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Introduction to Probability Models

Introduction to Probability Models. Course Focus Textbook Approach Why Study This?. Analysis of Stochastic Systems. Analytical models Deductive Descriptive Insight Stochastic = random (uncertain) Process: time element Systems Multiple interacting parts. Textbook Orientation.

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Introduction to Probability Models

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  1. Introduction to Probability Models Course Focus Textbook Approach Why Study This? Chapter 0

  2. Analysis of Stochastic Systems • Analytical models • Deductive • Descriptive • Insight • Stochastic = random (uncertain) • Process: time element • Systems • Multiple interacting parts Chapter 0

  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 Chapter 0

  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 Chapter 0

  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 Chapter 0

  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 Chapter 0

  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 Chapter 0

  8. Analytical vs. Simulation Models Chapter 0

  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 Chapter 0

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