Copyright c 2009 by Karl Sigman
1
Discrete-time renewal processes
Imagine busloads of passengers, where the ith bus contains Hi passengers, and the {Hi } are iid
with pmf
p(k) = P (H = k), k ≥ 1.
(1)
Imagine further that the seats on a bus are labeled 1,
IEOR 6711: Stochastic Models I First Midterm Exam, Chapters 1-2, October 7, 2008
Justify your answers; show your work. 1. A sequence of Events. (10 points) Let cfw_Bn : n 1 be a sequence of events in a probability space (, F, P ). (a) Explain what that me
IEOR 6711: Stochastic Models I First Midterm Exam, Chapters 1-2, October 10, 2010
Justify your answers; show your work. 1. Exponential Random Variables (23 points) Let X1 and X2 be independent exponential random variables with means E[X1 ] 1/1 and E[X2 ]
IEOR 6711, Stochastic Models, I: Final Exam
Fall 2013, SOLUTIONS
There are three questions, each with several parts.
1. Customers Coming to a Group of Automatic Teller Machines
(35 points: 3 points each for rst 5 parts; 4 points each for last 5 parts)
Cus
IEOR 6711: Stochastic Models, I Fall 2012, Professor Whitt, Final Exam SOLUTIONS
There are four questions, each with several parts. 1. Customers Coming to an Automatic Teller Machine (ATM) (30 points) Customers arrive one at a time at a single automatic t
IEOR 6711: Stochastic Models, I Fall 2010, Professor Whitt, Final Exam SOLUTIONS
There are four questions, each with several parts. Question 2 is longer than the others and thus counts more. 1. The Eight (8) Subway Line. A new subway line has been added t
IEOR 6711: Stochastic Models, I Fall 2012, Professor Whitt, Final Exam
There are four questions, each with several parts. 1. Customers Coming to an Automatic Teller Machine (ATM) (30 points) Customers arrive one at a time at a single automatic teller mach
IEOR 6711: Stochastic Models, I
Fall 2013, Professor Whitt, Final Exam
There are three questions, each with several parts.
1. Customers Coming to a Group of Automatic Teller Machines (35 points)
Customers arrive one at a time to a group of 4 ATMs (automat
IEOR 6711: Stochastic Models, I Fall 2010, Professor Whitt, Final Exam
There are four questions, each with several parts. Questions 2 and 4 are longer than the others and thus count more. 1. The Eight (8) Subway Line. (20 points) A new subway line has bee
IEOR 6711: Stochastic Models, I: Final Exam Fall 2007, SOLUTION NOTES
1. The GIN Barbershop (20 points) Ioannis Giannakakis, Gisli Ingimundarson and Behzad Nouri have joined together to form the GIN barbershop. The GIN barbershop has room for at most five
IEOR 6711: Stochastic Models, I Fall 2007, Professor Whitt, Final Exam
1. The GIN Barbershop (20 points) Ioannis Giannakakis, Gisli Ingimundarson and Behzad Nouri have joined together to form the GIN barbershop. The GIN barbershop has room for at most fiv
IEOR 6711: Professor Whitt
Notes on Laplace Transforms and Their Inversion
The shortest path between two truths in the real domain passes through the
complex domain; Jacques Hadamard (1865-1963).
1. Basic Denition
Let X be a nonnegative real-valued random
IEOR 6711: Stochastic Models I, Professor Whitt
Solutions to Homework Assignment 13
Problem 5.12 (a) Since
P0 =
it follows that
1/
=
1/ + 1/
+
N (t)
0
1
=
+
.
t
t
+ +
lim
(b) The expected total time spent in state 0 by t is
t
0
P00 (s)ds =
t+
1 e(+)t
+
IEOR 6711: Stochastic Models I
Fall 2013, Professor Whitt
Numerical Transform Inversion Homework: Tuesday, September 10
You have FIVE WEEKS: Due Tuesday, October 15.
1. Write a program implementing the algorithm Euler, which does the Fourier-series
algori
IEOR 6711: Stochastic Models I
Fall 2013, Professor Whitt
Homework Assignment 13, Tuesday, November 26
Chapter 5: Continuous-Time Markov Chains
Due on Thursday, December 5.
Problems from Chapter 5 of Stochastic Processes, second edition, by Sheldon Ross.
IEOR 6711: Stochastic Models I, Professor Whitt
Solutions to Homework Assignment 12
Problem 5.3 (a) Let N (t) denote the number of transitions be t. It is easy to show in this case
that
(M t)j
P(N (t) n)
eM t
j!
j =n
and thus P(N (t) < ) = 1.
(b) Let Xn+
IEOR 6711: Stochastic Models I SOLUTIONS to First Midterm Exam, October 10, 2010
Justify your answers; show your work. 1. Exponential Random Variables (23 points) Let X1 and X2 be independent exponential random variables with means E[X1 ] 1/1 and E[X2 ] 1
SOLUTIONS to the First Midterm Exam, October 7, 2012 IEOR 6711: Stochastic Models I,
1. Poisson Process and Transforms (30 points) [grading scheme: On all questions, partial credit will be given. Up to 4 points off for errors on parts (a)-(g); up to 8 poi
IEOR 6711: Stochastic Models I
First Midterm Exam, Chapters 1-2, October 6, 2013
There are ve questions, each with multiple parts.
Justify your answers; show your work.
1. Random Hats (15 points)
At a party n people each come wearing a hat. When they leav
EDU> % Escaping Markov Mouse Example EDU> EDU> % 9 transient states and 3 absorbing states EDU> EDU> % Q from EscapingMouseQ.dat EDU> % R from EscapingMouseR.dat EDU> % program absorbing(Q,R) in absorbing.m EDU> EDU> absorbing(Q, R) Q= 0 0.5000 0 0.5000 0
IEOR 6711: Stochastic Models I
SOLUTIONS to the Second Midterm Exam, November 17, 2013
1. Random Movement on a Chessboard (25 points)
The king (a chess piece) is placed on one corner square of an empty 8 8 = 64-square
chessboard. The king then makes a seq
IEOR 6711: Stochastic Models I SOLUTIONS to Second Midterm Exam, Chs. 3-4, November 20, 2007
Justify your answers; show your work. 1. Selling Flour: The Grainery. (30 points) The Grainery is a store that sells flour to bakers. Suppose that bakers come to
IEOR 6711: Stochastic Models I Second Midterm Exam, Chapters 3 and 4, November 20, 2007
Justify your answers; show your work. 1. Selling Flour: The Grainery. (30 points) The Grainery is a store that sells flour to bakers. Suppose that bakers come to the G
IEOR 6711: Stochastic Models I
Second Midterm Exam, Chapters 3-4, November 17, 2013
Justify your answers; show your work.
1. Random Movement on a Chessboard (25 points)
The king (a chess piece) is placed on one corner square of an empty 8 8 = 64-square
ch
IEOR 6711: Stochastic Models I Second Midterm Exam, Chapters 3-4, November 18, 2012 SOLUTIONS
Justify your answers; show your work. 1. Forecasting the Weather (12 points) Consider the following probability model of the weather over successive days. First,