Recent Developments in
Simulation Modeling and
Analysis
James R. Wilson
Department of Industrial Engineering
North Carolina State University
Raleigh, NC 27695-7906
<www.ie.ncsu.edu/jwilson>
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Overview
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Introduction to Simulation
Fundamental Simulation Concepts
Simple Applications
Modeling and Generating Inputs
Designing and Analyzing Experiments
Steps in a Successful Simulation Study
An Interesting Recent Application
Wrap-Up / Q&A
2
Introduction
• Simulation is no longer the “method of last
resort” in OR/MS/IE studies.
– A 1978 survey of M.S.O.R. graduates of Case
Western Reserve ranked simulation 5th in value
after graduation.
– Longitudinal study of corporate practice (1983,
1989, 1993) ranked simulation 2nd following
statistical analysis.
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Basic Simulation Objectives
• We use simulation to:
– Describe and analyze the behavior of a complex
system,
– Ask “what if” questions about the system, and
– Aid in the design or improvement of the
system.
• The system could be existing or conceptual.
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Fundamental Simulation Concepts
• Simulation: building a mathematical-logical
model of a system and experimenting with
it on a computer.
• In a discrete-event simulation, variables
defining system status are constant between
event times at which they may jump to
reflect changes in the system.
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• In a process-interaction simulation, entities
(traffic units, customers, parts) flow through
a queueing network, contending for service
at the workstations in the network.
• In a continuous simulation, system status
variables change continuously over time
according to a differential or difference
equation or an explicit function of time.
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• In a combined discrete-continuous
simulation, system status variables may
make jumps at event times but are constant
or changing continuously between those
event times.
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Reasons to Simulate
• To see if a new proposed system can meet design
requirements;
• To redesign an old system to meet new
requirements;
• To resolve disputes about system operation;
• To solve urgent problems (e.g., bottlenecks);
• To sell a new idea; and
• To create a system specification or action plan.
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Key Simulation Application Areas
• Manufacturing—material handling,
scheduling, inventory control, staffing
• Supply chain management
• Service industries—banks, restaurants,
theme parks
• Health care—emergency, operating rooms
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Application Areas (Cont’d)
• Computer/communications systems
• Military operations—war games, logistical
support, command and control
• Governmental operations—operation of
court, prison, and probation/parole systems
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Simple Simulation Examples
• A single-server queue at a fast-food
restaurant
– Customers arrive randomly, wait for service
first-come-first-served
– Constant service rate when server is busy
– Want to estimate server utilization, avg. queue
length, avg. waiting time for food
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A Discrete-Event Model of Restaurant
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A Process-Interaction Model of Restaurant
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Combined Discrete-Continuous Model
• Consider a refinery that is fed from storage tanks
that are supplied by ocean-going tankers.
Storage Tank
Waiting Tankers
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Discrete-Change System State Variables
• Number of Tankers Waiting
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Continuous-Change System State Variables
• Crude Oil Level in the Storage Tank
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Modeling and Generating
Probabilistic Simulation Inputs
• Methods for generating random numbers—
that is, random samples from the uniform
distribution on the unit interval [0, 1];
• Methods for generating nonuniform random
variables;
• Methods for selecting input models.
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Generating Random Numbers
• Multiplicative-congruential generator (MCG):
Generate “random” integers {Zi : i = 1, 2, … } in the
range 0, 1, …, m–1, starting with the seed Z0 via
Zi = a * Zi–1 ( mod m ) for i = 1, 2, …,
with multiplier a = 16,807 and modulus m = 231–1 =
2,147,483,647.
The associated “random” numbers {Ui } are
Ui = Zi / m for i = 1, 2, … .
• Used in many popular simulation languages; has
cycle length m–1 = 2,147,483,646 ≈ 2.1 * 109.
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Generating Random Numbers (Cont’d)
• A combined multiple recursive generator
(CMRG)
Starting with seeds X0, X–1, X–2, Y0, Y–1, Y–2, operate two
MCGs in parallel,
Xi = ( 1403580*Xi–2 – 810728*Xi–3 ) ( mod 4294967087 )
Yi = ( 527612*Yi–2 – 1370589*Yi–3 ) ( mod 4294944443 );
and then combine these streams of “random” integers via
Zi = ( Xi – Yi ) ( mod 4294967087 ) for i = 1, 2, … .
The associated “random” numbers {Ui } are
Ui = Zi / 4294967088, if Zi > 0; and
Ui = 4294967087 / 4294967088, if Zi = 0.
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Generating Random Numbers (Cont’d)
• See L’Ecuyer et al. (Operations Research, 2002).
• This CMRG has cycle length ≈ 3.1 * 1057.
• Run time per random number is only slightly
greater than the standard MCG.
• Even under Moore’s law (computers double in
speed every 1.5 years), this CMRG would require
require 219 years to cycle on leading-edge PCs.
• Used in Arena 5.0; see Kelton et al., Simulation
with Arena (2002).
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Generating Nonuniform Random Variables
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Inverse Transform Method
Acceptance/Rejection Method
Composition Method
Convolution Method
Alias Method
Special Properties
Focus on these
(continuous rv)
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Inverse Transform Method
• Continuous Case (similar for discrete, mixed random
variables):
– Random variable X has cumulative distribution function
(cdf) F(x) = Pr{X ≤ x} that is increasing for all real x.
cdf) F(x) = Pr{X ≤ x} that is increasing for all real x.
df) F(x) = Pr{X ≤ x} that is increasing for all real x.
f) F(x) = Pr{X ≤ x} that is increasing for all real x.
) F(x) = Pr{X ≤ x} that is increasing for all real x.
F(x) = Pr{X ≤ x} that is increasing for all real x.
x0
F(x) = Pr{X ≤ x} that is increasing for all real x.
(x) = Pr{X ≤ x} that is increasing for all real x.
real x.
rate U ~ Uniform(0, 1) using the random-number generator.
F(X) = U for X and return the solution as the sampled X
ampled
mpled XX
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Example of Inverse Transform
• Weibull(α, β) distribution with parameters α > 0 and β > 0
• Probability density function (pdf) is
αβ −α xα −1e −( x / β )
f ( x) = 
0
α
otherwise
1 − e −( x / β )
F ( x) = ∫ f (t )dt = 
−∞
0
x
• Cdf is
if x > 0
α
if x > 0
otherwise
• Solve U = F(X) for X:
U = 1− e
− ( X / β )α
⇒ X = β [ − ln(1 − U ) ]
1/ α
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Example of Inverse Transform (Cont’d)
• Algorithm:
1. Generate U ~ Uniform(0, 1)
1/ α
2. Return X = β ( − ln(1 − U ) )
• Weibull(α = 1.5, β = 6) example
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⇒
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Acceptance-Rejection Method
• Generally used when speed is an issue and the cdf is
analytically and numerically intractable.
• Goal: Generate X with pdf f(x).
Specify a function t(x) that majorizes f(x), i.e., t(x) ≥ f(x)
for all x
t (x )
f (x )
x
∞
∞
−∞
−∞
• Then t(x) ≥ 0 for all x, but C ≡ ∫ t ( x) dx ≥
∫ f ( x) dx = 1.
• If C is finite, then r(x) = t(x)/C is a pdf.
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Acceptance-Rejection Method (Cont’d)
• Acceptance-Rejection Algorithm:
1. Generate Y having pdf r(·).
2. Generate U ~ Uniform(0, 1) (independent of Y in Step
1).
3. If U ≤ f(Y)/t(Y), return X ← Y and stop;
else go to Step 1.
• Since Pr{Y ∈ [x, x+∆x] } ≈ r(x) ∆x and
Y ∈ [x, x+∆x] } ≈ f(x)/t(x) , we see that
f(Y)/t(Y) } ≈ r(x)∆x f(x)/t(x) ∝ f(x) ∆x,
f(x)
must
be∆x,
able to generate Y ~ r(·) easily.
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Acceptance-Rejection Method (Cont’d)
• Beta(4,3) distribution, density is f(x) = 60 x3 (1 – x)2 for 0 ≤ x ≤ 1.
• Peak of density is f(0.6) = 2.0736 (exactly), so let t(x) = 2.0736 for
0 ≤ x ≤ 1.
• Thus, c = 2.0736, and r(x) is the Uniform(0, 1) pdf.
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Acceptance-Rejection Method (Cont’d)
• Algorithm:
1. Generate Y ~ Uniform(0,1)
2. Generate U ~ Uniform(0,1) independent of Y
3. If U ≤ 60 Y3 (1 – Y)2 / 2.0736, return X ← Y
and stop; else go to step 1.
• On each iteration of step 3, Pr(acceptance) =
1/2.0736 = 0.48.
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Intuitive Basis for Acceptance-Rejection Method
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A Fatal Variate-Generation Scheme
• Box-Müller acceptance-rejection method for
generating pairs of normal random variables is
widely used but does not work well with some
MCGs.
• Scatter plots of pairs of normal variates should
show points symmetrically distributed about the
origin since bivariate normal pdf is bell-shaped.
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Selecting Valid Input Models
• Selecting a valid input model can be difficult.
– Availability of data: “feast or famine” syndrome.
– Many data sets arising in simulation studies cannot be
adequately modeled by standard pdfs.
• Good input-modeling packages are available; see
<www.informs-cs.org/wscpapers.html>.
• For a state-of-the-art review, see Barton et al. in
Proceedings of the 2002 Winter Simulation
Conference. Slides available via
<www.ie.ncsu.edu/jwilson>.
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Designing & Analyzing Simulation
Experiments
• Analyzing simulation output is difficult. Consider
consecutive customer waiting times.
– Not normally distributed – usually skewed;
– Not identically distributed – may have
transients or cyclic effects (e.g., warm-up or
time-of-day effects); and
– Not independent – usually correlated.
• Can’t analyze via “usual” statistical methods.
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Designing & Analyzing Simulation
Experiments (Cont’d)
• Two general cases to consider
– Terminating Simulations
• Interested in short-term behavior
• Example: Avg. customer waiting time in a bank
over the course of a day
– Steady-State Simulations
• Interested in long-term behavior
• Example: Continuously running assembly line
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Analyzing Terminating Simulations
• Method of Independent Replications
– Make independent runs (replications) of the
simulation, each under identical conditions.
– The sample means from each replication are
assumed to be approximately i.i.d. normal.
– Use classical statistical techniques on the
sample means (not the original observations).
– For handling nonnormal data, see Chapter 6 in
Computer Performance Modeling Handbook
(Lavenberg 1983).
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Analyzing Steady-State Simulations
• First deal with initialization (warm-up) bias.
– Usually we must “warm up” the simulation before
collecting data.
– Failure to do so can ruin the statistical analysis.
• Many methods for dealing with steady-state data
–
–
–
–
Batch Means,
Spectral Analysis,
Standardized Time Series, and
Regenerative Analysis.
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Analyzing Steady-State Simulations
(Cont’d)
• Method of Batch Means
– Make one long run (vs. many shorter runs).
– Warm up simulation before collecting data.
– Divide remaining observations into adjacent
batches of equal size.
– Assume the sample means from each batch are
approximately i.i.d. normal.
– Use classical statistical techniques on the batch
means (not on the original observations).
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Method of Batch Means
• Getting a confidence interval (CI) for the mean of
a highly correlated autoregressive process:
– CIs via classical statistics (“CLT” method on next
page) result in severe undercoverage.
– Look at batch means vs. CLT.
– For recent extensions of batch means that promise to
provide much more reliable CIs in practice, see
• Steiger and Wilson, Management Science (2002).
• Steiger et al., Proceedings of the 2002 Winter Simulation
Conference. Available via <www.informs-cs.org>.
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Key Steps in a Successful Simulation
Study
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Formulate the problem.
Specify the system and the simulation objectives.
Formulate and build the simulation model.
Verify (debug) the simulation model.
Validate the simulation model.
Design, perform, and analyze experiments with
the simulation model.
• Present recommendations and document the
results.
• Implement recommendations, disseminate model.
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An Interesting Recent Application
• Since 1995, United Network for Organ Sharing
(UNOS) has sponsored a series of simulation
models for evaluation of organ-allocation policies
in terms of
– Equity (fairness, justice);
– Medical Utility (efficiency); and
– Availability of organs for transplantation.
• ULAM (UNOS Liver Allocation Model) was the
first such simulation model; see Pritsker, OR/MS
Today (1998).
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ULAM Architecture
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ULAM Animation
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ULAM Animation (Cont’d)
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Wrap-Up / Q&A
• No longer the “method of last resort,” simulation
is now one of the most popular and effective tools
for studying complex systems.
• For state-of-the-art simulation work, see the
Proceedings of the Winter Simulation Conference
online at
<www.informs-cs.org/wscpapers.html>.
• Questions?
• Have a great conference!
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