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 Event Algebra (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 1100
1215 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) TA Narasimha
Shashidhar
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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.

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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 (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

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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

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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.

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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 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)

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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.

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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.

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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
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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

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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

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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

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Example (contd..)

• Collectively exhaustive events
• Defining each sample point to be an event