A versatile Markovian point process,” J. Appl. Probab. 16, 764–779, (1979).

    Google Scholar 

  52. Neuts, M.F., Matrix-Geometric Solutions in Stochastic Models, Johns Hopkins University Press, Baltimore, 1981.

    Google Scholar 

  53. Neuts, M.F., Structured Stochastic Matrices of the M/G/1 Type and Their Applications, Marcel Dekker, New York, 1989.

    Google Scholar 

  54. Rogers, L.C.G., “Fluid models in queueing theory and Wiener-Hopf factorisation of Markov chains,” Ann. Appl. Probab. 4, 390–413, (1994).

    Google Scholar 

  55. Rootzén, H., “Maxima and exceedances of stationary Markov chains,” Adv. Appl. Probab. 20, 371–390, (1988).

    Google Scholar 

  56. Sadowsky, J. and Szpankowski, W., “Maximum queue length and waiting time revisited: GI/G/c queue,” Prob. Eng. Inf. Sci. 6, 157–170, (1995).

    Google Scholar 

  57. Serfozo, R.F., “Extreme values of birth and death processes and queues,” Stoch. Proc. Appl. 27, 291–306, (1988a).

    Google Scholar 

  58. Serfozo, R.F., “Extreme values of queue lengths in M/G/1 and GI/M/1 systems,” Math. Oper. Res. 13, 349–357, (1988b).

    Google Scholar 

  59. Sigman, K., “Queues as Harris recurrent Markov chains,” Queueing Systems 3, 179–198, (1988).

    Google Scholar 

  60. Szpankowski, W., “On the maximum queue length with applications to data structure analysis,” Proceedings of the 27th Annual Allerton Conference on Communication, Control and Computing, University of Illinois, Urbana-Champaign,, 263–272, (1989).

  61. Smith, R.L., “The extremal index for a Markov chain,” J. Appl. Probab. 29, 37–45, (1992).

    Google Scholar 

  62. Smith, W.L., “Distribution of queueing times,” Proc. Cambridge Philos. Soc 49, 449–461, (1953).

    Google Scholar 

  63. Sundt, B. and Teugels, J.L., “Ruin estimates under interest force,” Insurance: Mathematics and Economics 16, 7–22, (1995).

    Google Scholar 

  64. Sundt, B. and Teugels, J.L., “The adjustment coefficient in ruin estimates under interest force,” Insurance: Mathematics and Economics (1996).

  65. Takács, L., “Application of Ballot theorems in the theory of queues,” Proc. Symp. on Congestion Theory, (W.L. Smith and W.E. Wilkinson, eds.), University of North Carolina Press, Chapel Hill, 1965.

    Google Scholar 

  66. Walrand, J., An Introduction to Queueing Networks, Prentice-Hall, Englewood Cliffs, N.J., 1988.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Statistics, Lund University, Box 118, 221 00, Lund, Sweden

    Søren Asmussen

Authors
  1. Søren Asmussen
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Asmussen, S. Extreme Value Theory for Queues Via Cycle Maxima. Extremes 1, 137–168 (1998). https://doi.org/10.1023/A:1009970005784

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1009970005784

Search

Navigation