CSE 221: Probabilistic Analysis of Computer Systems

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CSE 221 Probabilistic Analysis of Computer
Systems
Topics covered Course outline and
schedule Introduction (Sec. 1.1-1.4)
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General information
CSE 221 Probabilistic Analysis of Computer
Systems Instructor Swapna S. Gokhale Phone
6-2772. Email ssg_at_engr.uconn.edu Offic
e ITEB 237 Lecture time Mon/Fri 1230
145 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.e
du/ssg/cse221.html
(Lecture notes, homeworks, and general
announcements will
be posted on the web page)
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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

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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.

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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.

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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.

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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.

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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)

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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

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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.

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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.

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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)

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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 (HW3) 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
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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
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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)

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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.

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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.

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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
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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

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Feedback
Please provide informal feedback early and often,
before the formal review process.
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Introduction and motivation

• Why study probability theory?
• Answer questions such as

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Probability model

• Examples of random/chance phenomenon
• What is a probability model?

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Sample space

• Definition
• Example Status of a computer system
• Example Status of two components CPU, Memory
• Example Outcomes of three coin tosses

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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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Events

• Definition of an event
• Example Sequence of three coin tosses
• Example System up.

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Events (contd..)

• Universal event
• Null event
• Elementary event