cse 221: probabilistic analysis of computer systems topics covered: course outline and schedule...
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CSE 221: Probabilistic Analysis of Computer Systems
Topics covered:Course outline and scheduleIntroduction (Sec. 1.1-1.4)
General information
CSE 221 : Probabilistic Analysis of Computer SystemsInstructor : Swapna S. GokhalePhone : 6-2772.Email : ssg@engr.uconn.eduOffice : ITEB 237Lecture time : Mon/Fri 12:30 – 1:45 pmOffice 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)
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. 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 (Aug. 28): 1. Aug. 28: Logistics, Introduction, Sample Space, Events 2. Sept. 1: Event algebra, Probability axioms, Combinatorial problemsWeek #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 methodsWeek #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 estimationWeek #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 midtermProject – 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
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
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