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AN OVERVIEW OF TES PROCESSES AND MODELING METHODOLOGY BENJAMIN MELAMED NEC USA, Inc. C&C Research Laboratories 4 Independence Way Princeton, New Jersey 08540 Abstract TES (Transform-Expand-Sample) is a versatile methodology for modeling station- ary time series with general marginal distributions and a broad range of dependence structures. From the viewpoint of Monte Carlo simulation, TES constitutes a new and flexible input analysis approach whose principal merit is its potential ability to simul- taneously capture first-order and second-order statistics of empirical time series. That is, TES is designed to fit an arbitrary empirical marginal distribution (histogram), and to simultaneously approximate the leading empirical autocorrelations. This paper is a tutorial introduction to the theory of TES processes and to the modeling methodology based on it. It employs a didactic approach which relies heavily on visual intuition as a means of conveying key ideas and an aid in building deep understanding of TES. This approach is in line with practical TES modeling which itself is based on visual interaction under software support. The interaction takes on the form of a heuristic search in a large parameter space, and it currently relies on visual feedback supplied by computer graphics. The tutorial is structured around an illustrative example both to clarify the modeling methodology and to exemplify its efficacy.
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360 1 INTRODUCTION TES (Transform-Expand-Sample) is a recent approach to modeling stationary time series [15, 7, 8, 9]. The TES approach is non-parametric in the sense that it makes no assumptions on marginal distributions, though the underlying temporal dependence structure is postu- lated to be Markovian, with a continuous state space. Nevertheless, its modeling scope is quite broad: Additional transformations may be applied, leading to non-Markovian pro- cesses; and stationary TES processes may be combined into new ones, e.g., via modulation of TES processes by another process, often a discrete-state Markov process. The main ap- plication of TES to date has been to create source models (of incoming traffic or workload), in order to drive Monte Carlo simulations [2, 10]. What is new about TES is its potential ability to capture (fit) both the marginal dis- tribution (a first-order statistic) and the autocorrelation function (a second-order statistic) of empirical data. Most importantly, TES aims to fit both marginals and autocorrelations simultaneously. This goal is not new; in fact, engineers have attempted such simultaneous fitting, mainly in the context of signal processing (see, e.g., [13] and references therein). The TES variation on this theme is to precisely fit the empirical marginal distribution (typ- ically an empirical histogram), and at the same time capture temporal dependence proxied by the autocorrelation function (a measure of linear dependence). Being able to do this is no small feat. In fact, other modeling approaches to time series are able to do either one or the other but not both. For example, autoregression can fit a variety of autocorrelation func-
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This note was uploaded on 04/16/2010 for the course SCE sysc5001 taught by Professor Lambadaris during the Spring '10 term at Carleton CA.

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