#### Transcript ppt - Dr. Wissam Fawaz

```Welcome to COE 755: Queuing
Theory
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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
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Course Description
Queuing Analysis
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is a vital tool used in evaluating system performances
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its application covers
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A wide spectrum
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from Bank automated teller machines
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to transportation and communications data networks
This queuing course
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focuses on queuing modeling techniques
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and its application in data networks
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Course synopsis
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The course covers
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Discrete/continuous random variables
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Birth and death process
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Machine Interference Problem
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Hypo/Hyper exponential distributions
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Markovian/semi-Markovian Queueing Systems
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Discrete and Continuous Markov Processes
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Queueing Networks
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Course learning objectives
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Students are expected to be able:
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gain a mastery of Markovian queuing systems
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construct queuing models for engineering problems and
solve queuing problems
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become familiar with the derivation techniques used to
measure system performance
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work with an emerging class of arrival processes
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Markov-modulated arrival process
use QNAP (Queuing Network Analysis Package) software
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To solve real world network related problems
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Topic presentation/Homeworks
15%
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Simulation projects
15%
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Exam I
20%
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Exam II
20%
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Final
30%
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Queuing definitions
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Kleinrock
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Bose
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“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
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“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
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The study of queuing
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is the study of waiting
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Customers may wait on a line
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Planes may wait in a holding pattern
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Jobs may wait for the attention of the CPU in an computer
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Packets may wait in the buffer of a node in a computer
network
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Telephone calls may wait to get through an exchange
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Real life queuing examples
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Call center service level
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Service statement (Regie (public service) du Quebec)
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If you call the Regie
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The waiting time to speak with an information clerk
 is 30 seconds in 75% of cases
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and you will never wait more than 3 minutes
This is one of the commitments
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that the Regie respect 95% of the time
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History lesson
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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
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I assume
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That you are familiar with the pre-requisite knowledge
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Probability theory
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Transform theory
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And matrices
Still, I will review
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some essential background knowledge
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of certain related disciplines
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Introduction to probability
theory
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Example 1
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Suppose that we toss 2 coins
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What is the probability that the first coin falls heads?
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What is the probability that the second coin falls heads?
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What is the probability that either the first or the second coin
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Solution
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Sample space (Ω)
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collection of all possible outcomes
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Ω = {(H, H), (H,T), (T, H), (T, T)}
Possible Events
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 E = {(H, H), (H, T)} => Pr (E) = 1/2
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 F = {(H, H), (T, H)} => Pr (F) = 1/2
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E  F : the event that either the first or second coin falls heads
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Probability: definition
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Probability function Pr ( )
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Define a mapping between Ω and set of real numbers
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Assigning a number to each event E in Ω
Furthermore, Pr must satisfy, for all A, B in Ω
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Pr(A) ≥ 0 - probability is a positive measure
 Pr(A) = (cardinality of A)/(cardinality of Ω)
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Pr(Ω) = 1 – probability is a finite measure
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A, B disjoint events => Pr( A  B )  Pr( A)  Pr( B )
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Properties
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Set of properties
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Property 1
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Property 2
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If A
B => Pr(A) <= Pr(B)
Property 3
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Pr (impossible event) = 0
Pr(Ac) = 1 – Pr(A)
Property 4: Pr( A  B)  Pr( A)  Pr( B)  Pr( A  B)
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Combinatorics
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Permutations
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K-permutation of a set of n elements
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Combinations
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K-combination of a set of n elements
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=> k-permutation / k! (where k! is the number of possible ways
to permute that combination)
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Combinatorics (cont’d)
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Binomial coefficients
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Binomial expansion
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Conditional probability
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It enables us
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to determine whether 2 events A, and B
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are related in the sense
 That knowledge about the occurrence of one
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alters the likelihood of occurrence of the other
Probability of A given B has occurred
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Pr( A | B ) 
Pr( A  B ) => Pr( A  B )  Pr( A | B ). Pr( B )
Pr( B )
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Conditional probability:
Example
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Example 2
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A family has two children
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What is the conditional probability that both are boys
 Given that at least one of them is a boy
Assumptions:
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Sample space S = {(b,b), (b,g), (g,b), (g,g)}
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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
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Let B denote
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Let A denote
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the event that both children are boys
the event that at least one of them is a boy
=>
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Pr( B  A)
Pr({( b, b)})
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Pr( B | A) 
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 4 
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Pr( A)
Pr({( b, b), (b, g ), ( g , b)})
3
4
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Independent events
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Two events A and B are independent if
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Pr(A|B) = Pr(A)
Pr( AB)  Pr( A). Pr( B )
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Example 3
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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.
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Analysis of the example 3
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P(E1∩F) = Pr({(4,2)}) = 1/36
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P(E1)P(F) = 5/36 x 1/6 = 5/216
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=> E1 and F are not independent
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Reason
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Our chance of getting a total of six
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Depends on the outcome of the first die
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Example of independent
events
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Let E2 be the event that the sum of dice = 7
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Is E2 independent of F?
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P(E2∩F) = Pr({(4,3)}) = 1/36
Equivalent
P(E2)P(F) = 1/6 x 1/6 = 1/36
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Conditional probabilities and
Independence
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Total probability
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Let B1, B2, …, and Bn be mutually exclusive events
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Disjoint events whose union equals Ω (partition of
sample space)
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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 )
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Baye’s formula
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Let {B1, B2, …, Bn } be a partition of a sample space
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The problem is to calculate P(Bi) given A has occurred
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it enables us to calculate a conditional probability
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When we know the reverse conditional probabilities
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Baye’s formula: example
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Example 4
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Three cards with different colours on different sides
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The upper side of a randomly drawn card is red
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What is the probability that the other side is blue?
Solution (using Baye’s formula)
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