Title: Introduction to Probability Models
1Introduction to Probability Models
- Course Focus
- Textbook Approach
- Why Study This?
2Analysis of Stochastic Systems
- Analytical models
- Deductive
- Descriptive
- Insight
- Stochastic random (uncertain)
- Process time element
- Systems
- Multiple interacting parts
3Textbook 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
4Controlling Inventories with Stochastic Item
Returns (Fleischmann et al., 2002)
- Situation
- Manufacturer combines returned products with new
products to meet demand - Stochasticity
- Demands
- Arrivals of returned products
- 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
5Play 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
6Can 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
7Approximating 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
8Analytical vs. Simulation Models
9Analytical 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