1. Modellierung großer Netze in der Logistik SFB 559 Initial Transient Period Detection Using Parallel Replications F. Bause, M. Eickhoff LS Informatik IV, Universität Dortmund, Germany This research was supported by the Deutsche Forschungsgemeinschaft as part of the Collaborative Research Center „Modelling of large logistic networks“ (559). Outline: 1.Introduction and Motivation 2.Simulation data and Transformation 3.Algorithm (AR/DA) 4.Examples 5.Conclusions

2. F. Bause, M. Eickhoff1/12 ESS 2002Initial Transient Period Detection using Parallel Replications Introduction and Motivation (1) Output analysis in discrete event simulation: -Problem of initialisation -Initialisation bias because of system warm-up Well-known advices: -Transient period – truncation point – steady-state period Gordon: „... the first part of each simulation run can be ignored.“ -Optimal initialisation state Law/Kelton: „... the optimal state for initialisation tends to be larger than the mean...“ -Convergence of the mean Pawlikowski: „Rules R4-R8 are based on the convergence of the mean... Other criteria of convergence are also possible.“ -Ratio of transient and steady-state period Law/Kelton: „..., where m is much larger than the warmup period l...“ -Up to now Alexopoulos/Sheila: „One of the hardest problems... is the removal of the initialisation bias.“ truncation pointlm initialisation mean value m >> l density functions over model time

3. F. Bause, M. Eickhoff2/12 ESS 2002Initial Transient Period Detection using Parallel Replications Introduction and Motivation (2) Known strategies: -long simulation run or -many replications -fixed dataset or -sequential/adaptive approaches Our work: -many replications: 1)problem is easy to parallelize 2)hardware is available -adaptive approach: 1)during the simulation 2)needed in practise

4. F. Bause, M. Eickhoff3/12 ESS 2002Initial Transient Period Detection using Parallel Replications Simulation data and Transformation n random numbers, k replications random sample distributions over model time

5. F. Bause, M. Eickhoff4/12 ESS 2002Initial Transient Period Detection using Parallel Replications Basic Idea transient periodsteady-state period Transient:density function is changing over time. Steady-state:density function is constant over time. Truncation point:first density function equal to the remaining density functions Problem:systematic error and random error

6. F. Bause, M. Eickhoff5/12 ESS 2002Initial Transient Period Detection using Parallel Replications Adaptive Replication/Deletion Approach (AR/DA) First aim: Find truncation point! -Ignore first part (Gordon). Choose transient-steady-state-ratio (parameter r). -Warm-up period is much smaller (Law/Kelton). Comparison: Kolmogoroff-Smirnoff two-sample test. -Other criteria of convergence (Pawlikowski). -Null-Hypothesis: Equality of cumulative distributions. -No demands on the random samples. -No restrictions on the size of the random samples. Set safety-level. -Percentage of the number of rejections of the null-hypothesis. Second aim: Estimate result values! -An independent result is calculated for each truncated replication. test sampleremaining 1. Collect 1+r new observations of each replication. (here r=3) 2. Shift test sample and compare it with the remaining. 3. To much difference?: goto 1. 4. Calculate result values. 13 truefalse 2/3 > safety-level 26 true

7. F. Bause, M. Eickhoff6/12 ESS 2002Initial Transient Period Detection using Parallel Replications Example: M/M/1 with medium utilisation density functions over model time truncation point (AR/DA) 02080observed model time model time of test sample results of KS-Test Parameter: = 0.8 Initialisation = 100 jobs r = 3 Safety-level = 0.05 Result: truncation point at 540 Comment: high initialisation -advice of Law/Kelton -obvious transient period

8. F. Bause, M. Eickhoff7/12 ESS 2002Initial Transient Period Detection using Parallel Replications Example: M/M/1 with high utilisation Parameter: = 0.95 Initialisation = 100 jobs r = 3 Safety-level = 0.05 Result: truncation point at 2850 Comment: more challenging, difference between systematic and random error not obvious. model time of test sample results of KS-Test density functions over model time truncation point (AR/DA) 011400observed model time

9. F. Bause, M. Eickhoff8/12 ESS 2002Initial Transient Period Detection using Parallel Replications Comparison with visual methods M/M/1 with high utilisation density functions over model time If the initial bias slowly vanishes, visual methods have problems. truncation point (AR/DA) ??? graphical procedure of Welch

10. F. Bause, M. Eickhoff9/12 ESS 2002Initial Transient Period Detection using Parallel Replications Comparison with statistical methods Theory -average population (M/M/1): Long Simulation Run (Pawlikowski, 1990) -initial transient period detection: Emshoff/Sisson (1970) -steady-state analysis: batch means Results: Comment: - AR/DA needs more data: factor 1.5; 2.7, but - AR/DA is faster in execution: factor 65; 38 E[N] = 4truncation pointmean populationused data Long Run50004.16 +/- 0.20136000 AR/DA5404.09 +/- 0.192080*100 E[N] = 19truncation pointmean populationused data Long Run350019.04 +/- 0.95965800 AR/DA285019.28 +/- 0.9625700*100

11. F. Bause, M. Eickhoff10/12 ESS 2002Initial Transient Period Detection using Parallel Replications A Non-Ergodic System (1) Presented on ESS´99 from Bause/Beilner. very long „stable“ beginning highly increasing population more replications: advantage! at random model time

12. F. Bause, M. Eickhoff11/12 ESS 2002Initial Transient Period Detection using Parallel Replications A Non-Ergodic System (2) Comment: -AR/DA gives additional hints to detect non-ergodicity. -Parameter r must be sufficiently large. density functions over model time 028000observed model time model time of test sample results of KS-Test

13. F. Bause, M. Eickhoff12/12 ESS 2002Initial Transient Period Detection using Parallel Replications Benefits of AR/DA 1.fast execution time 2.other criteria than the convergence of the mean (equality of cumulative distributions) 3.proper choice of parameter r avoids poor results 4.gives additional hints for non-ergodicity 5.visualisation of available data („density functions“) might be helpful Future Work 1.reduce user-specified parameters for AR/DA 2.examine benefits of different initial states for AR/DA