Probability and Statistics with Reliability,
Queuing and Computer Science
Applications
Second edition
by K.S. Trivedi
PublisherJohn Wiley & Sons
Chapter 4 :Expectation
Dept. of Electrical & Computer engineering
Duke University
Email:kst@ee.duke.edu
URL:
Probability and Statistics with
Reliability, Queuing and Computer
Science Applications:
Second edition
by K.S. Trivedi
PublisherJohn Wiley & Sons
Chapter 3: Continuous Random Variables
Dept. of Electrical & Computer Engineering
Duke University
Email: kst
Probability and Statistics with
Reliability, Queuing and
Computer Science Applications:
second edition
by K.S. Trivedi
PublisherJohn Wiley & Sons
Chapter 2:Discrete Random Variables
Dept. of Electrical & Computer engineering
Duke University
Email: kst@ee
Probability and Statistics with
Reliability, Queuing and Computer
Science Applications
second edition
by K.S. Trivedi
PublisherJohn Wiley & Sons
Chapter 1: Introduction
Dept. of Electrical & Computer
Engineering
Duke University
Email: kst@ee.duke.edu
URL
Lecture 1 Formation of
Queues
Sec 4.3 Inside a Router (Kurose &
Ross)
Introduction to the Course
The objective of this course is to apply queueing
theory concepts to performance evaluation of
computer networks
First some material from text by Kurose & R
Lecture 9 Discrete Time
Queueing Systems
Robertazzi Ch 6 Secs 6.1, 6.2
6.3, 6.5
Introduction
In all the models considered so far, time has been
assumed to be continuous
That is, customers can arrive at any time and service
can start / finish at any time
Lecture 8 Networks of
Queues
Robertazzi Ch 2 Sec 2.4
Ch 3 Secs 3.1,3.2
Reversibility
A stochastic process X(t) is reversible if:
p(X(t1),X(t2),.) is the same as
p(X(t1), X(t2),) for any ,t1,t2.
Let qij be the transition rate from state i to
state j.
Lecture 7 M/G/1 Queue and
its Special Cases
Robertazzi Ch 2 Secs 2.6 2.9
Motivation for M/G/1 Queue
Recall that in the M/M/1 queue, the arrivals are assumed
to be Poisson and service times are assumed to be
exponential
In many realistic queues, it is r
Lecture 6 Other types of
M/M/. Queues
Robertazzi Ch 2 Secs 2.6 2.9
M/M/1/N Queue
This is called the Finite Buffer Case since the
number of customers in the system (including
the one receiving service, if any) is limited to N
Once the system is full, an
Lecture 5 Basic Queueing
Theory and M/M/1 Queue
Robertazzi Ch 2 Secs 2.1 2.3
Motivation for Queueing Theory
Remember from the previous course
Latency = Propagation delay + transmission delay +
queueing delay
The first two components can be precisely ca
Lecture 4  Introduction to
Probability (Contd)
Trivedi Chapters 4  6
Expectation
The expectation (also called expected
value, mean or average) of a r.v. is given
by:
E(x) = i xi p(xi) if x is discrete
or,
E(x) = + xp(x)dx if x is continuous
Moments an
Lecture 3 Introduction to
Probability
Trivedi Chapters 1  3
Probability Axioms
Let S be the sample space of a random
experiment and A be an event in S. The
axioms are:
A1: For any event A, P(A) >= 0
A2: P(S) = 1
A3: P(AUBUC ) = P(A) +P(B)+P(C)+.
for mut
Lecture 2 Scheduling
Mechanisms
Sec 7.6 Beyond Best Effort
(Kurose & Ross)
Network Scenarios
1.5 Mbps link shared by 2 streams H1H3
and H2H4
Scenario 1 : 1 Mbps Audio + FTP
 Audio packets should get priority
because they are delay sensitive
Scenario
Queuing Theory Exercises. E11
1
Course coordinator: Armin Halilovic
Exercises. E 11
Mixed problems with M/M/ queueing systems
Some formulas


Q 1. We consider an M/M/3/3 queueing system with three types of customers.
Type 1 customers with arrival rate 1
Simulation Modeling
J. M. Akinpelu
1
What is Simulation?
Simulation is the use of a computer to
evaluate a system model numerically, in
order to estimate the desired true
characteristics of the system.
Simulation is useful when a realworld
system is too
Scheduling Theory
J. M. Akinpelu
Discussion
1. For the M/G/1 queue that we have
considered, what things do we know about
the system? What dont we know?
2. What are some service policies that are
independent of service times?
FCFS
LCFS (Last Come First Ser
Reliability Theory
J. M. Akinpelu
System Reliability
System
a collection of interacting or interdependent
components, organized to provide a function or
functions
Components
can be unique
can be redundant
2
Example  Communications System
3
Source: htt
Queueing Systems
Part II
J. M. Akinpelu
M/M/1/K Queue
M/M/1/K queueing system
exponential interarrival distribution
exponential service distribution
1 server
finite queue (K spaces)
FCFS service discipline
2
M/M/1/K Queue
j1=
j=
j
j1
j+1
j=
j+1=
Pj = Pj
Queueing Systems
Part I
J. M. Akinpelu
Queueing Systems
Examples
Customers in a grocery
store
Cars on a highway
Packets in a router
network
Call requests in a
telephone network
2
Queueing Systems
Components
waiting lines (queues)
servers
arriving cu
Queueing Systems
Part I
J. M. Akinpelu
Queueing Systems
Examples
Customers in a grocery
store
Cars on a highway
Packets in a router
network
Call requests in a
telephone network
2
Queueing Systems
Components
waiting lines (queues)
servers
arriving cu
Markov Processes
and
BirthDeath Processes
J. M. Akinpelu
Exponential Distribution
Definition. A continuous random variable X has an
exponential distribution with parameter > 0 if its
probability density function is given by
e x ,
f ( x) =
0,
x0
x < 0.
Markov Chains
J. M. Akinpelu
Stochastic Processes
A stochastic process is a collection of random variables
X = cfw_ X t , t T .
The index t is often interpreted as time.
X t is called the state of the process at time t.
The set T is called the index s
Forecasting Theory
Part II
J. M. Akinpelu
1
Time Series Forecasting
Singlevariable (time series) forecasting
We use past history of the variable of
interest to predict the future.
Predictions exploit correlations between past
history and the future.
P
Forecasting Theory
J. M. Akinpelu
1
What is Forecasting?
Forecast  to calculate or predict some future
event or condition, usually as a result of rational
study or analysis of pertinent data
Websters Dictionary
2
What is Forecasting?
Forecasting methods
Conditional Probability and
Conditional Expectation
J. M. Akinpelu
Discussion
If a fair die is tossed, what is the probability of
getting a 6?
What are you assuming about the possible
outcomes?
S = cfw_1, 2, 3, 4, 5, 6
Now suppose I tell you that the o
Baynesian Networks
J. M. Akinpelu
1
Conditional Probability
1.
P( E F  G ) = P( E  F G ) P( F  G )
2. If E and F are independent, then
P( E  F ) = P( E )
3. Law of Total Probability
i =1
i =1
P ( E ) = P ( E Fi ) = P ( E  Fi ) P( Fi )
2
Conditional P
ECEC632901 Solutions to HW # 6
6.1
a) Utilization of buffer A = 1 + 3
b) Both full : a2
(A packet arrives at both)
One full 2a(1a)
(A packet arrives at one but not the other)
2
Both empty (1a) (No packet arrives at either)
c) This can happen in 3 diff
ECEC632901 Homework # 6
Do the following problems at the end of Ch 6 in Robertazzis text.
Q1. Problems 6.1
Q2. Problem 6.3
Q3. Problem 6.4
Q4. Problem 6.38
Q5. Problem 6.39