IEOR 4106: Introduction to Operations Research: Stochastic Models
Spring 2011, Professor Whitt
Numerical Part of Homework Assignment 3: Tuesday, February 1
Markov Chains
Due on Tuesday, February 15), an extra week is given, To be discussed at
the recitati
IEOR 4106
Intro to OR: Stochastic Models
Prof. Mariana Olvera-Cravioto
Lecture #19
March 31, 2016
Page 1 of 2
Lecture #19
Reading from Rosss Introduction to Probability Models:
Chapter 7, Section 7.5.1
Key concepts:
Alternating renewal processes: A regen
IEOR 4106, HMWK 7, Professor Sigman
1. Printer with jams: Jobs arrive to a computer printer according to a Poisson process at
rate . Jobs are printed one at a time requiring iid printing times that are exponentially
distributed with rate . Jobs wait in a
Stochastic Models, Homework 2, 9/28/16
Ch 2: 43, 44; Ch 3: 37, 38, 39, 40, 41, 44, 50, 74
Problem 2.43
The value of is binary and is 1 if a red ball is picked up before any black is chosen and 0 otherwise.
Given that we seek the total number of red balls
IEOR 4106
Intro to OR: Stochastic Models
Prof. Mariana Olvera-Cravioto
Lecture #21
April 11, 2016
Page 1 of 2
Lecture #21
Reading from Rosss Introduction to Probability Models:
Chapter 8, Sections 8.2.2, 8.3.1, 8.3.2, 8.3.3
Key concepts:
Steady-state pro
IEOR 4106
Intro to OR: Stochastic Models
Prof. Mariana Olvera-Cravioto
Review Session
May 2, 2016
Page 1 of 1
Review Session
1. Suppose that a population consists of a fixed number, say, m, of genes in any generation.
Each gene is one of two possible gene
\
i
i
i
i
i
MW .4 mi
l
U \
"WV cfw_ v: a, C)
O K
W\ E? (3.3
g! x q
cfw_)k 7 (gray: in 1 if! i ' "\ (Ix. 2 r x. .J
cfw_.5
VK\
'N. - \
\fW .J (15(WWWA';L \
x L ' 2#_1
j:- a
cfw_33 U
1 i is k,
Nl
Wit: \ 3 mp
'/ :5, </\fr\) ( I" \ <. \fmvl
1'." " W
IEOR 4106
Intro to OR: Stochastic Models
Prof. Mariana Olvera-Cravioto
Lecture #18
March 30, 2016
Page 1 of 2
Lecture #18
Reading from Rosss Introduction to Probability Models:
Chapter 7, Sections 7.4, 7.5, 7.5.1
Key concepts:
The age and excess of N (t)
IEOR 4106
Intro to OR: Stochastic Models
Prof. Mariana Olvera-Cravioto
Lecture #17
March 23, 2016
Page 1 of 2
Lecture #17
Reading from Rosss Introduction to Probability Models:
Chapter 7, Sections 7.3 and 7.4.
Key concepts:
Stopping times: Let cfw_Xi i1
IEOR 4106
Intro to OR: Stochastic Models
Prof. Mariana Olvera-Cravioto
Lecture #23
April 18, 2016
Page 1 of 2
Lecture #23
Reading from Rosss Introduction to Probability Models:
Chapter 8, Sections 8.4.1, 8.4.2
Key concepts:
Open Jackson Networks: Conside
Stochastic Models, Homework 5, 10/11/16
Ch 4: 2,3,5,6,20,24,25,35,45,46
Problem 4.2
There are a total of eight states. Allow the ordered tuple (1 , 2 , 3 ) to define a given a given state
where 1 is the weather two days ago, 2 is the weather yesterday, an
Stochastic Models, Homework 1, 9/23/16
Ch 1: 36, 37, 43, 44; Ch 2: 20, 21, 30, 36, 37
Problem 1.36
This is an example of branching probability and combined probability within the branching possibilities;
basically, we first choose a box and then choose a
IEOR 4106, HMWK 8, Professor Sigman
1. Recall Problem 2 from HMWK 7:
Consider 4 iPhones, each independently having a battery lifetime that is exponentially
distributed with mean 2 years (hence rate = 0.5). Once a battery breaks down, the
iPhone immediatel
IEOR 4106, HMWK 9, Professor Sigman
1. For a renewal process with iid interarrival times cfw_Xn with E(X) = 1/, give an
expression for
1 t 2
lim
A (s)ds,
t t 0
that involves only the moments, E(X n ), n 1.
To do so: (1) Graph cfw_A2 (t) : t 0, and look a
Stochastic Models, Homework 4, 10/11/16
Ch 5: 44, 50, 85, 86
Problem 5.44a
This is the same as asking the probability that there will be no cars for the next T units right now which in
turn analogous to saying that the next vehicle will arrive T or more u
Stochastic Models, Homework 3, 10/11/16
Ch 5: 4, 6, 22, 43, 47(a,b)
Problem 5.4a
If the service time is definitely ten minutes, then there is no way that person A will remain in the post
office after the other two have left. This is because people A and B
Stochastic Models, Homework 4, 10/11/16
Ch 5: 44, 50, 85, 86
Problem 5.44a
This is the same as asking the probability that there will be no cars for the next T units right now which
in turn analogous to saying that the next vehicle will arrive T or more u
Stochastic Models, Homework 5, 10/11/16
Ch 4: 2,3,5,6,20,24,25,35,45,46
Problem 4.2
There are a total of eight states. Allow the ordered tuple
X 1 is the weather two days ago,
where
( X1 , X2 , X3)
to define a given a given state
X 2 is the weather yester
Stochastic Models, Homework 2, 9/28/16
Ch 2: 43, 44; Ch 3: 37, 38, 39, 40, 41, 44, 50, 74
Problem 2.43
X i is binary and is 1 if a red ball is picked up before any black is chosen and 0 otherwise.
The value of
Given that we seek the total number of red ba
Stochastic Models, Homework 1, 9/23/16
Ch 1: 36, 37, 43, 44; Ch 2: 20, 21, 30, 36, 37
Problem 1.36
This is an example of branching probability and combined probability within the branching possibilities;
basically, we first choose a box and then choose a
Stochastic Models, Homework 3, 10/11/16
Ch 5: 4, 6, 22, 43, 47(a,b)
Problem 5.4a
If the service time is definitely ten minutes, then there is no way that person A will remain in the post
office after the other two have left. This is because people A and B
IEOR 4106
Intro to OR: Stochastic Models
Prof. Mariana Olvera-Cravioto
Lecture #25
April 25, 2016
Page 1 of 1
Lecture #25
Reading from Rosss Introduction to Probability Models:
Chapter 10, Section 10.2, 10.3, 10.4.
Key concepts:
Brownian motion with drif
IEOR 4106
Intro to OR: Stochastic Models
Prof. Mariana Olvera-Cravioto
Lecture #22
April 13, 2016
Page 1 of 2
Lecture #22
Reading from Rosss Introduction to Probability Models:
Chapter 8, Sections 8.3.1, 8.3.2, 8.3.3, 8.4.1
Key concepts:
Measures of perf
Stochastic Models, Homework 11, 12/6/16
Ch 7: 2, 4, 15, 22, 25
Problem 7.2a
Considering the discrete probability function given, it is readily evident that
S n is Poisson with mean
.
Problem 7.2b
P ( N ( t )=n )=P ( N ( t ) n )P ( N ( t ) n+1 )=P ( S n t
IEOR4106 Intro to OR - Stoch Models
David D. Yao
Practice Final Examination
(180 minutes)
All problems are equally weighted.
1. Consider the M/M/1 in steady state; let and denote the arrival and service rates. Let Wq
be the waiting time in queue of a new