Modellierung großer Netze in der Logistik

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

2. 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 • ofthe 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.“ density functions over model time m >> l initialisation mean value l truncation point m

3. Introduction and Motivation (2) • Known strategies: • long simulation run or • many replications • fixed dataset or • sequential/adaptive approaches • Our work: • many replications: • problem is easy to parallelize • hardware is available • adaptive approach: • during the simulation • needed in practise

4. Simulation data and Transformation n random numbers, k replications random sample distributions over model time

5. Basic Idea transient period steady-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. 2 6 1 3 true true true true true true 2/3 > safety-level true false false 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. 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. test sample remaining

7. results of KS-Test model time of test sample Example: M/M/1 with medium utilisation truncation point (AR/DA) density functions over model time 0 observed model time 2080 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. results of KS-Test model time of test sample Example: M/M/1 with high utilisation truncation point (AR/DA) density functions over model time 0 observed model time 11400 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.

9. ? ? ? Comparison with visual methods M/M/1 with high utilisation density functions over model time truncation point (AR/DA) graphical procedure of Welch If the initial bias slowly vanishes, visual methods have problems.

10. 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

11. A Non-Ergodic System (1) Presented on ESS´99 from Bause/Beilner. highly increasing population very long „stable“ beginning more replications: advantage! at random model time

12. A Non-Ergodic System (2) density functions over model time 0 observed model time 28000 results of KS-Test model time of test sample • Comment: • AR/DA gives additional hints to detect non-ergodicity. • Parameter r must be sufficiently large.

13. Benefits of AR/DA • fast execution time • other criteria than the convergence of the mean(equality of cumulative distributions) • proper choice of parameter r avoids poor results • gives additional hints for non-ergodicity • visualisation of available data („density functions“) might be helpful Future Work • reduce user-specified parameters for AR/DA • examine benefits of different initial states for AR/DA