IOE 316 Winter 2013 Homework 1 Solution
Due 10:30 am on March 19, 2013
1. (25 points) The professor of an IOE class may end the lecture early, on time, or late on
any given day. Students have noticed that they can forecast the probability of ending the
cl

IOE 316 Fall 2016 Homework 4
Due 9:10 am on November 10, 2016
1. (5 points) The lifetime of deck stain is exponentially distributed with mean of 6 years. If your
deck was stained 10 years ago and the deck is still OK, what is the probability that it will

IOE 316 Fall 2012 Homework 3
Due 9:10 am on November 27, 2012
1. (10 points) The lifetime of a TV is exponentially distributed with mean of 3 years. If you
buy a four-year-old TV, what is the probability that it will be working after an additional 3
years

IOE 316 Winter 2011 Homework 1
Due 10:30 am on March 15, 2011
1. (40 points) Let X and Y be two discrete random variables with joint density function given
by
1
3
1
1
12
1
12
1
12
2
1
12
1
6
1
6
3
x
y
2
1
12
1
4
0
Compute the probability of the following

IOE 316 Fall 2012 Homework 4 Solution
Due 9:10 am on December 4, 2012
1. (35 points) The Dance Marathon is a 30 hour event during which people can make online
or cash donations. Assume that 70 percent of the donations are made online and all other
donatio

IOE 316 Fall 2012 Homework 5 Solution
Due MONDAY December 10, 2012 12:30pm
Note: To turn in your homework, please bring it to the GSIs oce hours on Monday,
December 12, 2011, between 8:30am and 12:30pm (IOE G610). No late homework will be
accepted.
1. (35

IOE 316 Fall 2012 Homework 3 Solution
Due 9:10 am on November 27, 2012
1. (10 points) The lifetime of a TV is exponentially distributed with mean of 3 years. If you
buy a four-year-old TV, what is the probability that it will be working after an additiona

IOE 316 Winter 2011 Homework 3 Solution
Due Monday, March 28, 2011 at 2:00pm
Note: To turn in your homework, please bring it to the GSIs oce hours on
Monday, March 28, 2011, between 10:00am and 2:00pm (IOE G610).
No late homework will be accepted
1. (20 p

IOE 316 Winter 2011 Homework 5
Due 10:30 am on April 12, 2011
1. (15 points) Let cfw_N (t), t 0 be a Poisson process with rate . Let Sn denote the time of the
nth event.
(a) (5 points) Find E [S4 ].
(b) (5 points) Find E [S4 |N (1) = 2].
(c) (5 points) Fi

IOE 316 Fall 2007 Homework 6
Due MONDAY December 10, 2007 NOON (12pm) Note: To turn in your homework, you can either slide it under instructor's office door (1783 IOE) by noon, or bring it to Arleigh's office hours that morning. No late homework wil

IOE 316 Introduction to Markov Processes Fall 2007 Midterm Exam 2
Tuesday, December 11, 2007 10:40 am - 12:00 pm
NAME UMID
SOLUTION
Engineering Honor Code (Make sure to sign it before turning in the exam): I have not received nor given any unautho

IOE 316 Homework 1 Solutions
Due October 30, 2007
1. (20 points) Three white and three black balls are distributed in two urns in such a way that each contains three balls. We say that the system is in state i, i = 0, 1, 2, 3, if the first urn con

IOE 316 Fall 2007 Homework 4 Solutions
Due November 20, 2007
1. (10 points) As you arrive to a single-server fast-food restaurant you notice that there are three customers waiting in line and one additional customer is currently being serviced. Yo

IOE 316 Fall 2007 Homework 6 SOLUTIONS
Due MONDAY December 10, 2007 NOON (12pm) Note: To turn in your homework, you can either slide it under instructor's office door (1783 IOE) by noon, or bring it to Arleigh's office hours that morning. No late ho

IOE 316 Fall 2007 Homework 5 Solution
Due December 4, 2007
1. (25 points) Cars cross a certain point in the highway in accordance with a Poisson process with rate = 4 per minute. (a) (5 points) If Kate blindly runs across the highway, then what i

IOE 316 Fall 2012 Homework 2 Solution
Due 9:10 am on November 13, 2012
1. (35 points) Traders of a stock on an exchange pay close attention to the tickets of a stock.
On any given day, a stock can trade on an up-tick, even tick, or down-tick, if the trade

IOE 316 Fall 2012 Homework 1 Solution
Due 9:10 am on November 6, 2012
1. (20 points) A skydiver purchases a reusable parachute. If the parachute works on a given
jump, it will work on the next jump with probability 0.9, it will be defective with probabili

IOE 316 Fall 2012 Homework 4
Due 9:10 am on December 4, 2012
1. (35 points) The Dance Marathon is a 30 hour event during which people can make online
or cash donations. Assume that 70 percent of the donations are made online and all other
donations are ma

IOE 316 Fall 2012 Homework 1
Due 9:10 am on November 6, 2012
1. (20 points) A skydiver purchases a reusable parachute. If the parachute works on a given
jump, it will work on the next jump with probability 0.9, it will be defective with probability
0.08,

IOE 316 Fall 2012 Homework 2
Due 9:10 am on November 13, 2012
1. (35 points) Traders of a stock on an exchange pay close attention to the tickets of a stock.
On any given day, a stock can trade on an up-tick, even tick, or down-tick, if the trade price
is

NAME: _
IOE 316 Winter 2011
Quiz #5 Solution
April 8, 2011
Customers arrive at Kroger with rate 20 customers/hour(assume consistent throughout the day).
1) (1pt) What is the probability of one customer arriving Kroger within the next 3 minutes?
2) (1pt) W

NAME: _Solutions_
IOE 316 Winter 2011
Quiz #1
March 11, 2011
1. True or False, students may submit their homework by 6pm on the day it is due.
FALSE
2. True or False, Fridays lectures are mandatory.
TRUE
3. What are the first names of the instructor and t

IOE 316
Introduction to Markov Processes
Winter 2011
Midterm Exam 1
Tuesday, March 29, 2011
10:40 am - 12:00 pm
NAME
SOLUTION
UMID
Engineering Honor Code (Make sure to sign it before turning in the exam): I have
not received nor given any unauthorized aid

NAME: _
IOE 316 Winter 2011
Quiz #3
March 25, 2011
People connect with each other on the Web through three networking applications: Facebook, Twitter and Linked-in.
Assume that students only use one the applications each month. Define states as follows: S

NAME: _
IOE 316 Fall 2011
Quiz #4 Solution
December 2, 2011
On the long road trip back from Thanksgiving break, you decide to count the number of vehicles passing
in the left lane. Since youre a conservative driver , vehicles pass according to a Poisson p

NAME: _
IOE 316 Winter 2011
Quiz #6 Solution
April 15, 2011
Jessica still drives around her very first car, which was purchased 8 years ago. It frequently breaks
down and must be repaired. Every time her car is repaired, it remains running for an exponent

Lecture 3 Handout (from W12 Exam 1):
Consider a Markov Chain with the one step transition matrix P defined below, where
the x in the matrix represents a positive entry. Let the states be labeled in order as
cfw_0, 1, 2, 3, 4, 5, 6, 7.
P=
0
1/2
0
0
1/3
0
2

Q: Given the transition matrix as following
0.8
0.1
0
0
0
0.2
0
0.25
0
0
0
0.9
0.75
0
0.9
0
0
0
1
0.1
0
0
0
0
0
a). Draw the Markov diagram, with state = 0, 1, 2, 3, 4
b). Figure out all the classes
c). Which class is transient, which is recurrent?
d). Is

NAME: _
IOE 316 Winter 2011
Quiz #2 Solution
March 18, 2011
The basketball team, the Miami Heat, have three high scoring players: LeBron James, Dwayne
Wade, and Chris Bosh. Its guaranteed that one of these three players will be the highest scorer
(in poin

NAME: _
IOE 316 Fall 2011
Quiz #2
November 11, 2011
A professional tennis player always hits cross-court or down the line. In order to
give himself a tactical edge, he never hits down the line two consecutive times,
but if he hits cross-court on one shot,

IOE 316 Fall 2016 Homework 4 Solution
Due 9:10 am on November 10, 2016
1. (5 points) The lifetime of deck stain is exponentially distributed with mean of 6 years. If your
deck was stained 10 years ago and the deck is still OK, what is the probability that

6
Continuous-Time Markov Chains
6.1
Introduction
Recall that discrete time Markov Chains transition from one state to another at each
discrete count (e.g. each day, every hour, year). Continuous-time Markov Chains
transition from one state to another at

8
Queueing Theory
8.1
Introduction
One of the most important Industrial Engineering application of birth and death process is Queueing Theory. Let the state of the system denote the the number of customers in the system, at any moment in time a new custo

IOE 316: Introduction to Markov Processes
Reading Recommendations
Chapter 6
8th Edition:
6.1 Read the entire section
6.2 Read the entire section
6.3 Skip Example 6.4 except its first paragraph. Read Examples 6.5 and 6.6. Skip the
rest of the section after

Example 1
Suppose youre at U-Gos buying a snack. There are two cashiers.
You arrive at a time when both are busy, but there is no one else
waiting in line. You will enter service when either cashier becomes
free. If service times for clerk i are exponenti