Introduction to Probability Models

1
Introduction to Probability Models

• Course Focus
• Textbook Approach
• Why Study This?

2
Analysis of Stochastic Systems

• Analytical models
• Deductive
• Descriptive
• Insight
• Stochastic random (uncertain)
• Process time element
• Systems
• Multiple interacting parts

3
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

4
Controlling 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

5
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

6
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

7
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

8
Analytical vs. Simulation Models
9
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