Transcript ppt - Dr. Wissam Fawaz

Welcome to COE 755: Queuing
Theory









Instructor:
Dr. Wissam F. Fawaz
Office
103, Bassil Bldg.
Email: [email protected]
Required text book:
Chee Hock Ng, Queuing Modelling Fundamentals, second
edition, John Wiley & Sons, 2008.
Course website:
http://www.wissamfawaz.com/queueing_theory.htm
1
Course Description
Queuing Analysis


is a vital tool used in evaluating system performances

its application covers

A wide spectrum

from Bank automated teller machines

to transportation and communications data networks
This queuing course


focuses on queuing modeling techniques

and its application in data networks
2
Course synopsis

The course covers

Discrete/continuous random variables

Birth and death process

Machine Interference Problem

Hypo/Hyper exponential distributions

Markovian/semi-Markovian Queueing Systems

Discrete and Continuous Markov Processes

Queueing Networks
3
Course learning objectives

Students are expected to be able:

gain a mastery of Markovian queuing systems

construct queuing models for engineering problems and
solve queuing problems

become familiar with the derivation techniques used to
measure system performance

work with an emerging class of arrival processes

Markov-modulated arrival process
use QNAP (Queuing Network Analysis Package) software


To solve real world network related problems
4
Assessment & grading

Topic presentation/Homeworks
15%

Simulation projects
15%

Exam I
20%

Exam II
20%

Final
30%
5
Queuing definitions

Kleinrock


Bose


“We study the phenomena of standing, waiting, and serving, and we
call this study Queueing Theory.”
“The basic phenomenon of queuing arises whenever a shared facility
needs
 to be accessed for service by a large number of jobs or
customers.”
Takagi

“A queue is formed when service requests arrive at a service facility
 and are forced to wait while the server is busy working on other
requests”
6
Wrap up

The study of queuing

is the study of waiting

Customers may wait on a line

Planes may wait in a holding pattern

Jobs may wait for the attention of the CPU in an computer

Packets may wait in the buffer of a node in a computer
network

Telephone calls may wait to get through an exchange
7
Real life queuing examples

Call center service level

Service statement (Regie (public service) du Quebec)

If you call the Regie

The waiting time to speak with an information clerk
 is 30 seconds in 75% of cases


and you will never wait more than 3 minutes
This is one of the commitments

that the Regie respect 95% of the time
8
History lesson

History goes back to primitive man
9
History of queuing theory
Markov Analysis
Andrey A. Markov (born 1856).
Created the Markov models and
produced first results for these
processes in 1906
10
History of queuing theory
Erlang Analysis
Agner Erlang (in 1909): A Danish
engineer in the Copenhagen
Telephone Exchange,
published his first paper on
queuing theory
11
Course prerequisite

I assume


That you are familiar with the pre-requisite knowledge

Probability theory

Transform theory

And matrices
Still, I will review

some essential background knowledge

of certain related disciplines
12
Introduction to probability
theory

Example 1

Suppose that we toss 2 coins

What is the probability that the first coin falls heads?

What is the probability that the second coin falls heads?

What is the probability that either the first or the second coin
falls heads?
13
Solution


Sample space (Ω)

collection of all possible outcomes

Ω = {(H, H), (H,T), (T, H), (T, T)}
Possible Events

The first coin falls heads
 E = {(H, H), (H, T)} => Pr (E) = 1/2

The second coin falls heads
 F = {(H, H), (T, H)} => Pr (F) = 1/2

E  F : the event that either the first or second coin falls heads
14
Probability: definition

Probability function Pr ( )

Define a mapping between Ω and set of real numbers


Assigning a number to each event E in Ω
Furthermore, Pr must satisfy, for all A, B in Ω

Pr(A) ≥ 0 - probability is a positive measure
 Pr(A) = (cardinality of A)/(cardinality of Ω)

Pr(Ω) = 1 – probability is a finite measure

A, B disjoint events => Pr( A  B )  Pr( A)  Pr( B )
15
Properties

Set of properties

Property 1


Property 2


If A
B => Pr(A) <= Pr(B)
Property 3


Pr (impossible event) = 0
Pr(Ac) = 1 – Pr(A)
Property 4: Pr( A  B)  Pr( A)  Pr( B)  Pr( A  B)
16
Combinatorics

Permutations

K-permutation of a set of n elements

Combinations

K-combination of a set of n elements

=> k-permutation / k! (where k! is the number of possible ways
to permute that combination)
17
Combinatorics (cont’d)

Binomial coefficients

Binomial expansion
18
Conditional probability

It enables us

to determine whether 2 events A, and B

are related in the sense
 That knowledge about the occurrence of one


alters the likelihood of occurrence of the other
Probability of A given B has occurred

Pr( A | B ) 
Pr( A  B ) => Pr( A  B )  Pr( A | B ). Pr( B )
Pr( B )
19
Conditional probability:
Example

Example 2

A family has two children


What is the conditional probability that both are boys
 Given that at least one of them is a boy
Assumptions:

Sample space S = {(b,b), (b,g), (g,b), (g,g)}

All outcomes are equally likely
 For instance, (b,g) means the older child is a boy and the
younger child is a girl
20
Solution

Let B denote


Let A denote


the event that both children are boys
the event that at least one of them is a boy
=>
1
Pr( B  A)
Pr({( b, b)})
1
Pr( B | A) 

 4 
3
Pr( A)
Pr({( b, b), (b, g ), ( g , b)})
3
4
21
Independent events

Two events A and B are independent if

Pr(A|B) = Pr(A)
Pr( AB)  Pr( A). Pr( B )

Example 3

Suppose we roll two fair dice. Let E1 denote the event
that the sum of the dice is six and F denote the event
that the first die equals four.
22
Analysis of the example 3

P(E1∩F) = Pr({(4,2)}) = 1/36

P(E1)P(F) = 5/36 x 1/6 = 5/216

=> E1 and F are not independent

Reason

Our chance of getting a total of six

Depends on the outcome of the first die
23
Example of independent
events

Let E2 be the event that the sum of dice = 7

Is E2 independent of F?

The answer is YES since:


P(E2∩F) = Pr({(4,3)}) = 1/36
Equivalent
P(E2)P(F) = 1/6 x 1/6 = 1/36
24
Conditional probabilities and
Independence
25
Total probability

Let B1, B2, …, and Bn be mutually exclusive events

Disjoint events whose union equals Ω (partition of
sample space)

Then, probability of any given event A in Ω
Pr( A)  Pr( A | B1 ). Pr( B1 )  Pr( A | B2 ). Pr( B2 )  ...  Pr( A | Bn ). Pr( Bn )
26
Baye’s formula

Let {B1, B2, …, Bn } be a partition of a sample space

The problem is to calculate P(Bi) given A has occurred

it enables us to calculate a conditional probability

When we know the reverse conditional probabilities
27
Baye’s formula: example

Example 4

Three cards with different colours on different sides

The upper side of a randomly drawn card is red


What is the probability that the other side is blue?
Solution (using Baye’s formula)
28