CSE 221: Probabilistic Analysis of Computer Systems Topics covered: Course outline and schedule Introduction (Sec )

General information CSE 221 : Probabilistic Analysis of Computer Systems Instructor: Swapna S. Gokhale Phone : 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: (Lecture notes, homeworks, and general announcements will be posted on the web page)

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. (Book will be available week of Sept. 6)

Course topics  Introduction (Ch. 1, Sec , ):  Sample space and events, Event algebra, Probability axioms, Combinatorial problems, Independent events, Bayes rule, Bernoulli trials  Discrete random variables (Ch. 2, Sec , , 2.5.5,2.5.7, ):  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 , 3.4.6,3.4.7):  Probability density function and cumulative distribution functions, Exponential and uniform distributions, Reliability and failure rate, Normal distribution

Course topics (contd..)  Expectation (Ch. 4, Sec , ):  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 ):  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, , ):  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 , 8.4.1):  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 (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)

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

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

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)

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.

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.

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

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