Introduction to Probability
Models
Course Focus
Textbook Approach
Why Study This?
Chapter 0
1
Analysis of Stochastic Systems
• Analytical models
– Deductive
– Descriptive
– Insight
• Stochastic = random (uncertain)
– Process: time element
• Systems
– Multiple interacting parts
Chapter 0
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Textbook Orientation
• Intuitive approach  probabilistic thinking
• Conditioning as a tool for understanding
and simplifying
– What additional knowledge would help to
answer this question?
• Similar structure in various applications
Chapter 0
3
Controlling Inventories with Stochastic
Item Returns (Fleischmann et al., 2002)
• Situation
– Manufacturer combines returned products with new products to meet demand
• Stochasticity
Procurement orders,
delivery time 
– Demands
– Arrivals of returned products
Returns with
intensity  R
Reusable
Inventory
Demand with
intensity  D
• Objective
– order policy minimizing the long-run expected average costs per unit time
• when, how much
• costs for ordering, holding, failing to satisfy demand on time
• Model/Technique: Poisson process
Chapter 0
4
Play It Again, Sam? (Swami, et al., 2001)
• Situation
– Theater manager decides weekly whether to keep or replace
currently showing movies
• Stochasticity
– Demand for movies as they “age”
– Timing of future releases
• Objective
– Replacement policy to maximize expected total revenue over a
planning period
• Given contractual obligations, ranks of all movies available
• Revenue-sharing arrangements with distributors
• Model/Technique: Markov decision process
Chapter 0
5
Can Difficult-to-Reuse Syringes Reduce
the Spread of HIV? (Caulkins, et al., 1998)
• Situation
– U.S. Surgeon General recommended that regular syringes be
replaced by DTR syringes to reduce sharing by injection drug
users
• Stochasticity
– whether or not a given syringe is infectious
– how many times a regular syringe is reused
• Objective
– Predict whether policy recommendation will work as intended
• Model/Technique: Markov chain, Circulation theory
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Approximating the Variance of Electric
Power Production Costs (Ryan, 1997)
• Situation
– Both the load (demand for power) and the availability of electric
power generating units vary over time
– If cheap units are unavailable when demand is high, then cost soars
• Stochasticity
– Availability of more or less expensive generating units over time
• Objective
– Efficiently estimate the variance of the cost to provide interval, not
just point, estimate of production cost
• Model/Technique: Continuous time Markov chain, renewal
reward, conditional variance
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Analytical vs. Simulation Models
Dimension
Analytical
Simulation
Complexity
Try to include only most
important aspects
Can be very complex and
detailed (+/-)
Flexibility
Easier to change but small
changes may have big
consequences
Hard to change once
built
Data
Needs less
Needs more
Transparency
Clear to analyst, may be
opaque to less well trained
Black box
Efficiency
Effort to get tractable
solution hard to estimate
Effort more “linear” and
predictable
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Analytical vs. Simulation: Summary
• Both are important!
– Use simulation to validate analytical approximations
– Use analysis to determine where to focus simulation effort
• For stochastic systems, both will be descriptive not prescriptive
– Analytical models usually easier to combine with optimization
– Ideal: closed form expression for performance in terms of parameter(s) – can
use calculus or search algorithm to optimize
– Simulation-based optimization is a growing field
• What is the purpose of the model?
–
–
–
–
–
Understanding: Gain insight into how variable affects performance
Teaching: Help managers/workers understand what factors affect performance
Improvement: Explore changes in parameters and rules
Optimization: Find an optimal combination of parameters
Decision Making: How to design and/or operate the system
• Discriminate effects of alternatives
• Project their impact over time
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