Transcript Place invariant

Assignment 4
Problems with parameterization (example:keeper usage):
average duration: 1.27, min: 0.106, max: 6.46
possible
outcome
for keeper
and crane
queues?
For validation, simulate long enough.
Half widths
Half width determination by Arena: statistical analysis of samples. Arena help file:
Half Width (Runtime Confidence Intervals—Within a Replication)
Some sections contain a column called "Half Width". This statistic is included to help you
determine the reliability of the results from your replication. Three results are possible in
the "Half Width" category:
Insufficient: The formula used to calculate half width requires the samples to be normally
distributed. That assumption may be violated if there is a small number (fewer than 320) of
samples. If that is the case, Arena will return the message "Insufficient" for that variable’s half
width, indicating there is insufficient data to accurately calculate the half width. Running the
simulation for a longer period of time should correct this.
Correllated: The formula used to calculate half width requires the samples to be
independently distributed. Data that is correlated (the value of one observation strongly
influences the value of the next observation) results in an invalid confidence interval
calculation. If data is determined to be correlated, the message "Correllated" is returned for
that variable’s half width. Running the simulation for a longer period of time
should correct this.
A value: If a value is returned in the Half Width category, this value may be interpreted by
saying "in 95% of repeated trials, the sample mean would be reported as
within the interval sample mean ± half width".
Half width calculation
Arena has to be trusted w.r.t. confidence values.
Computations "inside Arena" not clear!
- determine average and variance s^2
- test for insufficient/correlated
- half width = C.s for some constant C.
?
Concepts from probability theory and statistics.
Homework:
Appendices B,C,D of lecture notes.
Randomization
Random generator is in fact deterministic!
Replaying same model gives same result.
Still, it has all characteristics of "true" random generator
(no "fairness").
DCT case: 15% increase BF trucks
should diminish keeper queues.
Short simulation: keeper queues might get longer!
(see half width)
Increase confidence in simulation.
Divide simulation run into subruns.
Add initial run (move away from initial state).
Replications
Make sure that replications/subruns are independent
(e.g. no queue length dependencies).
When done right, division into subruns allows to compute
confidence intervals.
Example computation: page 30 of lecture notes.
n = 30 subruns, each with sample of measure x
(occupation rate).
1
2
3
4
5
6
7
8
9
10
0.914
0.964
0.934
0.978
0.912
0.956
0.958
0.934
0.978
0.976
11
12
13
14
15
16
17
18
19
20
0.894
0.962
0.973
0.984
0.923
0.932
0.967
0.924
0.945
0.936
Sample average/stddev:
x = 0.9408, s = 0.02485.
a - confidence interval:
21
22
23
24
25
26
27
28
29
30
0.898
0.912
0.943
0.953
0.923
0.914
0.923
0.936
0.945
0.934
how do you compute s?
s a 
x
.z  
n 2
Confidence interval matching
Computed 0.95 - confidence interval should match
Arena's half width.
Arena half width should (for large n) approximate
s  0.05 
.z 
  1.96(s / n )
n  2 
This example: 1.96(0.02485 / 5.51)=0.00451
So with 95% probability, occ.rate in [0.936,0.945]
cf. Chapter 6 of lecture notes
a - confidence interval:
s a 
x
.z  
n 2
Function z: normal distribution surface (table lookup).
Consequences:
- You can be 99.99% confident, but not 100%.
- Four times longer simulation halves confidence int.
- high variance = low confidence
Comparisons
Many simulation studies (e.g. DCT example) are about
relative shortage of resources, leading to queues.
Compare possible solutions through simulation.
Simulation yields to following reports:
Sol1:
Number waiting
res1.Queue
Average
4.18
Half Width
1.20
Sol2:
Number waiting
res1.Queue
Average
4.58
Half Width
1.39
Sol1 better?
Number waiting
res1.Queue
Average
4.18
Half Width
1.20
Number waiting
res1.Queue
Average
4.58
Half Width
1.39
Longer simulation needed to get following result:
Number waiting
res1.Queue
Average
4.11
Half Width
0.40
Number waiting
res1.Queue
Average
4.69
Half Width
0.48
Variance and confidence
Large half widths caused by high subrun variance,
require very long simulations for acceptable confidence.
For instance, compare
1
2
3
4
5
6
7
8
9
10
0.934
0.944
0.936
0.958
0.932
0.946
0.938
0.934
0.948
0.936
11
12
13
14
15
16
17
18
19
20
0.924
0.932
0.933
0.944
0.923
0.932
0.947
0.934
0.945
0.936
21
22
23
24
25
26
27
28
29
30
0.928
0.932
0.943
0.953
0.933
0.944
0.933
0.936
0.945
0.934
1
2
3
4
5
6
7
8
9
10
0.914
0.964
0.934
0.978
0.912
0.956
0.958
0.934
0.978
0.976
11
12
13
14
15
16
17
18
19
20
0.894
0.962
0.973
0.984
0.923
0.932
0.967
0.924
0.945
0.936
21
22
23
24
25
26
27
28
29
30
0.898
0.912
0.943
0.953
0.923
0.914
0.923
0.936
0.945
0.934
First samples:
less variance, higher confidence, shorter simulation run
Variance reduction 1
Large half widths caused by high subrun variance,
require very long simulations for acceptable confidence.
Various techniques to reduce subrun variance,
e.g. antithetic random numbers (see lecture notes)
Subrun result depends on sequence of random
numbers r1 , r2 ,...,rn
Next subrun: use antithetic sequence: 1  r1 ,1  r2 ,...,1  rn
Frequent arrivals, long processing times in a subrun
compensated by infrequent arrivals and short times
in next subrun.
Variance reduction 2
Second approach: sharing of e.g. arrival patterns.
Less risk of adopting inferior solution with fewer arrivals.
Both approaches introduce dependency in subruns.
A third approach: compensate for the number of entities.
sorted
subruns
sr
1
2
3
4
5
6
7
8
9
10
mq
0.934
0.944
0.936
0.958
0.932
0.946
0.938
0.934
0.948
0.936
#ent
3721
3696
3712
3754
3688
3718
3702
3694
3751
3734
sr
5
8
2
7
3
6
1
10
9
4
mq
0.932
0.934
0.944
0.938
0.936
0.946
0.934
0.936
0.948
0.958
combined
#ent
3688
3694
3696
3702
3712
3718
3721
3734
3751
3754
sr
5,4
8,9
2,10
7,1
3,6
mq
0.945
0.941
0.940
0.936
0.941
#ent
7442
7445
7430
7423
7439
Sensitivity analysis
Simulation model based on assumptions;
both modeling and parameters.
Assess dependency of simulation result on assumptions.
Simulate with modified parameters and compare.
Sensitive parameters / uncertain assumptions:
show various outcomes.
avql sol A
avql sol B
current demand
5.754
5.978
10% increase
7.872
7.901
15% increase
8.869
8.724