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Introduction to Probability Models

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Model/Technique: Continuous time Markov chain, renewal reward, ... Try to include only most important aspects. Complexity. Simulation. Analytical. Dimension ... – PowerPoint PPT presentation

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Title: 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
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