Marvin K. Nakayama

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

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.

Techniques for fast simulation of models of highly dependable systems

With the ever-increasing complexity and requirements of highly dependable systems, their evaluation during design and operation is becoming more crucial. Realistic models of such systems are often not amenable to analysis using conventional analytic or numerical methods. Therefore, analysts and designers turn to simulation to evaluate these models. However, accurate estimation of dependability measures of these models requires that the simulation frequently observes system failures, which are rare events in highly dependable systems. This renders ordinary Simulation impractical for evaluating such systems. To overcome this problem, simulation techniques based on importance sampling have been developed, and are very effective in certain settings. When importance sampling works well, simulation run lengths can be reduced by several orders of magnitude when estimating transient as well as steady-state dependability measures. This paper reviews some of the importance-sampling techniques that have been developed in recent years to estimate dependability measures efficiently in Markov and nonMarkov models of highly dependable systems.

Likelihood Ratio Sensitivity Analysis for Markovian Models of Highly Dependable Systems

This paper discusses the application of the likelihood ratio gradient estimator to simulations of large Markovian models of highly dependable systems. Extensive empirical work, as well as some mathematical analysis of small dependability models, suggests that in this model setting the gradient estimators are not significantly more noisy than the estimates of the performance measures themselves. The paper also discusses implementation issues associated with likelihood ratio gradient estimation, as well as some theoretical complements associated with application of the technique to continuous-time Markov chains.

Fast simulation of highly dependable systems with general failure and repair processes

An approach for simulating models of highly dependable systems with general failure and repair time distribution is described. The approach combines importance sampling with event rescheduling in order to obtain variance reductions in such rare event simulations. The approach is general in nature and allows a variety of features commonly arising in dependability modeling to be simulated effectively. It is shown how the technique can be applied to systems with redundant components and/or periodic maintenance. For different failure time distributions, the effect of the maintenance period on the steady-state availability is explored. The amount of component redundancy needed to achieve a certain reliability level is determined. >

Fast simulation of dependability models with general failure, repair and maintenance processes

An approach to simulating models of highly dependable systems with general failure and repair time distributions is described. The approach combines importance sampling with event rescheduling in order to obtain variance reduction in such rare event simulations. The approach is general in nature and allows effective simulation of a variety of features commonly arising in dependability modeling. For example, it is shown how the technique can be applied to systems with periodic maintenance. The effects on the steady-state availability of the maintenance period and of different failure time distributions are explored. Some of the trade-offs involved in the design of specific rescheduling rules are described, and their potential effectiveness in simulations of systems with nonexponential failure and repair time distributions are demonstrated. It is found that an effective method for selecting the rescheduling distribution is to keep the probability of a failure transition in the range between 0.1 and 0.5.<<ETX>>

A characterization of the simple failure-biasing method for simulations of highly reliable Markovian Systems

Simple failure biasing is an importance-sampling technique used to reduce the variance of estimates of performance measures and their gradients in simulations of highly reliable Markovian systems. Although simple failure biasing yields bounded relative error for the performance measure estimate when the system is balanced, it may not provide bounded relative error when the system is unbalanced. In this article, we provide a characterization of when the simple failure-biasing method produces estimators of a performance measure and its derivatives with bounded relative error. We derive a necessary and sufficient condition on the structure of the system for when the performance measure can be estimated with bounded relative error when using simple failure biasing. Furthermore, a similar condition for the derivative estimators is established. One interesting aspect of the conditions is that it shows that to obtain bounded relative error, not only the most likely paths to system failure must be examined but also some secondary paths leading to failure as well. We also show by example that the necessary and sufficient conditions for a derivative estimator do not imply those for the performance measure estimator; i.e., it is possible to estimate a derivative more efficiently than the performance measure when using simple failure biasing.

A Markovian Dependability Model with Cascading Failures

We develop a continuous-time Markov chain model of a dependability system operating in a randomly changing environment and subject to probabilistic cascading failures. A cascading failure can be thought of as a rooted tree. The root is the component whose failure triggers the cascade, its children are those components that the roots failure immediately caused, the next generation are those components whose failures were immediately caused by the failures of the roots children, and so on. The amount of cascading is unlimited. We consider probabilistic cascading in the sense that the failure of a component of type i causes a component of type j to fail simultaneously with a given probability, with all failures in a cascade being mutually independent. Computing the infinitesimal generator matrix of the Markov chain poses significant challenges because of the exponential growth in the number of trees one needs to consider as the number of components failing in the cascade increases. We provide a recursive algorithm generating all possible trees corresponding to a given transition, along with an experimental study of an implementation of the algorithm on two examples. The numerical results highlight the effects of cascading on the dependability of the models.

Confidence intervals for quantiles when applying variance-reduction techniques

Quantiles, which are also known as values-at-risk in finance, frequently arise in practice as measures of risk. This article develops asymptotically valid confidence intervals for quantiles estimated via simulation using variance-reduction techniques (VRTs). We establish our results within a general framework for VRTs, which we show includes importance sampling, stratified sampling, antithetic variates, and control variates. Our method for verifying asymptotic validity is to first demonstrate that a quantile estimator obtained via a VRT within our framework satisfies a Bahadur-Ghosh representation. We then exploit this to show that the quantile estimator obeys a central limit theorem (CLT) and to develop a consistent estimator for the variance constant appearing in the CLT, which enables us to construct a confidence interval. We provide explicit formulae for the estimators for each of the VRTs considered.

Using permutations in regenerative simulations to reduce variance

We propose a new estimator for a large class of performance measures obtained from a regenerative simulation of a system having two distinct sequences of regeneration times. To construct our new estimator, we first generate a sample path of a fixed number of cycles based on one sequence of regeneration times, divide the path into segments based on the second sequence of regeneration times, permute the segments, and calculate the performance on the new path using the first sequence of regeneration times. We average over all possible permutations to construct the new estimator. This strictly reduces variance when the original estimator is not simply an additive functional of the sample path. To use the new estimator in practice, the extra computational effort is not large since all permutations do not actually have to be computed as we derive explicit formulas for our new estimators. We examine the small-sample behavior of our estimators. In particular, we prove that for any fixed number of cycles from the first regenerative sequence, our new estimator has smaller mean squared error than the standard estimator. We show explicitly that our method can be used to derive new estimators for the expected cumulative reward until a certain set of states is hit and the time-average variance parameter of a regenerative simulation.

Statistical analysis of simulation output

We discuss methods for statistically analyzing the output from stochastic discrete-event or Monte Carlo simulations. Terminating and steady-state simulations are considered.