Bahar Biller

Modeling and generating multivariate time-series input processes using a vector autoregressive technique

We present a model for representing stationary multivariate time-series input processes with marginal distributions from the Johnson translation system and an autocorrelation structure specified through some finite lag. We then describe how to generate data accurately to drive computer simulations. The central idea is to transform a Gaussian vector autoregressive process into the desired multivariate time-series input process that we presume as having a VARTA (Vector-Autoregressive-To-Anything) distribution. We manipulate the autocorrelation structure of the Gaussian vector autoregressive process so that we achieve the desired autocorrelation structure for the simulation input process. We call this the correlation-matching problem and solve it by an algorithm that incorporates a numerical-search procedure and a numerical-integration technique. An illustrative example is included.

Fitting Time-Series Input Processes for Simulation

Providing accurate and automated input-modeling support is one of the challenging problems in the application of computer simulation of stochastic systems. The models incorporated in current input-modeling software packages often fall short because they assume independent and identically distributed processes, even though dependent time-series input processes occur naturally in the simulation of many real-life systems. Therefore, this paper introduces a statistical methodology for fitting stochastic models to dependent time-series input processes. Specifically, an automated and statistically valid algorithm is presented to fit autoregressive-to-anything (ARTA) processes with marginal distributions from the Johnson translation system to stationary univariate time-series data. ARTA processes are particularly well suited to driving stochastic simulations. The use of this algorithm is illustrated with examples.

Accounting for Parameter Uncertainty in Large-Scale Stochastic Simulations with Correlated Inputs

This paper considers large-scale stochastic simulations with correlated inputs having normal-to-anything (NORTA) distributions with arbitrary continuous marginal distributions. Examples of correlated inputs include processing times of workpieces across several workcenters in manufacturing facilities and product demands and exchange rates in global supply chains. Our goal is to obtain mean performance measures and confidence intervals for simulations with such correlated inputs by accounting for the uncertainty around the NORTA distribution parameters estimated from finite historical input data. This type of uncertainty is known as the parameter uncertainty in the discrete-event stochastic simulation literature. We demonstrate how to capture parameter uncertainty with a Bayesian model that uses Sklars marginal-copula representation and Cookes copula-vine specification for sampling the parameters of the NORTA distribution. The development of such a Bayesian model well suited for handling many correlated inputs is the primary contribution of this paper. We incorporate the Bayesian model into the simulation replication algorithm for the joint representation of stochastic uncertainty and parameter uncertainty in the mean performance estimate and the confidence interval. We show that our model improves both the consistency of the mean line-item fill-rate estimates and the coverage of the confidence intervals in multiproduct inventory simulations with correlated demands.

Copula-Based Multivariate Input Models for Stochastic Simulation

As large-scale discrete-event stochastic simulation becomes a tool that is used routinely for the design and analysis of stochastic systems, the need for input-modeling support with the ability to represent complex interactions and interdependencies among the components of multivariate time-series input processes is more critical than ever. Motivated by the failure of independent and identically distributed random variables to represent such input processes, a comprehensive framework called Vector-Autoregressive-To-Anything (VARTA) has been introduced for multivariate time-series input modeling. Despite its flexibility in capturing a wide variety of distributional shapes, we show that VARTA falls short in representing dependence structures that arise in situations where extreme component realizations occur together. We demonstrate that it is possible to extend VARTA to work for such dependence structures via the use of the copula theory, which has been used primarily for random vectors in the simulation input-modeling literature, for multivariate time-series input modeling. We show that our copula-based multivariate time-series input model, which includes VARTA as a special case, allows the development of statistically valid fitting and fast sampling algorithms well suited for driving large-scale stochastic simulations.

Improved Inventory Targets in the Presence of Limited Historical Demand Data

Most of the literature on inventory management assumes that the demand distribution and the values of its parameters are known with certainty. In this paper, we consider a repeated newsvendor setting where this is not the case and study the problem of setting inventory targets when there is a limited amount of historical demand data. Consequently, we achieve the following objectives: (1) to quantify the inaccuracy in the inventory-target estimation as a function of the length of the historical demand data, the critical fractile, and the shape parameters of the demand distribution; and (2) to determine the inventory target that minimizes the expected cost and accounts for the uncertainty around the demand parameters estimated from limited historical data. We achieve these objectives by using the concept of expected total operating cost and representing the demand distribution with the highly flexible Johnson translation system. Our procedures require no restrictive assumptions about the first four moments of the demand random variables, and they can be easily implemented in practical settings with reduced expected total operating costs.

Answers to the top ten input modeling questions

In this tutorial we provide answers to the top ten input-modeling questions that new simulation users ask, point out common mistakes that occur and give relevant references. We assume that commercial input-modeling software will be used when possible, and only suggest non-commercial options when there is little else available. Detailed examples will be provided in the tutorial presentation.

Introduction to simulation input modeling

In this tutorial we first review introductory techniques for simulation input modeling. We then identify situations in which the standard input models fail to adequately represent the available input data. In particular, we consider the cases where the input process may (i) have marginal characteristics that are not captured by standard distributions; (ii) exhibit dependence; and (iii) change over time. For case (i), we review flexible distribution systems, while we review two widely used multivariate input models for case (ii). Finally, we review nonhomogeneous Poisson processes for the last case. We focus our discussion around continuous random variables; however, when appropriate references are provided for discrete random variables. Detailed examples will be illustrated in the tutorial presentation.

A subset selection procedure under input parameter uncertainty

This paper considers a stochastic system simulation with unknown input distribution parameters and assumes the availability of a limited amount of historical data for parameter estimation. We investigate how to account for parameter uncertainty - the uncertainty that is due to the estimation of the input distribution parameters from historical data of finite length - in a subset selection procedure that identifies the stochastic system designs whose sample means are within a user-specified distance of the best mean performance measure. We show that even when the number of simulation replications is large enough for the stochastic uncertainty to be negligible, the amount of parameter uncertainty in output data imposes a threshold on the user-specified distance for an effective use of the subset selection procedure for simulation. We demonstrate the significance of this effect of parameter uncertainty for a multi-item inventory system simulation in the presence of short demand histories.

Chapter 5 Multivariate Input Processes

Abstract Representing uncertainty in a simulation study is referred to as input modeling, and is often characterized as selecting probability distributions to represent the input processes. This is a simple task when the input processes can be represented as sequences of independent random variables with identical distributions. However, dependent and multivariate input processes occur naturally in many service, communications, and manufacturing systems. This chapter focuses on the development of multivariate input models which incorporate the interactions and interdependencies among the inputs for the stochastic simulation of such systems.

Evaluation of the ARTAFIT Method for Fitting Time-Series Input Processes for Simulation

Time-series input processes occur naturally in the stochastic simulation of many service, communications, and manufacturing systems, and there are a variety of time-series input models available to match a given collection of properties, typically a marginal distribution and an autocorrelation structure specified via the use of one or more time lags. The focus of this paper is the situation in which the collection of properties are not “given,” but data are available from which a time-series input model is to be estimated. The input model we consider is the very flexible autoregressive-to-anything (ARTA) model of Cario and Nelson [Cario, M. C., B. L. Nelson. 1996. Autoregressive to anything: Time-series input processes for simulation. Oper. Res. Lett.19 51--58]. Recently, we developed a statistically valid algorithm (ARTAFIT) for fitting this model to stationary univariate time-series data using marginal distributions from the Johnson translation system. In this paper, we perform a comprehensive numerical study to assess the performance of our algorithm relative to the two most commonly used approaches: (a) fitting the marginal distribution but ignoring the autocorrelation structure, and (b) fitting separately the marginal distribution as in (a) and the autocorrelation structure using the sample autocorrelation function. We find that ARTAFIT, which fits the marginal distribution and the autocorrelation structure jointly, outperforms both (a) and (b), and we demonstrate the importance of taking dependencies into account while developing input models for stochastic simulation.