1. The AutoSimOA Project Katy Hoad, Stewart Robinson, Ruth Davies Warwick Business School WSC 07 A 3 year, EPSRC funded project in collaboration with SIMUL8 Corporation. http://www.wbs.ac.uk/go/autosimoa

2. Objective To provide an easy to use method, that can be incorporated into existing simulation software, that enables practitioners to obtain results of a specified accuracy from their discrete event simulation model. (Only looking at analysis of a single scenario)

3. OUTLINE Introduction Methods in literature Our Algorithm Test Methodology & Results Discussion & Summary

4. Underlying Assumptions  Any warm-up problems already dealt with.  Run length (m) decided upon.  Modeller decided to use multiple replications to obtain better estimate of mean performance. Response measure of interest summary statistic from each replication Perform N replications

5. QUESTION IS… How many replications are needed? Limiting factors: computing time and expense. 4 main methods found in the literature for choosing the number of replications N to perform.

6. 1. Rule of Thumb (Law & McComas 1990) Run at least 3 to 5 replications. Advantage: Very simple. Disadvantage: Does not use characteristics of model output. No measured precision level.

7. 2. Simple Graphical Method (Robinson 2004) Advantages: Simple Uses output of interest in decision. Disadvantages: Subjective No measured precision level.

8. 3. Confidence Interval Method (Robinson 2004, Law 2007, Banks et al. 2005). Advantages: Uses statistical inference to determine N. Uses output of interest in decision. Provides specified precision. Disadvantage: Many simulation users do not have the skills to apply approach.

9. 4. Prediction Formula (Banks et al. 2005) Decide size of error ε that can be can tolerated. Run ≥ 2 replications - estimate variance s 2. Solve to predict N. Check desired precision achieved – if not recalculate N with new estimate of variance. Advantages: Uses statistical inference to determine N. Uses output of interest in decision. Provides specified precision. Disadvantage: Can be very inaccurate especially for small number of replications.

10. AUTOMATE Confidence Interval Method: Algorithm interacts with simulation model sequentially.

11. is the student t value for n-1 df and a significance of 1-α, s n is the estimate of the standard deviation, calculated using results X i (i = 1 to n) of the n current replications. Where n is the current number of replications carried out, We define the precision, d n, as the ½ width of the Confidence Interval expressed as a percentage of the cumulative mean: is the cumulative mean, ALGORITHM DEFINITIONS

12. Stopping Criteria Simplest method: Stop when d n 1st found to be ≤ desired precision, d required. Recommend that number of replications, Nsol, to user. Problem: Data series could prematurely converge, by chance, to incorrect estimate of the mean, with precision d required, then diverge again. ‘Look-ahead’ procedure: When d n 1st found to be ≤ d required, algorithm performs set number of extra replications, to check that precision remains ≤ d required.

13. ‘Look-ahead’ procedure kLimit = ‘look ahead’ value. Actual number of replications checked ahead is Relates ‘look ahead’ period length with current value of n.

14. Nsol Nsol + f(kLimit) f(kLimit) Precision ≤ 5% 95% confidence limits Cumulative mean, Replication Algorithm

15. Precision≤ 5% Precision> 5% Precision ≤ 5% f(kLimit) Nsol 2 Nsol 2 + f(kLimit) Nsol 1

16. 24 artificial data sets: Left skewed, symmetric, right skewed; Varying values of relative st.dev (st.dev/mean). 100 sequences of 2000 data values. 8 real models selected. Different lengths of ‘look ahead’ period tested: kLimit values = 0 (i.e. no ‘look ahead’), 5, 10, 25. d required value kept constant at 5%. TESTING METHODOLOGY

17. 5 performance measures 1.Coverage of the true mean 2.Bias 3.Absolute Bias 4.Average Nsol value 5.Comparison of 4. with Theoretical Nsol value For real models: ‘true’ mean & variance values - estimated from whole sets of output data (3000 to 11000 data points).

18. Results Nsol values for individual algorithm runs are very variable. Average Nsol values for 100 runs per model close to the theoretical values of Nsol. Normality assumption appears robust. Using a ‘look ahead’ period improves performance of the algorithm.

19. Mean bias significantly different to zero Failed in coverage of true mean Mean est. Nsol significantly different to theoretical Nsol (>3) No ‘look- ahead’ period Proportion of Artificial models 4/242/249/18 Proportion of Real models 1/8 3/5 kLimit = 5 Proportion of Artificial models 1/2401/18 Proportion of Real models 000

20. % decrease in absolute mean bias kLimit = 0 to kLimit = 5 kLimit = 5 to kLimit = 10 kLimit = 10 to kLimit = 25 Artificial Models 8.76%0.07%0.26% Real Models 10.45%0.14%0.33% Impact of different look ahead periods on performance of algorithm

21. Number of times the Nsol value changes (out of 100 runs of the algorithm per model) because of the lengthening of the ‘look ahead’ period. Model ID kLimit = 0 to kLimit = 5 kLimit = 5 to kLimit = 10 kLimit = 10 to kLimit = 25 R1000 R3200 R52401 R82441 A53013 A62663 A15100 A172201 A212521 A243700

22. Model ID kLimitNsolTheoretical Nsol (approx) Mean estimate significantly different to the true mean? A904112Yes 5120No A2403755Yes 5718No R70310Yes 58No R4036Yes 57No R80345Yes 546No Examples of changes in Nsol & improvement in estimate of true mean

23. DISCUSSION kLimit default value set to 5. Initial number of replications set to 3. Multiple response variables - Algorithm run with each response - use maximum estimated value for Nsol. Different scenarios - advisable to repeat algorithm every few scenarios to check that precision has not degraded significantly. Implementation into Simul8 simulation package.

24. SUMMARY Selection and automation of Confidence Interval Method for estimating the number of replications to be run in a simulation. Algorithm created with ‘look ahead’ period - efficient and performs well on wide selection of artificial and real model output. ‘Black box’ - fully automated and does not require user intervention.

25. ACKNOWLEDGMENTS This work is part of the Automating Simulation Output Analysis (AutoSimOA) project (http://www.wbs.ac.uk/go/autosimoa) that is funded by the UK Engineering and Physical Sciences Research Council (EP/D033640/1). The work is being carried out in collaboration with SIMUL8 Corporation, who are also providing sponsorship for the project. Katy Hoad, Stewart Robinson, Ruth Davies Warwick Business School WSC 07