PowerPoint-Pr

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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).
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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.

density functions over model time
m gtgt l
initialisation
mean value
truncation point
l
m
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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

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Simulation data and Transformation
n random numbers, k replications
random sample distributions over model time
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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
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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 1r new observations of each
replication. (here r3)
2. Shift test sample and compare it with the
remaining.
3. To much difference? goto 1.
4. Calculate result values.
test sample
remaining
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Example M/M/1 with medium utilisation
truncation point (AR/DA)
density functions over model time
0
2080
observed model time
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

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Example M/M/1 with high utilisation
truncation point (AR/DA)
density functions over model time
0
11400
observed model time
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.
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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.
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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

EN 4 truncation point mean population used data
Long Run 5000 4.16 /- 0.20 136000
AR/DA 540 4.09 /- 0.19 2080100
EN 19 truncation point mean population used data
Long Run 3500 19.04 /- 0.95 965800
AR/DA 2850 19.28 /- 0.96 25700100

• Comment
• - AR/DA needs more data factor 1.5 2.7, but
• AR/DA is faster in execution factor 65 38

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A Non-Ergodic System (1) Presented on ESS99 from
Bause/Beilner.
highly increasing population
very long stable beginning
more replications advantage!
at random model time
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A Non-Ergodic System (2)
density functions over model time
0
28000
observed model time
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.

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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