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 : Probabilistic Analysis of Computer Systems Instructor : Swapna S. Gokhale Phone : 6-2772. Email : firstname.lastname@example.org 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
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: • 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