IE 143 Long Exam 3 Coverage
Continuous-Time Markov Chains
and Queueing Theory
Angelo C. Ani
Department of Industrial Engineering
UP Los Baos
Properties of Exponential
Memoryless Property
Let X be an exponential random variable.
P( X < s + t | X > s) = P(
IE 143 Long Exam 2 Coverage
Discrete-Time Markov Chains
and Poisson Process
Angelo C. Ani
Department of Industrial Engineering
UP Los Baos
Review of Matrix Operations
Multiplication and Exponentiation
0.3 0.2 0.5
A 0.6 0.2 0.2
0.2 0.4 0.5
0.1 0.5 0.4
B 0
IE 143 Long Exam 4 Coverage
Queueing Optimization
and Renewal Theory
Angelo C. Ani
Department of Industrial Engineering
UP Los Baos
Queueing Optimization
Example
(Winston 4ed) A bank is trying to determine which of two
machines should be rented to process
IE 143 LONG EXAM 4
26 May 2015
Name: _
Section: 1:00-2:30
Instruction. Encircle the letter of the BEST answer. Do not linger on a single item.
As a suggestion, finish first the items that can be answered easily before going back
to the difficult ones.
Par
STATISTICS VS. PROBABILITY
STATISTICS
Given the information in your
hand, what is in the pail?
Given data, predict model
PROBABILITY
Given the information in the
pail, what is in your hand?
Given model, predict data
AXIOMS OF PROBABILITY
Probability the n
arrival rate
service rate
servers
queue cap
threshold time
service level
P(X=0)
P(customer will wait)
P(customer will balk)
P(C=2)
ave server util
ave no. of cust in sys
ave waiting time in queue
M/M/1
20
30
1
inf
M/M/s+c
60
20
5
4
M/M/s
M/M/s
40
70
20
60
Homework (Use A4 to answer all the problems. Box your final answer)
Deadline: May 16, 2016, 12nn at IED Faculty Room
1. Apple Crisp Foods signed a contract some years ago for maintenance services on its fleet of trucks and cars. The
contract is up for ren
6/30/11
Locus of Equations
Functions and
Relations
The locus of an equation is a curve
containing those points, whose
coordinates satisfy the equation.
A point lies on a curve if and only if its
coordinates satisfy the equation of the
curve.
"
Exam
6/30/11
Quiz
1. What is a sequence?
2. Give an expression for the n-th term of
the sequence whose first four terms are
Monotone
Convergence
Theorem
3. Use the definition to show that
Convergent Sequences
Convergent Sequences
Monotone Convergence
Theorem
E
6/23/11
Sequences
Sequences
of Real Numbers
List of numbers in a definite order
Function whose domain is the set of natural
numbers
Behavior of the sequence as n tends to
infinity
Examples
Limit of a Sequence
The sequence
we write
has the limit L and
i
7/3/11
Quiz
1. When is a sequence increasing?
Parabola
Decreasing?
2. What is a lower bound of a sequence?
Upper bound?
3. Show that
is convergent.
"
Standard Equations
Example
Vertex:
vertex
p
distance of vertex from focus
and directrix
Example
Opening u
6/18/11
Peanos Postulates
Axiomatic Development
of the
Real Number System
P1: 1 is an element of the set X.
P2: For each element
there is a
unique
called the successor of n.
P3: For each
P4: If
are distinct, then
P5: Let
and
If for all
then
Construction o
6/18/11
Well-Ordering Principle
Every non-empty subset of the set of
natural numbers has a least element.
Well-Ordering
Principle
This follows from the Peanos fifth
postulate.
Well-Ordering Principle
Proof:
By contradiction
Suppose that there is a non-emp
7/6/11
Ellipse
Ellipse
An ellipse is the set of points on a
plane the sum of whose distances from
two fixed points is a constant.
"
Standard Equation
Properties
The points (a,0) and (-a,0) are called
2c is the distance between the two
foci
2a is the co
IE 150 SYSTEMS EVALUATION
PRACTICE SET
1. Julie has a low credit rating, plus she was furloughed from her job 2 months ago. She has a new job starting
next week and expects a salary to start again in a couple of weeks. Since she is a little short on money
IE 150 SYSTEMS EVALUATION
PRACTICE SET
1. Julie has a low credit rating, plus she was furloughed from her job 2 months ago. She has a
new job starting next week and expects a salary to start again in a couple of weeks. Since
she is a little short on money
VALIN, Kevin Ardie S.
2008-19255
Assignment 1
Aug. 13, 2015
IE 150 U
Career Opportunities for BS Agricultural Engineering Students:
The responsibilities of an Agricultural Engineer include:
Work in the agricultural industry to design structures, equipment
VALIN, Kevin Ardie S.
2008-19255
Assignment 1
Aug. 13, 2015
IE 150 U
Career Opportunities for BS Agricultural Engineering Students:
The responsibilities of an Agricultural Engineer include:
Work in the agricultural industry to design structures, equipmen