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Induced rare events: analysis via large deviations and time reversal

Part of: Special processes

Published online by Cambridge University Press:  01 July 2016

Adam Shwartz  and
Alan Weiss
Adam Shwartz*
Affiliation:
Technion-Israel Institute of Technology
Alan Weiss*
Affiliation:
AT&T Bell Laboratories, Murray Hill
*
∗ Postal address: Electrical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel.
∗∗ Postal address: AT&T Bell Laboratories, 600 Mountain Avenue, Room 2C-118, Murray Hill, NJ 07974, USA.
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Abstract

When a subsystem goes into an infrequent state, how does the remainder of the system behave? We show how to calculate the relevant distributions using the notions of reversed time for Markov processes and large deviations. For ease of exposition, most of the work deals with a specific queueing model due to Flatto, Hahn, and Wright. However, we show how the theorems may be applied to much more general jump-Markov systems.

We also show how the tools of time-reversal and large deviations complement each other to yield general theorems. We show that the way a constant coefficient process approaches a rare event is roughly by following the path of another constant coefficient process. We also obtain some properties, including apriori bounds, for the change of measure associated with some large deviations functionals; these are useful for accelerating simulations.

Keywords

QUEUEING SYSTEMSINFREQUENT STATESJUMP-MARKOV SYSTEMS

MSC classification

Primary: 60K25: Queueing theory
Type
Research Article
Information
Advances in Applied Probability , Volume 25 , Issue 3 , September 1993 , pp. 667 - 689
Copyright
Copyright © Applied Probability Trust 1993 

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References

[1] Anantharam, V. (1988) How large delays build up in a GI/G/1 queue. QUESTA 5, 345–368.Google Scholar
[2] Anantharam, V., Heidelberger, P. and Tsoucas, P. (1990) Analysis of rare events in continuous time Markov chains via time reversal and fluid approximation. IBM Research Report #RC 16280.Google Scholar
[3] Bucklew, J. A. (1990) Large Deviations Techniques in Decision, Simulation and Estimation. Wiley Interscience, New York.Google Scholar
[4] Chernoff, H. (1952) A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Ann. Math. Statist. 23, 493–507.Google Scholar
[5] Donsker, M. D. and Varadhan, S. R. S. (1983) Asymptotic evaluation of certain Markov process expectations for large time, IV. Commun. Pure Appl. Math. 36, 183–212.Google Scholar
[6] Dupuis, P., Ellis, R. and Weiss, A. (1991) Large deviations for Markov processes with discontinuous statistics, I: general upper bounds. Ann. Prob. 19, 1280–1297.CrossRefGoogle Scholar
[7] Dupuis, P., Ishii, H. and Soner, H. M. (1990) A viscosity solution approach to the asymptotic analysis of queueing systems. Ann. Prob. 18, 226–255.CrossRefGoogle Scholar
[8] Ellis, R. and Weiss, A. Unpublished manuscript.Google Scholar
[9] Ethier, S. N. and Kurtz, T. G. (1986) Markov Processes: Characterization and Convergence. Wiley, New York.Google Scholar
[10] Flatto, L. and Hahn, S. (1984) Two parallel queues created by arrivals with two demands I. SIAM J. Appl. Math. 44, 1041–1052.Google Scholar
[11] Frater, M. R., Kennedy, R. A. and Anderson, B. D. O. (1989) Reverse-time modeling, optimal control and large deviations. Syst. Control. Lett. 12, 351–356.CrossRefGoogle Scholar
[12] Harrison, J. M. and Reiman, M. I. (1981) Reflected Brownian motion on an orthant. Ann. Prob. 9, 302–308.CrossRefGoogle Scholar
[13] Kelly, F. P. (1979) Reversibility and Stochastic Networks. Wiley, New York.Google Scholar
[14] Kurtz, T. G. (1971) Limit theorem for sequences of jump Markov processes approximating ordinary differential processes. J. Appl. Prob. 8, 344–356.Google Scholar
[15] Parekh, S. and Walrand, J. (1989) Quick simulation method for excessive backlogs in networks of queues. IEEE Trans. Autom. Control 34, 54–66.Google Scholar
[16] Prabhu, N. U. (1980) Stochastic Storage Processes. Springer-Verlag, New York.Google Scholar
[17] Shwartz, A. and Weiss, A. Large Deviations for Performance Analysis. Book in preparation.Google Scholar
[18] Stroock, D. W. and Varadhan, S. R. S. (1979) Multidimensional Diffusion Processes. Springer-Verlag, New York.Google Scholar
[19] Wright, P. E. (1992) Two parallel processors with coupled inputs. Adv. Appl. Prob. 24, 986–1007.Google Scholar