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 s 2, the squaring function prevents a similar thing, and a naive s 2 calculation results in a biased estimate –s 2 underestimates  2 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 What is the accumulation rate of queuing time for this system? Q1=8 Q2=1/3 Q3=3

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   ’s are biased, consistent estimates the bias is small an iid confidence interval of {  g, 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  and  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 R i –hat be the sample autocorrelation of lag i Assume WLOG that b 0 =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 b s ’s, SO...

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

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  X with s X Create confidence interval for 

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