1. Chapter 01 Introduction to Probability Models Course Focus Textbook Approach Why Study This?

2. Chapter 02 Analysis of Stochastic Systems Analytical models –Deductive –Descriptive –Insight Stochastic = random (uncertain) –Process: time element Systems –Multiple interacting parts

3. Chapter 03 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. Chapter 04 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. Chapter 05 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. Chapter 06 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. Chapter 07 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. Chapter 08 Analytical vs. Simulation Models DimensionAnalyticalSimulation ComplexityTry to include only most important aspects Can be very complex and detailed (+/-) FlexibilityEasier to change but small changes may have big consequences Hard to change once built DataNeeds lessNeeds more TransparencyClear to analyst, may be opaque to less well trained Black box EfficiencyEffort to get tractable solution hard to estimate Effort more “linear” and predictable

9. Chapter 09 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