CSE 3504: Probabilistic Analysis of Computer Systems

Topics covered:Course outline and scheduleIntroductionEvent Algebra (Sec. 1.1-1.4)

General information

CSE 3504 : Probabilistic Analysis of Computer SystemsInstructor : Swapna S. GokhalePhone : 6-2772.Email : ssg@engr.uconn.eduOffice : ITEB 237Lecture time : Wed/Fri 9:30 – 10:45 am 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/cse3054.html (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.

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 21: Logistics, Introduction, Sample Space, Events, Event algebra 2. Jan 23: Probability axioms, combinatorial problems Week #2 (Jan. 28): 3. Jan 28: Conditional probability, Independent events, Bayes rule, Bernoulli trials 4. Jan 30: Discrete random variables, Probability mass and Distribution function Week #3 (Feb. 4): 5. Feb. 4: Special discrete distributions 6. Feb. 6: Poisson pmf, Uniform pmf, Probability Generating Function Week #4 (Feb. 11): 7. Feb. 11: Discrete random vectors, Independent random variables 8. Feb. 13: Continuous random variables, Uniform and Normal distributionsWeek #5 (Feb. 18): 9. Feb. 18: Exponential distribution, reliability and failure rate 10. Feb. 20: Expectations of random variables, moments

Course topics and exams calendar (contd..)

Week #6 (Feb. 25): 11. Feb. 25: Multiple random variables, transform methods 12. Feb. 27: Moments and transforms of special distributions Week #7 (Mar. 4): 13. Mar 4: Stochastic processes, Bernoulli and Poisson processes 14. Mar 6: Discrete time Markov chains Week #8 (Mar. 11): Spring break, no class. Week #9 (Mar. 18): 15. Mar 18: Discrete time Markov chains (contd..) 16. Mar 20: Analysis of software reliability and performanceWeek #10 (Mar. 25): 17. Mar 25: Statistical inference 18. Mar 27: Statistical inference (contd..)Week #11 (Apr. 1): 19. Apr. 1: Confidence intervals 20. Apr. 3: Hypothesis testing

Course topics and exams calendar (contd..)

Week #12 (Apr. 8): 21. Apr. 8: Hypothesis testing (contd..) 22. Apr. 10: Continuous time Markov Chains Week #13 (Apr. 15): 23. Apr. 14: Continuous time Markov chains, applications (contd..) 24. Apr. 18: Simple queuing modelsWeek #14: (Apr. 22) 25. Apr. 22: Pure death processes, availability models 26. Apr. 24: Lognormal distribution and its applications Week #15: (Apr. 29) Apr. 29: Make up class May 1: 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.

Exam logistics

Exams will have problems similar to that of the homeworks.

Midterm exam: Before Spring break Final exam: Handed on the last day of classes, due

when the exam for the class is scheduled. 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. Midterm - 20%Project – 25% - Two to three problems. Final - 55% - 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