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General conditions for bounded relative error in simulations of highly reliable Markovian systems

Part of: Markov processes Probabilistic methods, simulation and stochastic differential equations Special processes

Published online by Cambridge University Press:  01 July 2016

Marvin K. Nakayama
Marvin K. Nakayama*
Affiliation:
New Jersey Institute of Technology
*
∗ Postal address: Department of Computer and Information Science, New Jersey Institute of Technology, Newark, NJ 07102, USA.
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Abstract

We establish a necessary condition for any importance sampling scheme to give bounded relative error when estimating a performance measure of a highly reliable Markovian system. Also, a class of importance sampling methods is defined for which we prove a necessary and sufficient condition for bounded relative error for the performance measure estimator. This class of probability measures includes all of the currently existing failure biasing methods in the literature. Similar conditions for derivative estimators are established.

Keywords

SIMULATIONIMPORTANCE SAMPLINGLIKELIHOOD RATIOSGRADIENT ESTIMATIONRELIABILITYMARKOV CHAINS

MSC classification

Primary: 65C05: Monte Carlo methods
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60K10: Applications (reliability, demand theory, etc.)
Type
General Applied Probablity
Information
Advances in Applied Probability , Volume 28 , Issue 3 , September 1996 , pp. 687 - 727
Copyright
Copyright © Applied Probability Trust 1996 

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References

[1] Carrasco, J. A. (1992) Failure distance based simulation of repairable fault tolerant systems. In Proc. 5th Int. Conf. on Modelling Techniques and Tools for Computer Performance Evaluation. ed. Balbo, G. and Serazzi, G.. Elsevier, Amsterdam. pp. 351–365.Google Scholar
[2] Carrasco, J. A. (1991) Efficient transient simulation of failure/repair Markovian models. In Proc. Tenth Symp. on Reliable Distributed Systems. IEEE Press, New York. pp. 152–161.Google Scholar
[3] Chang, C. S., Heidelberger, P., Juneja, S. and Shahabuddin, P. (1994) Effective bandwidth and fast simulation of ATM intree networks. Perf. Eval. 20, 45–66.Google Scholar
[4] Conway, A. E. and Goyal, A. (1987) Monte Carlo simulation of computer system availability/reliability models. In Proc. 17th lnt. Symp. on Fault Tolerant Computing. IEEE Computer Society Press, Los Alamitos, CA. pp 230–235.Google Scholar
[5] Cottrell, M., Fort, J. C. and Malgouyres, G. (1983) Large deviations and rare events in the study of stochastic algorithms. IEEE Trans. Autom. Control AC-28, 907–920.Google Scholar
[6] Glasserman, P. (1990) Gradient Estimation Via Perturbation Analysis. Kluwer, Norwell, MA.Google Scholar
[7] Glynn, P. W. (1990) Likelihood ratio derivative estimators for stochastic systems. Commun. ACM 33, 75–84.Google Scholar
[8] Glynn, P. W. and Iglehart, D. L. (1989) Importance sampling for stochastic simulations. Management Sci. 35, 1367–1393.CrossRefGoogle Scholar
[9] Goyal, A., Heidelberger, P. and Shahabuddin, P. (1987) Measure specific dynamic importance sampling for availability simulations. In Proc. 1987 Winter Simulation Conf. ed. Thesen, A., Grant, H. and Kelton, W. D.. IEEE Press, New York. pp 351–357.Google Scholar
[10] Goyal, A. and Lavenberg, S. S. (1987) Modeling and analysis of computer system availability. IBM J. Res. Develop. 31, 651–664.Google Scholar
[11] Goyal, A., Shahabuddin, P., Heidelberger, P., Nicola, V. F. and Glynn, P. W. (1992) A unified framework for simulating Markovian models of highly dependable systems. IEEE Trans. Comput. C-41, 36–51.CrossRefGoogle Scholar
[12] Hammersley, J. M. and Handscomb, D. C. (1964) Monte Carlo Methods. Methuen, London.Google Scholar
[13] Heidelberger, P., Nicola, V. F. and Shahabuddin, P. (1992) Simultaneous and efficient simulation of highly dependable systems with different underlying distributions. In Proc. 1992 Winter Simulation Conf. ed. Swain, J. J., Goldsman, D., Crain, R. C. and Wilson, J. R.. pp. 458–465.CrossRefGoogle Scholar
[14] Heidelberger, P., Shahabuddin, P. and Nicola, V. F. (1994) Bounded relative error in estimating transient measures of highly dependable non-Markovian systems. ACM Trans. Mod. Comput. Simul. 4, 137–164.CrossRefGoogle Scholar
[15] Juneja, S. and Shahabuddin, P. (1992) Fast simulation of Markovian reliability/availability models with general repair policies. In Proc. 22nd Int. Symp. on Fault Tolerant Computing. IEEE Computer Society Press, Los Alamitos, CA. pp 150–159.Google Scholar
[16] L'Ecuyer, P. (1990) A unified view of the IPA, SF, and LR gradient estimation techniques. Management Sci. 36, 1364–1383.Google Scholar
[17] Lewis, E. E. and Böhm, F. (1984) Monte Carlo simulation of Markov unreliability models. Nucl. Eng. Design 77, 49–62.Google Scholar
[18] Nakayama, M. K. (1994) A characterization of the simple failure biasing method for simulations of highly reliable Markovian systems. ACM Trans. Model. Comput. Simul. 4, 52–88.Google Scholar
[19] Nakayama, M. K. (1995) Asymptotics of likelihood ratio derivative estimators in simulations of highly reliable Markovian systems. Management Sci. 41, 524–554.CrossRefGoogle Scholar
[20] Nakayama, M. K., Goyal, A. and Glynn, P. W. (1994) Likelihood ratio sensitivity analysis for Markovian models of highly dependable systems. Operat. Res. 42, 137–157.Google Scholar
[21] Nicola, V. F., Heidelberger, P. and Shahabuddin, P. (1992) Uniformization and exponential transformation: Techniques for fast simulation of highly dependable non-Markovian systems. In Proc. 22nd Int. Symp. on Fault-Tolerant Computing. IEEE Computer Society Press, Los Alamitos, CA. pp 130–139.Google Scholar
[22] Nicola, V. F., Nakayama, M. K., Heidelberger, P. and Goyal, A. (1993) Fast simulation of highly dependable systems with general failure and repair processes. IEEE Trans. Comput. 42, 1440–1452.Google Scholar
[23] Nicola, V. F., Shahabuddin, P., Heidelberger, P. and Glynn, P. W. (1993) Fast simulation of steady-state availability in non-Markovian highly dependable systems. In Proc. 23rd Int. Symp. on Fault-Tolerant Computing. IEEE Computer Society Press, Los Alamitos, CA. pp. 38–47.Google Scholar
[24] Reiman, M. I. and Weiss, A. (1989) Sensitivity analysis for simulations via likelihood ratios. Operat. Res. 37, 830–844.Google Scholar
[25] Shahabuddin, P. (1994) Importance sampling for the simulation of highly reliable Markovian systems. Management Sci. 40, 333–352.Google Scholar
[26] Shahabuddin, P. (1994) Fast transient simulation of Markovian models of highly dependable systems. Perf. Eval. 20, 267–286.Google Scholar
[27] Shahabuddin, P. and Nakayama, M. K. (1993) Estimation of reliability and its derivatives for large time horizons in Markovian systems. In Proc. 1993 Winter Simulation Conf. ed. Evans, G. W., Mollaghasemi, M., Russell, E. C. and Biles, W. E.. IEEE Press, New York. pp 422–429.Google Scholar
[28] Shahabuddin, P., Nicola, V. F., Heidelberger, P., Goyal, A. and Glynn, P. W. (1988) Variance reduction in mean time to failure simulations. In Proc. 1988 Winter Simulation Conf. ed Abrams, M. A., Haigh, P. L. and Comfort, J. C.. IEEE Press, New York. pp. 491–499.Google Scholar