Computer ppt

1. CSE 221: Probabilistic Analysis of Computer Systems
Topics covered:
Course outline and schedule
Introduction
Event Algebra
(Sec. 1.1-1.4)

2. General information
CSE 221 : Probabilistic Analysis of Computer Systems
Instructor : Swapna S. Gokhale
Phone : 6-2772.
Email : ssg@engr.uconn.edu
Office : ITEB 237
Lecture time : Mon/Fri 11:00 – 12:15 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.edu/~ssg/cse221.html
(Lecture notes, homeworks, and general announcements
will be posted on the web page)
TA : Narasimha Shashidhar

3. Course 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

4. Expected 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.

5. Expected 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.

6. Expected 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.

7. Expected 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.

8. Textbooks
Required text book:
1. K. S. Trivedi, Probability and Statistics with Reliability, Queuing and
Computer Science Applications, Second Edition, John Wiley.

9. Course 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

10. Course 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.

11. Course 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.

12. Course 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

13. Course 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

14. Course 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.

15. Assignment/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)

16. Exam 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.

17. Project 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.

18. Grading 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

19. Attendance 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

20. Feedback
Please provide informal feedback early and often, before the formal
review process.

21. Introduction and motivation
 Why study probability theory?
 Answer questions such as:

22. Probability model
 Examples of random/chance phenomenon:
 What is a probability model?

23. Sample space
 Definition:
 Example: Status of a computer system
 Example: Status of two components: CPU, Memory
 Example: Outcomes of three coin tosses

24. Types of sample space
 Based on the number of elements in the sample space:
 Example: Coin toss
 Countably finite/infinite
 Countably infinite

25. Events
 Definition of an event:
 Example: Sequence of three coin tosses:
 Example: System up.

26. Events (contd..)
 Universal event
 Null event
 Elementary event

27. Example
 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

28. Example (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

29. Example (contd..)
 Collectively exhaustive events:
 Defining each sample point to be an event