Title: CSE 221: Probabilistic Analysis of Computer Systems
1CSE 221 Probabilistic Analysis of Computer
Systems
Topics covered Course outline and
schedule Introduction Event Algebra (Sec.
1.1-1.4)
2General information
CSE 221 Probabilistic Analysis of Computer
Systems Instructor Swapna S. Gokhale Phone
6-2772. Email ssg_at_engr.uconn.edu Offic
e ITEB 237 Lecture time Mon/Fri 1100
1215 pm Office hours By appointment
(I will hang around for a few
minutes at the end of
each class). Web page http//www.engr.uconn.e
du/ssg/cse221.html
(Lecture notes, homeworks, and general
announcements will
be posted on the web page) TA Narasimha
Shashidhar
3Course goals
- Appreciation and motivation for the study of
probability theory. - Definition of a probability model
- Application of discrete and continuous random
variables - Computation of expectation and moments
- Application of discrete and continuous time
Markov chains. - Estimation of parameters of a distribution.
- Testing hypothesis on distribution parameters
4Expected learning outcomes
- Sample space and events
- Define a sample space (outcomes) of a random
experiment and identify events of interest and
independent events on the sample space. - Compute conditional and posterior probabilities
using Bayes rule. - Identify and compute probabilities for a sequence
of Bernoulli trials. - Discrete random variables
- Define a discrete random variable on a sample
space along with the associated probability mass
function. - Compute the distribution function of a discrete
random variable. - Apply special discrete random variables to
real-life problems. - Compute the probability generating function of a
discrete random variable. - Compute joint pmf of a vector of discrete random
variables. - Determine if a set of random variables are
independent.
5Expected learning outcomes (contd..)
- Continuous random variables
- Define general distribution and density
functions. - Apply special continuous random variables to real
problems. - Define and apply the concepts of reliability,
conditional failure rate, hazard rate and inverse
bath-tub curve. - Expectation and moments
- Obtain the expectation, moments and transforms of
special and general random variables. - Stochastic processes
- Define and classify stochastic processes.
- Derive the metrics for Bernoulli and Poisson
processes.
6Expected learning outcomes (contd..)
- Discrete time Markov chains
- Define the state space, state transitions and
transition probability matrix - Compute the steady state probabilities.
- Analyze the performance and reliability of a
software application based on its architecture. - Statistical inference
- Understand the role of statistical inference in
applying probability theory. - Derive the maximum likelihood estimators for
general and special random variables. - Test two-sided hypothesis concerning the mean of
a random variable.
7Expected learning outcomes (contd..)
- Continuous time Markov chains
- Define the state space, state transitions and
generator matrix. - Compute the steady state or limiting
probabilities. - Model real world phenomenon as birth-death
processes and compute limiting probabilities. - Model real world phenomenon as pure birth, and
pure death processes. - Model and compute system availability.
8Textbooks
- Required text book
- K. S. Trivedi, Probability and Statistics with
Reliability, Queuing and - Computer Science Applications, Second Edition,
John Wiley. -
9Course topics
- Introduction (Ch. 1, Sec. 1.1-1.5, 1.7-1.11)
- Sample space and events, Event algebra,
Probability axioms, Combinatorial problems,
Independent events, Bayes rule, Bernoulli trials - Discrete random variables (Ch. 2, Sec. 2.1-2.4,
2.5.1-2.5.3, 2.5.5,2.5.7,2.7-2.9) - Definition of a discrete random variable,
Probability mass and distribution functions,
Bernoulli, Binomial, Geometric, Modified
Geometric, and Poisson, Uniform pmfs, Probability
generating function, Discrete random vectors,
Independent events. - Continuous random variables (Ch. 3, Sec. 3.1-3.3,
3.4.6,3.4.7) - Probability density function and cumulative
distribution functions, Exponential and uniform
distributions, Reliability and failure rate,
Normal distribution
10Course topics (contd..)
- Expectation (Ch. 4, Sec. 4.1-4.4, 4.5.2-4.5.7)
- Expectation of single and multiple random
variables, Moments and transforms - Stochastic processes (Ch. 6, Sec. 6.1, 6.3 and
6.4) - Definition and classification of stochastic
processes, Bernoulli and Poisson processes. - Discrete time Markov chains (Ch. 7, Sec.
7.1-7.3) - Definition, transition probabilities, steady
state concept. Application of discrete time
Markov chains to software performance and
reliability analysis - Statistical inference (Ch. 10, Sec. 10.1, 10.2.2,
10.3.1) - Motivation, Maximum likelihood estimates for the
parameters of Bernoulli, Binomial, Geometric,
Poisson, Exponential and Normal distributions,
Parameter estimation of Discrete Time Markov
Chains (DTMCs), Hypothesis testing.
11Course topics (contd..)
- Continuous time Markov chains (Ch. 8, Sec.
8.1-8.3, 8.4.1) - Definition, Generator matrix, Computation of
steady state/limiting probabilities, Birth-death
process, M/M/1 and M/M/m queues, Pure birth and
pure death process, Availability analysis.
12Course topics and exams calendar
Week 1 (Jan. 21) 1. Jan 25 Logistics,
Introduction, Sample Space, Events, Event
algebra Week 2 (Jan. 28) 2. Jan 28
Probability axioms, combinatorial problems
3. Feb. 1 Conditional probability, Independent
events, Bayes rule, Bernoulli trials Week 3
(Feb. 4) 4. Feb. 4 Discrete random
variables, Probability mass and Distribution
function. 5. Feb. 8 Special discrete
distributions Week 4 (Feb. 11) 6. Feb.
11 Poisson pmf, Uniform pmf, Probability
Generating Function 7. Feb. 15 Discrete
random vectors, Independent random variables Week
5 (Feb. 18) 8. Feb. 18 Continuous
random variables, Uniform and Normal
distributions 9. Feb. 22 Exponential
distribution, reliability and failure rate
13Course topics and exams calendar (contd..)
Week 6 (Feb. 25) 10. Feb. 25
Expectations of random variables, moments
11. Feb. 29 Multiple random variables,
transform methods Week 7 (Mar. 3) 12.
Mar 3 Moments and transforms of special
distributions 13. Mar 7 Stochastic
processes, Bernoulli and Poisson processes Week
8 (Mar. 10) Spring break, no class.
Week 9 (Mar. 17) 14. Mar 17 Discrete
time Markov chains 15. Mar 21 Discrete
time Markov chains (contd..) Week 10 (Mar. 24)
16. Mar 24 Analysis of software
reliability and performance 17. Mar 28
Statistical inference Week 11 (Mar. 31)
18. Mar 31 Statistical inference (contd..)
19. Apr. 4 Confidence intervals
14Course topics and exams calendar (contd..)
Week 12 (Apr. 7) 20. Apr. 7 Hypothesis
testing 21. Apr. 11 Hypothesis testing
(contd..) Week 13 (Apr. 13) Apr. 14 No
class 22. Apr. 18 Continuous time Markov
chains Week 14 (Apr. 20) 23. Apr. 21
Simple queuing models 24. Apr. 25 Pure
death processes, availability models Week 15
(Apr. 27) Apr. 27 Make up class
May 2 Final exam handed.
15Assignment/Homework logistics
- There will be one homework based on each topic
(approximately) - One week will be allocated to complete each
homework - Homeworks will not be graded, but I encourage you
to do homeworks since the exam problems will be
similar to the homeworks. - Solution to each homework will be provided after
a week. - Homework schedule is as follows
- HW 1 (Handed Feb. 1, Lectures 1-3 )
- HW 2 (Handed Feb. 15, Lectures 4 - 7)
- HW 3 (Handed Feb. 22, Lectures 8 - 9)
- HW 4 (Handed Mar 2, Lectures 10 - 12 )
- HW 5 (Handed Mar. 24, Lectures 13 - 16)
- HW 6 (Handed Apr. 11, Lectures 17 - 21)
- HW 7 (Handed Apr. 25, Lectures 22 - 24)
16Exam logistics
- Exams will have problems similar to that of the
homeworks. - Exam I (Feb. 29)
- Lectures 1 through 9
- Exam II (Apr. 11)
- Lectures 10 through 19
- Exams will be take-home.
17Project logistics
- Project will be handed in the week first week of
April, and and will be due in the last week of
classes. - 2-3 problems
- Experimenting with design options to explore
tradeoffs and to determine which system has
better performance/reliability etc. - Parameter estimation, hypothesis testing with
real data. - May involve some programming (can be done using
Java, Matlab etc.) - Project report must describe
- Approach used to solve the problem.
- Results and analysis.
18Grading system
Homeworks 0 - Ungraded homeworks.
Midterms - 30 - Three midterms, 15 per
midterm Project 25 - Two to three problems.
Final - 45 - Heavy emphasis on the
final
19Attendance policy
- Attendance not mandatory.
- Attending classes helps!
- Many examples, derivations (not in the book) in
the class - Problems, examples covered in the class fair game
for the exams. - Everything not in the lecture notes
20Feedback
Please provide informal feedback early and often,
before the formal review process.
21Introduction and motivation
- Why study probability theory?
- Answer questions such as
22Probability model
- Examples of random/chance phenomenon
- What is a probability model?
23Sample space
- Definition
- Example Status of a computer system
- Example Status of two components CPU, Memory
- Example Outcomes of three coin tosses
24Types of sample space
- Based on the number of elements in the sample
space - Example Coin toss
- Countably finite/infinite
- Countably infinite
25Events
- Definition of an event
- Example Sequence of three coin tosses
- Example System up.
26Events (contd..)
- Universal event
- Null event
- Elementary event
27Example
- Sequence of three coin tosses
- Event E1 at least two heads
- Complement of event E1 at most one head (zero
or one head) - Event E2 at most two heads
28Example (contd..)
- Event E3 Intersection of events E1 and E2.
- Event E4 First coin toss is a head
- Event E5 Union of events E1 and E4
- Mutually exclusive events
29Example (contd..)
- Collectively exhaustive events
- Defining each sample point to be an event