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Welcome to COE 755: Queuing Theory Instructor:

Dr. Wissam F. Fawaz

Office 103, Bassil Bldg. Email: wissam.fawaz@lau.edu.lb

Required text book: Chee Hock Ng, Queuing Modelling Fundamentals, second

edition, John Wiley & Sons, 2008.

Course website: http://www.wissamfawaz.com/queueing_theory.htm

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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

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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

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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

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Assessment & grading

Topic presentation/Homeworks 15%

Simulation projects 15%

Exam I 20%

Exam II 20%

Final 30%

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Queuing definitions Kleinrock

“We study the phenomena of standing, waiting, and serving, and we call this study Queueing Theory.”

Bose “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”

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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

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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

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History lesson History goes back to primitive man

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History of queuing theory

Markov Analysis

Andrey A. Markov (born 1856). Created the Markov models and produced first results for these processes in 1906

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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

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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

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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?

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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

: the event that either the first or second coin falls heads

FE

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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()Pr()Pr( BABA

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Properties

Set of properties Property 1

Pr (impossible event) = 0

Property 2 If A B => Pr(A) <= Pr(B)

Property 3 Pr(Ac) = 1 – Pr(A)

Property 4: )Pr()Pr()Pr()Pr( BABABA

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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)

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Combinatorics (cont’d)

Binomial coefficients

Binomial expansion

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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(

)Pr()|Pr(

B

BABA

)Pr().|Pr()Pr( BBABA

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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

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Solution

Let B denote the event that both children are boys

Let A denote the event that at least one of them is a boy

=>

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1

4341

)}),(),,(),,Pr({(

)}),Pr({(

)Pr(

)Pr()|Pr(

bggbbb

bb

A

ABAB

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Independent events

Two events A and B are independent if Pr(A|B) = Pr(A)

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.

)Pr().Pr()Pr( BAAB

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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

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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

P(E2)P(F) = 1/6 x 1/6 = 1/36

Equivalent

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Conditional probabilities and Independence

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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().|Pr(...)Pr().|Pr()Pr().|Pr()Pr( 2211 nn BBABBABBAA

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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

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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)