Objects. The ~ b s t rac t ~ b j ect forms the fundamental base class for the entire design and all other classes are derived from this base class. The Abstract Object class defines and characterizes all the essential properties every class in this 404 OBJECT-ORIENTED SIMULATION

Handbook of Simulation

This paper surveys efficient techniques for estimating, via simulation, the probabilities of certain rare events in queueing and reliability models. The rare events of interest are long waiting times or buffer overflows in queueing systems, and system failure events in reliability models of highly dependable computing systems. The general approach to speeding up such simulations is to accelerate the occurrence of the rare events by using importance sampling. In importance sampling, the system is simulated using a new set of input probability distributions, and unbiased estimates are recovered by multiplying the simulation output by a likelihood ratio. Our focus is on describing asymptotically optimal importance sampling techniques. Using asymptotically optimal importance sampling, the number of samples required to get accurate estimates grows slowly compared to the rate at which the probability of the rare event approaches zero. In practice, this means that run lengths can be reduced by many orders of magnitude, compared to standard simulation. In certain cases, asymptotically optimal importance sampling results in estimates having bounded relative error. With bounded relative error, only a fixed number of samples are required to get accurate estimates, no matter how rare the event of interest is. The queueing systems studied include simple queues (e.g., GI/GI/1), Jackson networks, discrete time queues with multiple autocorrelated arrival processes that arise in the analysis of Asynchronous Transfer Mode communications switches, and tree structured networks of such switches. Both Markovian and non-Markovian reliability models are treated.

/pdf/fast-simulation-of-rare-events-in-queueing-and-reliability-fc1evjiciz.pdf

Fast simulation of rare events in queueing and reliability models

This paper reviews statistical methods for analyzing output data from computer simulations. First, it focuses on the estimation of steady-state system parameters. The estimation techniques include the replication/deletion approach, the regenerative method, the batch means method, and methods based on standardized time series. Second, it reviews recent statistical procedures to find the best system among a set of competing alternatives.

Proceedings of the 2000 winter simulation conference

Two important questions that must be answered whenever a Markov chain Monte Carlo (MCMC) algorithm is used are (Q1) What is an appropriate burn-in? and (Q2) How long should the sampling continue after burn-in?Developing rigorous answers to these questions presently requires a detailed study of the convergence properties of the underlying Markov chain. Consequently, in most practical applications of MCMC, exact answers to (Q1)and (Q2) are not sought. The goal of this paper is to demystify the analysis that leads to honest answers to (Q1) and (Q2). The authors hope that this article will serve as a bridge between those developing Markov chain theory and practitioners using MCMC to solve practical problems. The ability to address (Q1) and (Q2) formally comes from establishing a drift condition and an associated minorization condition, which together imply that the underlying Markov chain is geometrically ergodic. In this article, we explain exactly what drift and minorization are as well as how and why these conditions can be used to form rigorous answers to (Q1) and (Q2). The basic ideas are as follows. The results of Rosenthal (1995) and Roberts and Tweedie (1999) allow one to use drift and minorization conditions to construct a formula giving an analytic upper bound on the distance to stationarity. A rigorous answer to (Q1) can be calculated using this formula. The desired characteristics of the target distribution are typically estimated using ergodic averages. Geometric ergodicity of the underlying Markov chain implies that there are central limit theorems available for ergodic averages (Chan and Geyer 1994). The regenerative simulation technique (Mykland, Tierney and Yu, 1995; Robert, 1995) can be used to get a consistent estimate of the variance of the asymptotic normal distribution. Hence, an asymptotic standard error can be calculated, which provides an answer to (Q2) in the sense that an appropriate time to stop sampling can be determined. The methods are illustrated using a Gibbs sampler for a Bayesian version of the one-way random effects model and a data set concerning styrene exposure.

/pdf/honest-exploration-of-intractable-probability-distributions-5f04r61s9a.pdf

Honest Exploration of Intractable Probability Distributions via Markov Chain Monte Carlo

Observations of a stationary stochastic process can be transformed into a standardized time series. This paper presents a lemma giving the asymptotic properties of this standardized series under quite general conditions. In particular, the conditions are satisfied by stationary discrete-event simulations. Confidence intervals can be constructed using this lemma. For illustration, we develop two easily computed interval estimators for the process mean. When independent replications of the series are available, such as in computer simulation experiments, these interval estimators may be combined with the classical confidence interval estimator. These interval estimators also tend to compensate for simulation initialization bias if the sign of the bias is known. In an empirical study using three elementary simulated processes, the interval estimators presented here compare favorably with the classical interval estimator. In a recent paper, Goldsman and Schruben Goldsman, D., L. Schruben. 1982. Asymptotic properties of some confidence interval estimators. Tech. Rep. 544, School of O.R.I.E., Cornell University, Ithaca, NY. show that the asymptotic properties of the confidence intervals presented in this paper strictly dominate those of classical confidence intervals.

Confidence Interval Estimation Using Standardized Time Series

A technique for simulating GI/G/s queues is shown to apply to simulations of discrete and continuous-time Markov chains. It is possible to address questions of simulation run duration and of starting and stopping simulations because of the existence of a random grouping of observations which produces independent identically distributed blocks from the start of the simulation. This grouping allows confidence intervals to be obtained for a general function of the steady-state distribution of the Markov chain. The technique is illustrated with simulation of an (s, S) inventory model in discrete time and the classical repairman problem in continuous time. Consideration is also given to determining system sensitivity to errors and uncertainty in the input parameters.

Simulating Stable Stochastic Systems, II: Markov Chains

The previous papers in this series developed a methodology for obtaining from certain simulations confidence intervals for parameters associated with the steady-state distribution. This methodology required the simulations to contain an embedded renewal process at whose epochs the simulation started from scratch. The present paper contains four approximation techniques for obtaining confidence intervals when the simulation does not contain the required renewal process.

Simulating Stable Stochastic Systems, IV: Approximation Techniques

: A new technique is introduced for analyzing simulations of stochastic systems in the steady state. From the viewpoint of classical statistics, questions of simulation run duration and of starting and stopping simulations are addressed. This is possible because of the existence of a random grouping of observations which produces independent identically distributed blocks from the start of the simulation. The analysis is presented in the context of the general multi-server queue, with arbitrarily distributed inter-arrival and service times. In this case, it is the busy period structure of the system which produces the grouping mentioned above. Numerical illustrations are given for the M/M/1 queue.

A New Approach to Simulating Stable Stochastic Systems: 1 - General Multi-Server Queues

: In simulating stable stochastic systems one is frequently confronted by the need to produce confidence intervals for the ratio of the expected values of two random variables (r.v.'s). This situation arises, for example, when one simulates the general many-server queue or stochastic systems modeled as Markov chains in either discrete or continuous time. The report derives a confidence interval for the ratio of two means which is a distinct improvement over an old method. Both the old and new confidence intervals are illustrated by numerical examples. (Author)

Confidence Intervals for the Ratio of Two Means with Application to Simulations.

The purpose of this paper is to provide an introduction to the regenerative method for simulation analysis. The simulations we are concerned with here are simulations of stochastic systems, i.e., systems with random elements. The regenerative approach leads to a statistical methodology for analyzing the output of those simulations which have the property of “starting afresh probabilistically” from time to time. The class of such simulations is very large and very important, including simulations of a broad variety of queues and queueing networks, inventory systems, inspection, maintenance, and repair operations, and numerous other situations.

A tutorial on the regenerative approach and discrete-event simulations

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