Homework 8
Solutions
8.1
(8)
E-[nu mber of arrivals] = E[E[nu1nber of arrivalslservice period is S]
= E [AS]
_ i
pi
(b)
P(0 arrivals) = E[P(0 arrivalslservice period is S)]
= E[P(N(S) = 0)]
= Ele—XS]
_ x -M we
—— l; c [1.8. d3
= I1
u + A
8.3
Let CM=Mary
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
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 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
IEOR 4106, HMWK 6, Professor Sigman
1. Suppose that the rat in the closed maze visits Room 4 at times cfw_tn : n 1 that form a
Poisson process at rate . You are not allowed to watch the rat, but each time it enters
Room 4 you are immediately told so. You
IEOR 4106, HMWK 5, Professor Sigman
1. End of Text Chapter 5, Page 354: Exercises 4,5, 18
2. You arrive at the West 96th Street Subway station to go Downtown. Suppose that the
time until the next Express train is exponentially distributed with rate 3 (per
IEOR 4106, HMWK 4, Professor Sigman
1. End of Text Chapter 4, Page 281: Exercises 41 and 42.
2. Consider a Markov chain with nite state space S = cfw_1, 2, . . . , b, and transition matrix
P . Suppose that it is irreducible, and that each column of P sums
IEOR 4106, HMWK 3, Professor Sigman
1. Each of the following transition matrices is for a Markov chain. For each, nd the communication classes for breaking down the state space, S = C1 C2 and for each class
Ci tell if it is recurrent or transient.
(a)
1/6
IEOR 4106, HMWK 2, Professor Sigman
1. Consider modeling the weather where we now assume that the weather today depends (at most) on the previous three days weather (recall the Text, Page 193
Example 4.4, where the weather was assumed to depend upon the p
IEOR 4106: Introduction to Operations Research: Stochastic Models
Spring 2011, Professor Whitt
Homework Assignment 10: Tuesday, April 12.
Due on Tuesday, April 19.
Chapter 7: More Renewal Theory
In Ross, read Examples 7.15 and 7.16 in Section 7.4. Read Se
IEOR 4106: Introduction to Operations Research: Stochastic Models
Spring 2011, Professor Whitt
Homework Assignment 9: Tuesday, April 5.
Due on Tuesday, April 12.
Chapter 7: Renewal Theory and its Applications
In Ross, read Sections 7.1-7.3 up to, but not
IEOR 4106: Introduction to Operations Research: Stochastic Models
Spring 2011, Professor Whitt
Homework Assignment 8: Tuesday, March 22
Due on Tuesday, March 29.
Chapter 6: More Continuous-Time Markov Chains
Read Sections 6.6-6.8 in Ross. Do the following
IEOR 4106: Introduction to Operations Research: Stochastic Models
Spring 2011, Professor Whitt
Homework Assignment 7: Tuesday, March 8
Due on Tuesday, March 22; to be discussed in the recitation on Monday,
March 21. Spring Break: March 14-18.
More of Chap
IEOR 4106: Introduction to Operations Research: Stochastic Models
Spring 2011, Professor Whitt
Homework Assignment 6: Tuesday, March 1
Due on Tuesday, March 8.
Chapter 6: Continuous-Time Markov Chains
Read Sections 6.1-6.5 in Ross. Do the following exerci
IEOR 4106: Introduction to Operations Research: Stochastic Models
Spring 2011, Professor Whitt
Homework Assignment 5: Tuesday, February 22
Due on Tuesday, March 1.
Read Chapter 5: Sections 5.1-5.3, up to (but not including) Example 5.16 on page 313.
You m
IEOR 4106: Introduction to Operations Research: Stochastic Models
Spring 2011, Professor Whitt
Homework Assignment 4: Tuesday, February 8.
Due on Tuesday, February 15.
Even More Markov Chains
Read Sections 4.5.1, 4.5.2, 4.6 and part of 4.8. In Section 4.8
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
Mathematical Programming:
An Overview
1
Management science is characterized by a scientic approach to managerial decision making. It attempts
to apply mathematical methods and the capabilities of modern computers to the difcult and unstructured
problems c
IEEM 225
Tutorial 4
10/20/15
1. The joint probability mass function of X and Y, p(x, y), is given by
P(1, 1)=1/9, P(2, 1)=1/3, P(3, 1)=1/9
P(1, 2)=1/9, P(2, 2)=0, P(3, 2)=1/18
P(1, 3)=0, P(3, 2)=1/6, P(3, 3)=1/9
Compute E[X|Y=I] for I =1, 2, 3.
2. The joi
Lecture Notes for Introductory Probability
Janko Gravner
Mathematics Department
University of California
Davis, CA 95616
gravner@math.ucdavis.edu
January 5, 2014
These notes were started in January 2009 with help from Christopher Ng, a student in
Math 135
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
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
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 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 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 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