Bounded relative error in estimating transient measures of highly dependable non-Markovian systems

This article deals with fast simulation techniques for estimating transient measures in highly dependable systems. The systems we consider of components with generally distributed lifetimes and repair times, with complex interaction among components. As is well known, standard simulation of highly dependable systems is very inefficient, and importance-sampling is widely used to improve efficiency. We present two new techniques, one of which is based on the uniformization approach to simulation, and the other is a natural extension of the uniformization approach which we call exponential transformation. We show that under certain assumptions, these techniques have the bounded relative error property, i.e., the relative error of the simulation estimate remains bounded as components become more and more reliable, unlike standard simulation in which it tends to infinity. This implies that only a fixed number of observations are required to achieve a given relative error, no matter how rare the failure events are.

[1]  Perwez Shahabuddin,et al.  Fast Transient Simulation of Markovian Models of Highly Dependable Systems , 1994, Perform. Evaluation.

[2]  Michael R. Frater,et al.  Optimally efficient estimation of the statistics of rare events in queueing networks , 1991 .

[3]  Mark Brown.  Error bounds for exponential approximations of geometric convolutions , 1990 .

[4]  J. George Shanthikumar.  Uniformization and Hybrid Simulation/Analytic Models of Renewal Processes , 1986, Oper. Res..

[5]  Kishor S. Trivedi,et al.  Ultrahigh Reliability Prediction for Fault-Tolerant Computer Systems , 1983, IEEE Transactions on Computers.

[6]  Marvin K. Nakayama.  A characterization of the simple failure-biasing method for simulations of highly reliable Markovian Systems , 1994, TOMC.

[7]  Kishor S. Trivedi,et al.  The hybrid automated reliability predictor , 1986 .

[8]  P. Glynn.  Importance sampling for markov chains: asymptotics for the variance , 1994 .

[9]  G. Shedler,et al.  Simulation of Nonhomogeneous Poisson Processes by Thinning , 1979 .

[10]  Boudewijn R. Haverkort,et al.  Fault Injection Simulation: A Variance Reduction Technique for Systems with Rare Events , 1992 .

[11]  Marvin K. Nakayama,et al.  General conditions for bounded relative error in simulations of highly reliable Markovian systems , 1996, Advances in Applied Probability.

[12]  Philip Heidelberger,et al.  Simultaneous and efficient simulation of highly dependable systems with different underlying distributions , 1992, WSC '92.

[13]  Richard E. Barlow,et al.  Statistical Theory of Reliability and Life Testing: Probability Models , 1976 .

[14]  Philip Heidelberger,et al.  Fast simulation of rare events in queueing and reliability models , 1993, TOMC.

[15]  Donald Gross,et al.  The Randomization Technique as a Modeling Tool and Solution Procedure for Transient Markov Processes , 1984, Oper. Res..

[16]  V. F. Nicola,et al.  Techniques for Fast Simulation of Highly Dependable Systems , 1993 .

[17]  Stephen S. Lavenberg,et al.  Modeling and Analysis of Computer System Availability , 1987, Computer Performance and Reliability.

[18]  P. Glynn,et al.  Discrete-time conversion for simulating finite-horizon Markov processes , 1990 .

[19]  Philip Heidelberger,et al.  A Unified Framework for Simulating Markovian Models of Highly Dependable Systems , 1992, IEEE Trans. Computers.

[20]  Juan A. Carrasco.  Efficient transient simulation of failure/repair Markovian models , 1991, [1991] Proceedings Tenth Symposium on Reliable Distributed Systems.

[21]  I. Gertsbakh.  Asymptotic methods in reliability theory: a review , 1984, Advances in Applied Probability.

[22]  Philip Heidelberger,et al.  Uniformization and exponential transformation: Techniques for fast simulation of highly dependable non-Markovian systems , 1992, [1992] Digest of Papers. FTCS-22: The Twenty-Second International Symposium on Fault-Tolerant Computing.

[23]  Donald L. Iglehart,et al.  Importance sampling for stochastic simulations , 1989 .

[24]  Philip Heidelberger,et al.  Fast simulation of dependability models with general failure, repair and maintenance processes , 1990, [1990] Digest of Papers. Fault-Tolerant Computing: 20th International Symposium.

[25]  Perwez Shahabuddin,et al.  Importance sampling for the simulation of highly reliable Markovian systems , 1994 .

[26]  J. Sadowsky.  Large deviations theory and efficient simulation of excessive backlogs in a GI/GI/m queue , 1991 .

[27]  Jean Walrand,et al.  A quick simulation method for excessive backlogs in networks of queues , 1989 .

[28]  N. Dijk.  On a simple proof of uniformization for continuous and discrete-state continuous-time Markov chains , 1990 .

[29]  Philip Heidelberger,et al.  Fast simulation of steady-state availability in non-Markovian highly dependable systems , 1993, FTCS-23 The Twenty-Third International Symposium on Fault-Tolerant Computing.

[30]  P. Shahabuddin,et al.  Estimation of reliability and its derivatives for large time horizons in Markovian systems , 1993, WSC '93.

[31]  P. Glynn.  A GSMP formalism for discrete event systems , 1989, Proc. IEEE.

[32]  Elmer E Lewis,et al.  Monte Carlo simulation of Markov unreliability models , 1984 .

[33]  Juan A. Carrasco.  Failure distance-based simulation of repairable fault-tolerant systems , 1992 .

[34]  M. Nakayama.  Asymptotics of likelihood ratio derivative estimators in simulations of highly reliable Markovian systems , 1995 .

[35]  J. Keilson.  Markov Chain Models--Rarity And Exponentiality , 1979 .

[36]  Robert Geist,et al.  Ultrahigh reliability estimates through simulation , 1989, Proceedings., Annual Reliability and Maintainability Symposium.

[37]  J. Hammersley,et al.  Monte Carlo Methods , 1965 .

[38]  Peter W. Glynn,et al.  Simulation and analysis of highly reliable systems , 1990 .

[39]  Sandeep Juneja,et al.  Fast simulation of Markovian reliability/availability models with general repair policies , 1992, [1992] Digest of Papers. FTCS-22: The Twenty-Second International Symposium on Fault-Tolerant Computing.

[40]  Philip Heidelberger,et al.  Efficient estimation of the mean time between failures in non-regenerative dependability models , 1993, WSC '93.