ORIE 361 Final Examination Spring 2008
Instructor: Mark E. Lewis
May 7, 2008
This is a closed book, closed notes exam. However, you are allowed to use a single 8.5 X 11 inch sheet of paper with whatever formulas or notes you deem appropriate. No other res

OR&IE 361/523 - Section 2
1. A total of m white and m black balls are distributed among two urns, with each urn containing m balls. At each stage, a ball is randomly selected from each urn and the two selected balls are interchanged. Let Xn denote t

ORIE 361/523 Homework 9
Instructor: Mark E. Lewis
due April 9, 2008 (drop box)
1. Each individual in a biological population is assumed to give birth at an exponential rate and to die at an exponential rate . In addition, there is an exponential rate of i

ORIE 361 Homework 1 (Introduction to Probability Theory)
Instructor: Mark E. Lewis
due January 30, 2008 (drop box)
This homework assignment is designed to give you some practice in probability. Some of the concepts should be review, others are leading you

ORIE 361/523 Homework 2
Instructor: Mark E. Lewis
due February 6, 2008 (drop box)
1. Consider a gene composed of d subunits, where d is some positive integer and each subunit is either normal or mutant in form. Consider a cell with a gene composed of m mu

ORIE 361 Homework 2 (Introduction to Discrete Time Markov Chains)
Instructor: Mark E. Lewis
due February 5, 2007 (drop box)
Answers. 1. The state space is = cfw_0, 1, . . . , d and Pi,j =
2i j 2d2i dj 2d d
j = 0, 1, . . . , d, i = 1, 2, . . . , d 1
0 and

ORIE 361/523 Homework 3
Instructor: Mark E. Lewis
due February 13, 2008 (drop box)
1. A fair coin is tossed repeatedly with results Y0 , Y1 , Y2 , . . . that are 0 and 1 with probability 1/2 each. For n 1, let Xn = Yn + Yn1 be the number of 1 s in the (n

ORIE 361 Homework 3 (Introduction to Discrete Time Markov Chains)
Instructor: Mark E. Lewis
due February 13, 2008 (drop box)
Answers. 1. No, the chain is not Markov. Note P (Xn = 2|Xn1 = 1, Xn2 = 2) = 0, since, cfw_Xn2 = 2 implies cfw_Yn2 = 1 and cfw_Xn1

ORIE 361/523 Homework 4 Solutions
Instructor: Mark E. Lewis February 26, 2008
1. For large n we will estimate P (Xn = 1|X0 = 1) by the steady state probability of state 1. Suppose the steady state probability or the stationary distribution is given by = (

ORIE 361/523 Homework 5
Instructor: Mark E. Lewis
due February 27, 2008 (drop box)
1. One light bulb at the Rhodes student lounge has a maximum life of m days. In each day, the bulb will go crashed with probability p, in which case a new bulb must be inst

ORIE 361 Homework 5
August 2, 2007
Answers. 1. (a) =
1 1/2
= 2. Then F (x) = 1 - e-2x .
P (X > 1/2) = 1 - F (1/2) = e-21/2 = e-1 (b) By the memoryless property of exponential distribution, the probability is the same as in part (a), that is e-1 2. Let T b

ORIE 361/523 Homework 6
Instructor: Mark E. Lewis
due March 6, 2008 (drop box)
1. Two basketball players A and B will take turns attempting a three-point shot until either of them has made one, and then the game is over. A, being the better player, shoots

ORIE 361/523 Homework 6
Instructor: Mark E. Lewis
due March 13, 2008 (drop box)
1. A machine has two critically important parts and is subject to 3 different types of shocks. Shocks of type i occur at times of a Poisson process with rate i . Shocks of typ

ORIE 361/523 Homework 8
Instructor: Mark E. Lewis
due April 2, 2008 (drop box)
1. During the winter, snowstorms hit Ithaca according to a Poisson process. People believe that on average, Ithaca receives 4 snowstorms during February. On a particular Februa

ORIE 361 Preliminary Examination Spring 2008
Instructor: Mark E. Lewis
March 4, 2008
This is a closed book, closed notes exam. However, you are only allowed to use a single 8.5 X 11 inch sheet of paper with whatever formulas or notes you deem appropriate.

Introductory Engineering Stochastic Processes, ORIE 361
Instructor: Mark E. Lewis, Associate Professor
School of Operations Research and Information Engineering Cornell University
Renewal Theory
1/ 30
Renewal Theory
I T1
x T2
x
T3
x
T4
x
A renewal process

ORIE 361 Homework 1 (Introduction to Probability Theory)
Instructor: Mark E. Lewis
due January 30, 2008 (drop box)
This homework assignment is designed to give you some practice in probability. Some of the concepts should be review, others are leading you

ORIE 361/523 Homework 9
Instructor: Mark E. Lewis
due April 16, 2008 (drop box)
1. Potential customers arrive at a a single-server station (which works at rate ) in accordance with a Poisson process with rate . However, if a customer finds n people alread

Introductory Engineering Stochastic Processes, ORIE 361
Instructor: Mark E. Lewis, Associate Professor
School of Operations Research and Information Engineering Cornell University
Introduction to Continuous-time Markov Chains
1/ 15
Monotone Increa

Introductory Engineering Stochastic Processes, ORIE 361 Instructor: Mark E. Lewis, Associate Professor
Introductory Engineering Stochastic Processes, ORIE 361
Instructor: Mark E. Lewis, Associate Professor
School of Operations Research and Informati

Introductory Engineering Stochastic Processes, ORIE 361
Instructor: Mark E. Lewis, Associate Professor
School of Operations Research and Information Engineering Cornell University
Markov Chains Introduction
1/ 11
Stochastic Processes
A stochastic

Introductory Engineering Stochastic Processes, ORIE 361
Instructor: Mark E. Lewis, Associate Professor
School of Operations Research and Information Engineering Cornell University
Markov Chains State Classifications
1/ 11
Motivation
In any syste

Introductory Engineering Stochastic Processes, ORIE 361
Instructor: Mark E. Lewis, Associate Professor
School of Operations Research and Information Engineering Cornell University
More on class Properties
1/ 24
Periodicity
Now that we know that i

Introductory Engineering Stochastic Processes, ORIE 361 Instructor: Mark E. Lewis, Associate Professor
Introductory Engineering Stochastic Processes, ORIE 361
Instructor: Mark E. Lewis, Associate Professor
School of Operations Research and Informati

Motivating Example
Suppose you are Houston Jones. You work for a well-known micro-chip developer Pintel. You are part of the logistics group and are responsible for making production quantity and shipping decisions. Today your boss walked in and said

ORIE 361
Handout 1
ORIE 361 Introductory Engineering Stochastic Processes Spring 2008
Basic Course Information
Instructor Information
Instructor: Mark E. Lewis Associate Professor Operations Research and Information Engineering Rhodes 226 255-0757

Introductory Engineering Stochastic Processes, ORIE 361
Instructor: Mark E. Lewis, Associate Professor
School of Operations Research and Information Engineering Cornell University
Spring, 2008
1/ 28
Disclaimer
This file can be used as a study gui

Some Background Material for ORIE 361
Instructor: Mark E. Lewis
1
Conditional Probability
Let A and B be events. Then P (A|B) = P (A B) . P (B)
Let X and Y be discrete random variables. Then for any values x and y P (X = x|Y = y) = P (X = x, Y = y) . P (Y

Introductory Engineering Stochastic Processes, ORIE 361
Instructor: Mark E. Lewis, Associate Professor
School of Operations Research and Information Engineering Cornell University
CTMCs II
1/ 1
Time Homogeneity
Definition A continuous-time stochastic proc