Chapter 0 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

Reusable Inventory

Procurement orders, delivery time

Returns with intensity R

Demand with intensity D

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 ModelsDimension Analytical Simulation

Complexity Try to include only most important aspects

Can be very complex and detailed (+/-)

Flexibility Easier to change but small changes may have big consequences

Hard to change once built

Data Needs less Needs more

Transparency Clear to analyst, may be opaque to less well trained

Black box

Efficiency Effort to get tractable solution hard to estimate

Effort more “linear” and predictable

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