Reasons Modeling is Difficult
System size & Complexity
Systems always changing
Relationships
Data or Info Gathered
Difficult to define
objectives
Act of observing error
Analysis subject to user
error
Analytical vs. Simulation
Analysis
Analytical -E
I E 413 Assignment 6 Jess McCall Ju Xiong
I n t roduction: We model a small manufacturing system, which consists of four manufacturing cells. T here are 3 different part types from customer 1: Part 1, 2 and 3. The interarr ival time between successive par
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IE 413
Lecture Notes
Week 8
Fall 2016
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Learning Objectives
IE 413
2
At the end of this week you will be able to
Calculate and interpret the steady-state behavior of ergodic
Markov chains
Calculate probabilities of which absorbing state is reached
in an
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IE 413
Lecture Notes
Week 10
Fall 2016
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Learning Objectives
n At
the end of this week, you will be able to
n Use
batch-means to estimate performance of a
simulation model using a single sample path
n Use resampling to improve the true coverage of the
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IE 413
Lecture Notes
Week 7
Fall 2016
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Learning Objectives
IE 413
2
At the end of this week, you'll be able to .
Determine when a discrete time Markov chain is an
appropriate model for an application
Write down a state space and transition matrix for a
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IE 413
Lecture Notes
Week 11-12
Fall 2016
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Learning Objectives
IE 413
2
At the end of week 11-12, you'll be able to
Model systems as a continuous time Markov chain (CTMC)
Calculate and interpret the stationary distribution of CTMCs
Model Markovian que
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IE 413
Lecture Notes
Week 13
Fall 2016
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Learning Objectives
IE 413
2
At the end of this week you will be able to
Use common random numbers (CRNs) to reduce simulation
output variance when comparing two or more systems
Use antithetic random variates, a
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IE 413
Lecture Notes
Week 14
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2
Purpose
(rather than specific learning objectives)
n
Think about the broader context of some of the issues
discussed over the semester, especially
n
Simulation project process
Simulation experimental design
n
Simulation
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IE 413
Lecture Notes
Week 2
Fall 2016
Fall 2016
IE 413
+
Learning Objectives
Use stochastic processes to model outputs that are
subject to uncertainties
Note that this objective will be addressed repeated
throughout the semester. This week, we will focu
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IE 413
Lecture Notes
Week 9
Fall 2016
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Learning Objectives
At
IE 413
2
the end of this week, you will be able to:
Estimate performance measures and construct
confidence intervals base on simulation replications
Identify appropriate initial conditions
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IE 413
Lecture Notes
Week 1
Fall 2016
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A Short Introduction
Hi! My name is Siggi Olafsson
Originally from Iceland and have lived
around here since 1994 (but the
accent is definitely still form Iceland!)
The view from my
window when I was a kid
2
A rive
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IE 413
Lecture Notes
Week 5
Fall 2016
Fall 2016
IE 413
+
Learning Objectives
At
2
the end of the week you will be able
to
Model
a simple system using appropriate
state variables and events
Generate a sample path for a simulation
program by hand
IE 41
+
IE 413
Lecture Notes
Week 5
Fall 2016
Fall 2016
IE 413
+
Learning Objectives
At
2
the end of the week you will be able
to
Implement
IE 413
a simple simulation using VBA
Fall 2016
+
Code by Hand?
Will I need to code by hand on the exam?
No!
(But expect
Arrival/Departure events - how does simulation handle
these events?
Primary and secondary events
Snapshots- Points in a simulation that when put together
make up the evolution of the simulation
Future Events List Mechanism for advancing simulation
time an
Arrival/Departure events - how does simulation handle
these events?
Primary and secondary events
Snapshots- Points in a simulation that when put together
make up the evolution of the simulation
Future Events List Mechanism for advancing simulation
time an
Markov Chains
Pij= Probability of going from state I to state j
in n steps
N Step Probabilities
To find how to move from I to j in n steps, take
the matrix ^n, then find from I to j
Probability of being state j at time n = Sum of
(Probability state is i)*
9/14/2011
Quicksummaryofthelastlectureonthe
Poissonprocess
Fromaslightlydifferentangle!
Queueof
Customers
Population
Server
Exit
PostedPoissonProcessinAdditionalHandoutsfolder
Manyarrivalprocessesaresuchthattheinterarrivaltimes
areexp()
i.e.
X1
Interarriv
INTERARRIVAL DISTRIBUTION OF CARS
Time between
Arrivals (min)
1
2
3
4
Probability
Cumulative
Probability
0.25
0.40
0.20
0.15
0.25
0.65
0.85
1.00
Random Number
Assignment
0.00R0.25
0.25<R0.65
0.65<R0.85
0.85<R1.00
SERVICE DISTRIBUTION OF ABLE
Service Time
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IE 413
Week 3
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Learning Objectives
IE 413
2
At the end of this week, you will be able to
Determine the appropriate method for finding the
distribution of a random variable
Use data to fit a distribution to data when appropriate
Generate a realization