CSE 221: Probabilistic Analysis of Computer Systems

Topics covered: Course outline and schedule Introduction Event Algebra (Sec. 1.1-1.4)

General information

CSE 221 Instructor Phone Email Office Lecture time Office hours Web page TA : Probabilistic Analysis of Computer Systems : Swapna S. Gokhale : 6-2772.

: [email protected]

: ITEB 237 : Mon/Fri 11:00 – 12:15 pm : By appointment (I will hang around for a few minutes at the end of each class).

: http://www.engr.uconn.edu/~ssg/cse221.html

(Lecture notes, homeworks, and general announcements will be posted on the web page) : Narasimha Shashidhar

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

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.

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.

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.

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.

Textbooks

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

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

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.

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.

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

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

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.

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)

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.

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.

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

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

Feedback

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

Introduction and motivation

 Why study probability theory?

 Answer questions such as:

Probability model

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

Sample space

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

Types of sample space

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

Events

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

Events (contd..)

 Universal event  Null event  Elementary event

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

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

Example (contd..)

 Collectively exhaustive events:  Defining each sample point to be an event