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Control and recovery from rare congestion events in a large multi-server system

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Abstract

We develop deterministic fluid approximations to describe the recovery from rare congestion events in a large multi-server system in which customer holding times have a general distribution. There are two cases, depending on whether or not we exploit the age distribution (the distribution of elapsed holding times of customers in service). If we do not exploit the age distribution, then the rare congestion event is a large number of customers present. If we do exploit the age distribution, then the rare event is an unusual age distribution, possibly accompanied by a large number of customers present. As an approximation, we represent the large multi-server system as an M/G/∞ model. We prove that, under regularity conditions, the fluid approximations are asymptotically correct as the arrival rate increases. The fluid approximations show the impact upon the recovery time of the holding-time distribution beyond its mean. The recovery time may or not be affected by the holding-time distribution having a long tail, depending on the precise definition of recovery. The fluid approximations can be used to analyze various overload control schemes, such as reducing the arrival rate or interrupting services in progress. We also establish large deviations principles to show that the two kinds of rare events have the same exponentially small order. We give numerical examples showing the effect of the holding-time distribution and the age distribution, focusing especially on the consequences of long-tail distributions.

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References

  1. J. Abate, G.L. Choudhury and W. Whitt, Waiting-time tail probabilities in queues with long-tail service-time distributions, Queueing Systems 16 (1994) 311–338.

    Article  Google Scholar 

  2. J. Abate and W. Whitt, Calculating transient characteristics of the Erlang loss model by numerical transform inversion, Stochastic Models, to appear.

  3. F. Baccelli and P. Brémaud, Elements of Queueing Theory (Springer, New York, 1994).

    Google Scholar 

  4. R.E. Barlow and F. Proschan, Statistical Theory of Reliability and Life Testing (Holt, Rinehart and Winston, New York, 1975).

  5. P. Billingsley, Convergence of Probability Measures (Wiley, New York, 1968).

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

  7. A.A. Borovkov, Asymptotic Methods in Queueing Theory (Wiley, New York, 1984).

  8. R. Cáceres, P.B. Danzig, S. Jamin and D.J. Mitzel, Characteristics of wide-area TCP/IP conversations, Computer Communication Review 21 (1991) 101–112.

    Article  Google Scholar 

  9. H. Chen and A. Mandelbaum, Discrete flow networks: bottleneck analysis and fluid approximation, Math. Oper. Res. 16 (1991) 408–446.

    Google Scholar 

  10. G.L. Choudhury, K.K. Leung and W. Whitt, An inversion algorithm to compute blocking probabilities in loss networks with state-dependent rates, IEEE/ACM Trans. Networking 3 (1995) 585–601.

    Article  Google Scholar 

  11. G.L. Choudhury, A. Mandelbaum, M.I. Reiman and W. Whitt, Fluid and diffusion limits for queues in slowly changing environments, Stochastic Models 13 (1997) 121–146.

    Google Scholar 

  12. M.E. Crovella and A. Bestavros, Self-similarity in World Wide Web traffic - evidence and possible causes, in: Proc. Sigmetrics '96 (1996) pp. 160–169.

  13. J.L. Davis, W.A. Massey and W. Whitt, Sensitivity to the service-time distribution in the nonstationary Erlang loss model, Management Sci. 41 (1995) 1107–1116.

    Google Scholar 

  14. A. Dembo and O. Zeitouni, Large Deviation Techniques and Applications (Jones and Bartlett, Boston, 1993).

    Google Scholar 

  15. N.G. Duffield, Conditioned asymptotics for tail probabilities in large multiplexers, Performance Evaluation, to appear.

  16. S.G. Eick, W.A. Massey and W. Whitt, The physics of the Mt=G=∞ queue, Oper. Res. 41 (1993) 731–742.

    Article  Google Scholar 

  17. J.A. Erdelyi, Asymptotic Expansions (Dover, New York, 1956).

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

  19. P.G. Glynn and W. Whitt, A new view of the heavy-traffic limit for infinite-server queues, Adv. Appl. Probab. 23 (1991) 188–209.

    Article  Google Scholar 

  20. A.G. Greenberg, R. Srikant and W. Whitt, Resource sharing for book-ahead and instantaneous-request calls. AT&T Laboratories (1996), submitted.

  21. T. Kamae, U. Krengel and G.L. O'Brien, Stochastic inequalities on partially ordered spaces, Ann. Probab. 5 (1977) 899–912.

    Google Scholar 

  22. E.V. Krichagina and A.A. Puhalskii, An asymptotic analysis of a closed queueing system with a GI/℞ service center, Institute for Problems in Information Transmission, Moscow, 1995.

    Google Scholar 

  23. S. Kullback, Information Theory and Statistics (Wiley, New York, 1959).

  24. W.E. Leland, M.S. Taqqu, W. Willinger and D.V. Wilson, On the self-similar nature of Ethernet traffic, IEEE/ACM Trans. Networking 2 (1994) 1–15.

    Article  Google Scholar 

  25. K.K. Leung, W.A. Massey and W. Whitt, Traffic models for wireless communication networks, IEEE J. Sel. Areas Commun. 12 (1994) 1353–1364.

    Article  Google Scholar 

  26. T. Lindvall, Lectures on the Coupling Method (Wiley, New York, 1992).

  27. W.A. Massey and W. Whitt, Networks of infinite-server queues with nonstationary Poisson input, Queueing Systems 13 (1993) 183–250.

    Article  Google Scholar 

  28. V. Paxson, Empirically derived analytical models of wide-area TCP connections, IEEE/ACM Trans. Networking 2 (1994) 316–336.

    Article  Google Scholar 

  29. D. Stoyan, Comparison Methods for Queues and Other Stochastic Models (Wiley, New York, 1983).

  30. W. Whitt, The renewal-process stationary-excess operator, J. Appl. Probab. 22 (1985) 156–167.

    Article  Google Scholar 

  31. W. Willinger, M.S. Taqqu, R. Sherman and D.V. Wilson, Self-similarity through high variability: statistical analysis of Ethernet LAN traffic at the source level, in: SIGCOMM Symp. on Commun. Arch. and Protocols (1995) pp. 100–113.

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Duffield, N., Whitt, W. Control and recovery from rare congestion events in a large multi-server system. Queueing Systems 26, 69–104 (1997). https://doi.org/10.1023/A:1019168821588

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