ORIE3510
Introduction to Engineering Stochastic Processes
Homework 5: Discrete Time Markov Chains Due 2:30pm, March 3, 2010 (drop box)
Spring 2010
Be sure to write your name and section number or day on your homework. In all questions, be sure to give the

ORIE 3510/5510 Introduction to Engineering Stochastic Processes I
Summer 2012
ASSIGNMENT 1. Given: July 2, 2012. Due: July 6, 2012. (Revised July 5, 2012)
1. If it is known that in a series of coin tosses, the 4th head occurred on the 12th trial, what is

ORIE3510
Introduction to Engineering Stochastic Processes
Homework 4: Discrete Time Markov Chains Due 2:30pm, February 24, 2010 (drop box)
Spring 2010
Be sure to write your name and section number or day&time on your homework. In all questions, be sure to

ORIE 3510
Introduction to Engineering Stochastic Processes
Spring 2014
Solutions to Homework 1
1. We are given that E[X] = 2 and V ar(X) = 9. Thus
a) E[6 2X] = 6 2E[X] = 2 and V ar(6 2X) = 4V ar(X) = 36.
b) E[(X 3)/4] = 41 E[X]
3
4
= 1/4 and V ar(X 3)/4)

ORIE 3510/5510
J. G. Dai
A sample write-up
March 3, 2017
The Wisdom of the Goal
Drew Ungerman
for an operations course at Stanford
taught by Professor Kumar
The Goal outlines new global principles in manufacturing and illustrates how cash flow can be maxi

ORIE 3510
Introduction to Engineering Stochastic Processes
Spring 2017
J. Dai
Solutions to Homework 4
1. (a) The fixed cost, variable cost, holding cost and back-order costs are: cf = 1000, cv = 50,
h = 20, b = 100. The optimal order-up-to quantity S is t

ORIE 3510
Introduction to Engineering Stochastic Processes I
Anton Braverman and J. Dai
Spring 2017
Homework 7 Solutions
1. (a) Yes, each state has period 2. Each state cannot have period 1 because Pii1 is 0
for all states i. However, looking at Pii2 yiel

ORIE 3510
Anton Braverman and J. Dai
Introduction to Engineering Stochastic Processes I
Spring 2017
Homework 3
Due: Wednesday, February 8
1. Let D be a discrete random variable with the following
1/10
1/10
2/5
P[D = k] =
3/10
1/10
0
p.m.f.
if k = 5
if k =

ORIE 3510
Introduction to Engineering Stochastic Processes
Spring 2017
Solutions to Homework 6
1. Recall that the DTMC is cfw_Xn , where Xn
is cfw_0, 1, 2, 3, and the transition matrix is
0
0
P =
0
2/6
is inventory at the end of the day. The state space

ORIE 3510
Stochastic Introduction to Engineering Stochastic Processes I
Spring 2017
Anton Braverman and Jim Dai
Solutions to Homework 8
1.
0
1/2
P =
0
1/3
0
1/2
0
0
1/3
1
1/2 0
0 1/2
0 1/2
1/3 0
0
0
0
0
1/2
.
0
0
Using matlab to compute P 100 , one obt

Laura Watson
ORIE 3510
Homework 2 The Goal
1. In The Goal, how is cash flow affected by the throughput, inventory, and customer
response time?
Definition: Throughput the rate the system generates money through sales
Not rate of production!
Definition: O

3510 Discussion Notes
02/07/2017
MatLab Code
If in a given period our level less than small s
Then we are going to make a order to the level s-d(i)
If it is greater than s
Then we are not going to make an order
Output: First column = demand, second colu

ORIE3510
Introduction to Engineering Stochastic Processes
Spring 2016
Homework 7: CTMC
Due 10:30am, Thursday April 28, 2016 (drop box)
Problem 1
Since a failed satellite will never become operational,
P01 (t) = 0
for all t 0
P00 (t) = 1
Let T be the remai

ORIE3510
Introduction to Engineering Stochastic Processes
Homework 2: Discrete Time Markov Chains Due February 10, 2010 (drop box)
Spring 2010
Be sure to write your name and section number or day&time on your homework. In all questions, be sure to give th

ORIE3510
Introduction to Engineering Stochastic Processes
Homework 3: Discrete Time Markov Chains Due 2:30pm, February 17, 2010 (drop box)
Spring 2010
Be sure to write your name and section number or day&time on your homework. In all questions, be sure to

ORIE3510
Introduction to Engineering Stochastic Processes
Homework 6: Poisson Processes Due 2:30pm, March 10, 2010 (drop box)
Spring 2010
Be sure to write your name and section number or day&time on your homework. In all questions, be sure to give the jus

ORIE 3510 Homework 1 (Introduction to Probability Theory)
Instructor: Mark E. Lewis
due January 30, 2012 (ORIE Hallway drop box)
This homework assignment is designed to give you some practice in probability. Some of the
concepts should be review, others a

ORIE 3510 Homework 2
Instructor: Mark E. Lewis
due 2PM, Wednesday February 8, 2012 (ORIE Hallway drop box)
1. Give the transition diagrams for the Markov chains with the given transition matrix:
(a)
P=
0.3 0.7
0.5 0.5
(b)
1
0
0
P = 1/2 0 1/2
1/2 1/4 1/4

ORIE 3510 Homework 3
Instructor: Mark E. Lewis
due 2PM, Wednesday February 15, 2012 (ORIE Hallway drop box)
1. Consider the stochastic process
Xn = min(Xn1 , Xn2 ) + n
with X0 = 0, X1 = 0 and n is a random variable taking values 1 and 1 with probability
1

ORIE 3510 Homework 4
Instructor: Mark E. Lewis
due 2PM, Wednesday February 22, 2012 (ORIE Hallway drop box)
1. Show that if state i is recurrent and state i does not communicate with state j , then
pij = 0. (This implies that once a process enters a recur