Name:
IE 8532 Midterm Exam
November 7, 2012
This exam has 9 numbered pages. Please check that all pages are present. Show all work.
1. Suppose that cfw_Xn is a Markov chain with state space S = cfw_1, 2, transition matrix
P=
1/4 3/4
2/3 1/3
,
and initial
IE 8532 Midterm Exam
November 2, 2011
Solutions
This exam has 8 numbered pages. Place your answers in the boxes provided and show all work.
1. Suppose that cfw_Xn is a Markov chain with state space S = cfw_1, 2, transition matrix
P=
1/4 3/4
1/3 2/3
,
and
IE 8532 Homework #1
Due: 12:20 PM, September 24, 2014.
When completing this homework assignment, you may choose to work in a group with at
most two other students. On your write-up, please indicate with whom you worked (if
anyone). You must write your own
IE 8532 Final Exam Solutions to Practice Problems
[12/5/12: This is not a complete exam. It is roughly 3/4 of an exam.]
1. Suppose cfw_N (t) and cfw_N (t) are independent Poisson processes with respective rates and .
Let T1 be the rst point of cfw_N (t),
IE 8532 Midterm Exam
SOLUTIONS
November 7, 2012
This exam has 9 numbered pages. Please check that all pages are present. Show all work.
1. Suppose that cfw_Xn is a Markov chain with state space S = cfw_1, 2, transition matrix
1/4 3/4
,
P=
2/3 1/3
and ini
Stochastic Processes and Queueing Systems
IE 8532, University of Minnesota, Fall 2014
Instructor:
William L. Cooper
Oce: ME 130F
Phone: (612) 624-4322
Email: billcoop@umn.edu
Oce Hours:
Tuesdays 11:00 AM 12:00 PM
or by appointment
Meeting Time, Location:
IE 8532 Homework #4 Selected Solutions
2. (a) If the number of people on the bus in the previous time period is i, then the number of
people on the bus this time period is
mincfw_c, i (# people who get o) + (# of potential riders).
The number of people wh
IE 8532 Homework #3 Selected Solutions
1. Solution to problem 21, chapter 3.
(a) X =
N
i=1 Ti .
(b) The random variable N has the geometric distribution with parameter p = P (T1 = 2) =
1/3. Therefore, EN = 3.
(c) Since N = mincfw_n : Tn = 2, it follows th
IE 8532 Homework #2 Selected Solutions
1. Solution to problem 73, chapter 2. For i = 1, . . . , r , k = 1, . . . , n let
1 if outcome i occurs on trial k
Ii,k =
0 otherwise.
Note that Ni =
(a) ENi = E
n
k =1 Ii,k .
n
k =1 Ii,k
(b) Var(Ni ) = Var(
=
n
k
IE 8532 Homework #1 Selected Solutions
3. Solution to problem 46(a), chapter 2.
Following the hint in the book, let In = 1(X n) for n = 1, 2, . . . . Then X =
P (X > n) .
P (X n) =
n=0
n=1
n=1
n=1
n=1
P (In = 1) =
EIn =
In =
EX = E
and
n=1 In ,
For a dier
IE 8532 Homework #6 Selected Solutions
1. Solution to problem 61, chapter 5. Let X denote the number of aws. By assumption,
X is Poisson-distributed with mean c. Fix t. Categorize a aw as Type 1 if it has caused a
failure by time t. Categorize a aw as Typ
IE 8532 Homework #1 Selected Solutions
3. Solution to problem 46(a), chapter 2.
Following the hint in the book, let In = 1(X n) for n = 1, 2, . . . . Then X =
P (X > n) .
P (X n) =
n=0
n=1
n=1
n=1
n=1
P (In = 1) =
EIn =
In =
EX = E
and
n=1 In ,
For a dier
IE 8532 Homework #2 Selected Solutions
1. Solution to problem 73, chapter 2. For i = 1, . . . , r , k = 1, . . . , n let
1 if outcome i occurs on trial k
Ii,k =
0 otherwise.
Note that Ni =
(a) ENi = E
n
k =1 Ii,k .
n
k =1 Ii,k
(b) Var(Ni ) = Var(
=
n
k
IE 8532 Homework #3 Selected Solutions
1. Solution to problem 21, chapter 3.
(a) X =
N
i=1 Ti .
(b) The random variable N has the geometric distribution with parameter p = P (T1 = 2) =
1/3. Therefore, EN = 3.
(c) Since N = mincfw_n : Tn = 2, it follows th
IE 8532 Homework #4 Selected Solutions
2. (a) If the number of people on the bus in the previous time period is i, then the number of
people on the bus this time period is
mincfw_c, i (# people who get o) + (# of potential riders).
The number of people wh
IE 8532 Homework #8 Selected Solutions
2. Solution to problem 38, chapter 8.
(b) We have E (n ) = E (Xn+1 ) E (Xn ) + 1 E (Yn ). The random variable Yn represents the
number of arrivals during the service time Sn+1 of the (n + 1)-st customer. The arrival