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CSE 221: Probabilistic Analysis of Computer Systems

CSE 221: Probabilistic Analysis of Computer Systems. Topics covered: Course outline and schedule Introduction (Sec. 1.1-1.4). General information. CSE 221 : Probabilistic Analysis of Computer Systems Instructor : Swapna S. Gokhale Phone : 6-2772. Email : ssg@engr.uconn.edu

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CSE 221: Probabilistic Analysis of Computer Systems

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  1. CSE 221: Probabilistic Analysis of Computer Systems Topics covered: Course outline and schedule Introduction (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 12:30 – 1:45 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)

  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: • K. S. Trivedi, Probability and Statistics with Reliability, Queuing and • Computer Science Applications, Second Edition, John Wiley. • (Book will be available week of Sept. 6)

  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 (Aug. 28): 1. Aug. 28: Logistics, Introduction, Sample Space, Events 2. Sept. 1: Event algebra, Probability axioms, Combinatorial problems Week #2 (Sept. 4): Sept. 4: Labor Day (no class) 3. Sept. 8: Combinatorial problems, Conditional probability, Independent events. Week #3 (Sept. 11): Sept. 11: No class. 4. Sept. 15: Bayes rule, Bernoulli trials (HW #1) Week #4 (Sept. 18): 5. Sept. 18: Discrete random variables, Mass and Distribution functions 6. Sept. 22: Bernoulli, Binomial and Geometric pmfs. Week #5 (Sept. 25): 7. Sept. 25: Poisson pmf, Probability Generating Function (PGF) 8. Sept. 29: Discrete random vectors, Independent random variables. (HW #2)

  13. Course topics and exams calendar (contd..) Week #6 (Oct. 2): 9. Oct. 2: Continuous random variables, Uniform & Normal distributions 10. Oct. 6: Exponential distribution, Reliability, Failure rate (HW#3) Week #7 (Oct. 9): 11. Oct 9: Expectation of random variables, Moments 12. Oct. 13: Multiple random variables, Transform methods Week #8 (Oct. 16): 13. Oct. 16: Moments and transforms of some distributions 14. Oct. 20: Stochastic process, Bernoulli and Poisson process (HW #4) Week #9 (Oct. 23): 15. Oct. 23: Discrete Time Markov Chains 16. Oct. 27: Discrete Time Markov Chains Week #10 (Oct. 30): 17. Oct. 30: Discrete Time Markov Chains (HW #5) 18. Nov. 3: Statistical inference, Parameter estimation Week #11 (Nov. 6): 19. Nov. 6: Statistical inference, Parameter estimation Nov. 10 – no class

  14. Course topics and exams calendar (contd..) Week #12 (Nov. 13): 20. Nov. 13: Hypothesis testing (HW #6) 21. Nov. 17: Continuous Time Markov Chains, Birth-Death process (Project) Week #13 (Nov. 20): Thanksgiving (no class) Week #14: (Nov. 27) 22. Nov. 27: Simple queuing models 23. Dec. 1: Simple queuing models (contd..) Week #15: (Dec. 4) 23. Dec. 4: Pure birth/pure death process, Availability analysis (HW #7) 24. Dec. 8: Overview

  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: Sept. 15, Lectures #1-#4) • HW #2 (Handed: Sept. 29, Lectures #5-#8) • HW #3 (Handed: Oct. 6, Lectures #9-#10) • HW #4 (Handed: Oct. 20, Lectures #11-#14) • HW #5 (Handed: Oct. 30, Lectures, #15-#17) • HW #6 (Handed: Nov. 13, Lectures #18-#20) • HW #7 (Handed: Dec. 4, Lectures #21-#24)

  16. Exam logistics • Exams will have problems similar to that of the homeworks. • Exam I: (Oct. 6) • Lectures 1 through 8 • Exam II: (Nov. 3) • Lectures 9 through 14 • Exam III: (Dec. 1) • Lectures 15 through 20 • Exams will be take-home.

  17. Project logistics • Project will be handed in the week before Thanksgiving, 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 - 45% - Three midterms, 15% per midterm Project – 25% - Two to three problems. Final - 30% - 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

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