1. STEADY-STATE SYSTEM SIMULATION(2)

2. REVIEW OF THE BASICS • Initial Transient a.k.a. Warm-Up Period

3. PROCEDURE • Y(i,j) is the ith sample of the jth replication • Confidence interval on {Y(i,*)} • Eyeball the diminution of drift (j* is where) • Make 3j* the truncation point • Restart the system for ONE LONG RUN • {Y(i), i>3j*} is a set of autocorrelated, identically distributed data

4. DEAL WITH AUTOCORRELATION • Batch Means • Regenerative Method • Jackknife • Time Series

5. BEFORE WE BEGIN That’s a joint distribution function for the whole set of n samples! It captures all of the correlation in the X’s.

7. WHAT DOES THAT MEAN? • The summation and the integral are interchanged • The joint density function reduces to the marginal distribution for Xi (the correlations are “marginaled out”) • The mean X-bar is unbiased, even when the data has correlation • Unfortunately, when we deal with s2, the squaring function prevents a similar thing, and a naive s2 calculation results in a biased estimate • s2 underestimates s2 when the autocorrelation is positive

8. BATCH MEANS • {Y(i), 0<i<=n} is the data (3j* already removed) • adjacent BATCHES of size b are formed and the batch average for each is calculated • (regularity conditions) As the batches become large, all correlation between them disappears • Treat the batch means as iid

9. REGENERATIVE METHOD • Suppose we could define events {T1, T2, ...} where we know that the system is memoryless (by system dynamics) • arrival to empty/idle system • all “clocks” are exponentially distributed • discrete event involving a geometric trial • Samples taken between Ti’s are independent

10. BUSY PERIOD EXAMPLE Q1=8 Q3=3 Q2=1/3 What is the accumulation rate of queuing time for this system?

11. SAMPLES • At arrival to an empty queue... • The inter-arrival process is sampled • The service time of the entering customer is sampled • No other activities are happening, no pending events • From the picture our sample is ... • 8/3, (1/3)/1, and 3/2 • which we can treat as iid • note this is not Q-bar/(inter-B)-bar

12. JACKKNIFE ESTIMATORS • Q-bar/(inter-B)-bar is a biased, consistent estimator • Its expected value is not E[Q/(inter-B)] • As the sample gets large, the bias dimishes to 0 • The bias comes from the dependency of Qi with its accompanying inter-Bi • We care because we want to relax the “memorylessness” property and use a ratio of mean estimates

13. JACKKNIFE • f’s are biased, consistent estimates • the bias is small • an iid confidence interval of {fg, g=1,2,..n}

14. TIME SERIES • Also called the Autoregressive Approach • Uses estimates of the coefficients of autocorrelation to create an iid sample with known relationship to m and s • Most well-studied by the statistician community

15. MECHANICS OF AUTOREGRESSIVE APPROACH • Assume Y’s autocorrelation vanishes after lag p • Create the sample X’s using the b’s • b’s chosen so that X’s have no autocorrelation

16. RESULT • Let Ri –hat be the sample autocorrelation of lag i • Assume WLOG that b0=1 • Then the b’s solve the system of p equations below:

17. IN THE LIMIT... Writing • where the JUNK is a term vanishing as n gets large; • where b is the sum of the bs’s, SO ...

18. so we get the variance we need for our estimate of m

19. RECIPE • Sample Y’s from the system, Calculate Y-bar • Feel how large p needs to be • Estimate R’s, s=1, 2, ..., p • Solve equation to get b’s, sum them • Create the sample of X’s and estimate sX with sX • Create confidence interval for m

20. DEAL WITH AUTOCORRELATION • Batch Means • Regenerative Method • Jackknife • Time Series