OUTPUT ANALYSIS FOR SIMULATIONS. Outline. Introduction Analysis of One System Terminating vs. Steady-State Simulations Analysis of Terminating Simulations Obtaining a Specified Precision Analysis of Steady-State Simulations Method of Batch Means. Introduction.
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Outline • Introduction • Analysis of One System • Terminating vs. Steady-State Simulations • Analysis of Terminating Simulations • Obtaining a Specified Precision • Analysis of Steady-State Simulations • Method of Batch Means
Introduction • After understanding the under laying process, collecting data, fitting data to a distribution, coding and debugging the simulation program • selecting a performance measure to evaluate the system • evaluating your design by runs • But by doing one or two runs, is it enough to evaluate your system? • Answer is No. • Because components driving your simulation include randomness, the output of simulation is also random • The output is not independent and identically distributed (i.i.d), we can not use classical statistical methods
What Outputs to Watch? • Performance measure - criteria that evaluate how god your system is • Average, and worst (longest) time in system • Average, and worst time in queue(s) • Average hourly production • Standard deviation of hourly production • Proportion of time a machine is up, idle, or down • Maximum queue length • Average number of parts in system
Types of Simulations with Regard to Output Analysis • Transient : A simulation where there is a specific starting and stopping condition that is part of the model. • transient performancemeasures: the performance of system finite horizon • Steady-state: A simulation where there is no specific starting and ending conditions. Here, we are interested in the steady-state behavior of the system. • Steady-stateperformance measures: the performance for infinite horizon “The type of analysis depends on the goal of the study.”
Analysis for Transient Simulations Objective: Obtain a point estimate and confidence interval for some parameter Examples: = E (average time in system for n customers) = E (machine utilization) = E (work-in-process) Reminder: Can not use classical statistical methods within a simulation run because observations from one run are not independently and identically distributed (i.i.d.)
Analysis for Transient Simulations • Make n independent replications of the model • Let Yi be the performance measure from the ith replication Yi = average time in system, or Yi = work-in-process, or Yi = utilization of a critical facility • Performance measures from different replications, Y1, Y2, ..., Yn, are i.i.d. • But, only one sample is obtained from each replication • Apply classical statistics to Yi’s, not to observations within a run • Select confidence level 1 – a (0.90, 0.95, etc.)
Analysis for Transient Simulations • Approximate 100(1 – a)% confidence interval for m: estimator of m estimator of Var(Yi) covers m with approximate probability (1 – a) is the Half-Width expression
Example Consider a single-server (M/M/1) queue. The objective is to calculate a confidence interval for the delay of customers in the queue. n = 10 replications of a single-server queue Yi = average delay in queue from ith replication Yi’s: 2.02, 0.73, 3.20, 6.23, 1.76, 0.47, 3.89, 5.45, 1.44, 1.23 For 90% confidence interval, = 0.10 = 2.64, = 3.96, t9, 0.95 = 1.833 Approximate 90% confidence interval is 2.64 ± 1.15, or [1.49, 3.79]
Analysis for Transient Simulations Interpretation: 100(1 – a)% of the time, the confidence interval formed in this way covers m Wrong Interpretation: “I am 90% confident that mis between 1.49 and 3.79”
Issue 1 • This confidence-interval method assumes Yi’s are normally distributed. In real life, this is almost never true. • Because of central-limit theorem, as the number of replications (n) grows, the coverage probability approaches 1– a. • In general, if Yi’s are averages of something, their distribution tends not to be too asymmetric, and the confidence- interval method shown above has reasonably good coverage.
Issue 2 • The confidence interval may be too wide In the M/M/1 queue example, the approximate 90% C.I. was: 2.64 ± 1.15, or [1.49, 3.79] The half-width is 1.15 which is 44% of the mean (1.15/2.64) That means that the C.I. is 2.64 44% which is not very precise. • To decrease the half-width: Increase n until is small enough (this is called Sequential Sampling) • There are two ways of defining the precision in the estimate Y: • Absolute precision • Relative precision
Obtaining a Specified Precision • Relative Precision:
Analysis for Steady-State Simulations Objective: Estimate the steady state mean Basic question: Should you do many short runs or one long run ?????
Analysis for Steady-State Simulations • Advantages: • Many short runs: • Simple analysis, similar to the analysis for terminating systems • The data from different replications are i.i.d. • One long run: • Less initial bias • No restarts • Disadvantages • Many short runs: • Initial bias is introduced several times • One long run: • Sample of size 1 • Difficult to get a good estimate of the variance
Analysis for Steady-State Simulations • Make many short runs: The analysis is exactly the same as for terminating systems. The (1 – a)% C.I. is computed as before. • Problem: Because of initial bias, may no longer be an unbiased estimator for the steady state mean, . • Solution: Remove the initial portion of the data (warm-up period) beyond which observations are in steady-state. Specifically pick l (warm-up period) and n (number of observations in one run) such that
Analysis for Steady-State Simulations • Make one Long run: Make just one long replication so that the initial bias is only introduced once. This way, you will not be “throwing out” a lot of data. Problem: How do you estimate the variance because there is only one run? Solution: Several methods to estimate the variance: • Batch means (only approach to be discussed) • Time-series models • Spectral analysis • Standardized time series
Method of Batch Means • Divide a run of length m into n adjacent “batches” of length k where m = nk. • Let be the sample or (batch) mean of the jth batch. • The grand sample mean is computed as
Method of Batch Means • The sample variance is computed as • The approximate 100(1 – a )% confidence interval for is
Method of Batch Means Two important issues: • Issue 1: How do we choose the batch size k? • Choose the batch size k large enough so that the batch means, are approximately uncorrelated. Otherwise, the variance, , will be biased low and the confidence interval will be too small which means that it will cover the mean with a probability lower than the desired probability of (1 – a ).
Method of Batch Means • Issue 2: How many batches n? • Due to autocorrelation, splitting the run into a larger number of smaller batches, degrades the quality of each individual batch. Therefore, 20 to 30 batches are sufficient.