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A new technique for analyzing large traffic systems

Published online by Cambridge University Press:  01 July 2016

Alan Weiss
Alan Weiss*
Affiliation:
AT&T Bell Laboratories
*
∗Postal address: AT&T Bell Laboratories, Murray Hill, NJ 07974, USA.
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Abstract

This paper presents a new technique for analyzing the frequency of a very large class of rare events in large traffic systems. The method is based on the theory of large deviations. If n is a large parameter, typically the number of potential traffic sources, then where I is the solution to an associated variational problem. We present a new analysis of a previously solved system as well as an analysis of a previously intractable system. As by-products of our analysis, we obtain estimates of the transient behavior of the system, and show how they may be used in analyzing some flow control schemes.

Keywords

QUEUINGLARGE DEVIATIONSEXTREME VALUES
Type
Research Article
Information
Advances in Applied Probability , Volume 18 , Issue 2 , June 1986 , pp. 506 - 532
Copyright
Copyright © Applied Probability Trust 1986 

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References

1. Anick, D., Mitra, D. and Sondhi, M. M. (1982) Stochastic theory of a data-handling system and multiple sources. Bell Syst. Tech. J. 61, 1871–1894.CrossRefGoogle Scholar
2. Chernoff, H. (1952) A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Ann. Math. Statist. 23, 494–507.CrossRefGoogle Scholar
3. Cottrell, M., Fort, J.-C. and Malgouyres, G. (1983) Large deviations and rare events in the study of stochastic algorithms. IEEE Trans. Autom. Control 28, 907–920.CrossRefGoogle Scholar
4. Elsgolc, L. E. (1962) Calculus of Variations. Addison-Wesley, Reading, Mass.Google Scholar
5. Kosten, L. (1974) Stochastic theory of a multi-entry buffer (1). Delft Progr. Report 1, 10–18.Google Scholar
6. Kurtz, T. (1978) Strong approximation theorems for density dependent Markov chains. Stoch. Proc. Appl. 6, 223–240.CrossRefGoogle Scholar
7. Newell, G. (1971) Applications of Queueing Theory. Chapman and Hall, London.Google Scholar
8. Ventsel’, A. D. (1976) Rough limit theorems on large deviations for Markov stochastic processes I. Theory Prob. Appl. 21, 227–242.CrossRefGoogle Scholar
9. Ventsel’, A. D. (1976) Rough limit theorems on large deviations for Markov stochastic processes II. Theory Prob. Appl. 21, 499–512.CrossRefGoogle Scholar
10. Ventsel’, A. D. and Freidlin, M. I. (1970) On small random perturbations of dynamical systems. Russ. Math. Surveys 25, 1–55.CrossRefGoogle Scholar
11. Ventsel’, A. D. and Freidlin, M. I. (1984) Random Perturbations of Dynamical Systems. Springer-Verlag, New York.Google Scholar
12. Weiss, A. (1985) The large deviations of a Markov process which models traffic generation.Google Scholar
13. Weinstock, R. (1974) Calculus of Variations. Dover, New York.Google Scholar