The Analysis of Simulation-Generated Time Series

This study applies spectral analysis to the study of time series data generated by simulated stochastic models. Because these data are autocorrelated, analysis by methods applicable to independent observations is not possible. Mathematical models known as covariance stationary stochastic processes are useful representations of autocorrelated time series. The increased publication of literature describing stochastic processes and spectral analysis, in particular, is making these ideas available to an increasing audience. Section I presents a rationale for our interest in time series models and spectral analysis. Section II describes the basic notions of covariance stationary processes. It emphasizes the equivalence of these processes in both the time and frequency domains; the compactness of frequency domain analysis seemingly recommends it over correlation analysis. Section III provides a heuristic background for understanding statistical spectral analysis. Simple frequency-domain statistical properties are emphasized and compared with the rather involved sampling properties of estimated correlograms. Several relevant statistical tests are described. Three simulated experiments are used as examples of how to apply spectral analysis. These are described in Section IV. They are (1) a single-server, first-come-first-served (FCFS) queueing problem with Poisson arrivals and exponentially distributed service times; (2) a similar model with the FCFS assignment rule replaced by one that chooses the job with the shortest service time; and (3) yet another similar model, but with constant service time. In this third example both assignment rules are equivalent. The state of the queue was observed and recorded at unit intervals for all three examples and forms the time series on which Section V is based; theoretical and statistical correlograms and spectra are compared for example 1; examples 1 and 2 are compared using estimated correlograms and spectra; example 3 serves to show some unusual properties of spectra. A particular conclusion of this paper is that differences in the statistical properties of queue length that result under various assumptions and operating rules are easily identified using spectral analysis. More generally, however, this estimation procedure provides a tool for (1) comparing simulated time series with real-world data, and (2) for understanding the implications that various alternative assumptions have on the output of simulated stochastic models.