RSM Estimation for Robust Design of Queueing Systems

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RSM Estimation for Robust Design of Queueing
Systems

• Russell R. Barton
• The Smeal College of Business Administration
• The Pennsylvania State University

Research Team Nirmal Govind, Intel (Primary
Author Doctoral Thesis) DJ Medeiros, Penn
State Lee Schruben, Berkeley
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RSM for Robust Design - Overview

• Robust design philosophy (vs RSM) and methods
• Queueing systems and the robust design issue
• A robust design example
• How to estimate parameters for RSM (metamodel)
  based robust design
• Results

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Robust Design Philosophy RD vs RSM

• Response Surface Methodology
• x (weld current, weld time), Y weld strength
• Max (E(Y(x))
• How build RSM metamodel, perform local
  optimization
• Y b0 xb xBx e, e i.i.d. N(0, s2)
• How to estimate parameters for local RSM
  metamodel?
• DOE (vary xs measure Ys) Least Squares

x
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Robust Design Philosophy

• Key assumption both mean and variability of
  system performance are important E(weld
  strength), Var(weld strength)
• Taguchis approach involved fractional factorial
  designs and selection of optimal operating
  conditions based on univariate (marginal)
  summaries
• Modern approach is based on fitting response
  models to the data and optimizing the response
  model

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-1
0
1
2000
3850
3250
1500
1750
550
500
600
1
2000
3200
Robust Design
1750
3000
Graphical Analysis
2850
200
2250
3750
1700
B
1100
0
600
1600
2850
2100
1850
1450
1050
100
1350
750
E
100
300
2700
-1
1150
1300
2300
D
150
500
C
A
Design Factors (x)
S
h
e
a
r

F
o
r
c
e
A
Welding Current
0

-

7
0
0

l
b
s
.
B
Cycle Time
7
0
1

-

1
4
0
0

l
b
s
.
Nuisance Factors (z)
1
4
0
1

-

2
1
0
0

l
b
s
.
C
Material Thickness
D
Air Pressure
o
v
e
r

2
1
0
0

l
b
s
.
E
Surface Cleanliness
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Robust Design vs RSM

• Robust Design Methodology
• Min Var(Y(x, Z))
  (1)
• s.t. E(Y(x)) S
• Z are hard to control noise factors that vary
  randomly during normal operation of the system
• Assume we can control them (set to value z)
  during experimentation
• Approach fit metamodel and do global
  optimization

x
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RSM-based Robust Design

• How to develop functional approximations for
  E(Y(x)) and Var(Y(x))
• Two-metamodel approach
• Y b0 xb xBx e, e i.i.d. N(0, s12)
  (2)
• Log Var(Y) a0 xa xAx d, d i.i.d. N(0,
  s22) (3)
• Problems with this approach (replications, proper
  scaling of z)
• Alternative single metamodel approach

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RSM-based Robust Design

• Single metamodel approach includes noise
  variables (z) explicitly
• Y b0 xb xBx zg xDz e, e i.i.d.
  N(0, se2) (4)
• THEN
• E(Y) b0 xb xBx
• and
• Var(Y) (g Dx)Sz (g Dx) se2
• Fit the model (4) and use the subsequent models
  to optimize (1)
• This is called RSM-based robust design

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RSM-based Robust DesignHow to Estimate
Parameters

• Design an experiment in which x and z are varied,
  typically using an RSM/factorial design
• Typical Robust Design Assumption

• Normal System Operation
• Z F

• Experimentation
• z -1, 1 OR Z F

Our NPM Assumption

• Normal System Operation
• Z F

• Experimentation

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Noise Plus Mean Model

• Noise plus mean model
• Consequence

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RSM-based Robust DesignHow to Estimate
Parameters

• V is a function of the parameters to be
  estimated!
• OLS/WLS can't be employed
• We propose Two-Stage Estimation

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Robust Design of Queueing Systems

• Performance metric Cycle time, Waiting time,
• Noise Inter-arrival time, Service time (previous
  work assumes rates)
• Mean performance widely used
• Correlation between mean and variance ? robust
  design often trivial!

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Robust Design in Queueing can be Trivial

• Consider determining a probabilistic routing
  parameter for a two-resource queueing system (old
  and new machines)
• Performance cycle time
• Costs
• Cw per time unit

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Robust Design in Queueing

• Robust design for this queueing application is
  not interesting expectation and variance are
  monotonically related RSM is simpler and yields
  the same result

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Robust Design in Queueing Example

• Consider a model with a fixed cost per customer
  (e.g. depreciation of the machine)
• Costs
• Ci per customer
• Cw per time unit

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Robust Design in Queueing Example

• Robust design for cost is interesting
• If S E(cost) lt 83 then p .64
• Q how to find these functions in general?

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RSM-based Robust DesignHow to Estimate
Parameters

• DOE for our example
• zs are l, m1, m2
• x is p
• Factorial design on x (3 levels) and z (2 levels
  each)
• Estimation approaches
• New Two-stage Approach
• Ordinary Least Squares (OLS) (problems as cited
  before)
• Regression

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Results
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Results
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Summary

• The NPM framework for robust design can provide
  more precise estimates for RSM-based robust
  design
• The NPM framework allows robust design of
  queueing systems with noise factors such as
  inter-arrival and service times
• The NPM framework is also effective in more
  conventional robust design setting (Ys
  independent, Sz not dependent on )

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Publications

• Govind, N., Barton, R. R., and Medeiros, D. J.
  (2004a), A Response
• Surface Framework for Robust Parameter Design
  with Imperfect Experimental Control of Noise,
  submitted for publication.
• Govind, N., Medeiros, D. J., Barton, R. R., and
  Schruben, L. W.
• (2004b), Variance Response Surface Estimation
  for Robust Design A
• Framework for Queueing Systems, to appear in
  Proceedings of the
• 2004 Industrial Engineering Research Conference,
  Institute of Industrial
• Engineers.
• Govind, N., Barton, R. R., Medeiros, D. J., and
  Schruben, L. W. (2004c),
• A Response Surface Framework for Robust Design
  of Queueing
• Systems, working paper.
• Duenyas, I. and Hopp, W. J. (1990), Estimating
  Variance of Output from Cyclic
• Exponential Queueing Systems, Queueing Systems,
  7, 337 354

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Using replicated design points