0. CSE 3504: Probabilistic Analysis of Computer Systems
Topics covered:
Course outline and schedule
Introduction
Event Algebra
(Sec )

1. General information
CSE : Probabilistic Analysis of Computer Systems
Instructor : Swapna S. Gokhale
Phone :
Office : ITEB 237
Lecture 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 :
(Lecture notes, homeworks, and general announcements
will be posted on the web page)

2. 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

3. 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.

4. 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.

5. 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.

6. 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.

7. Textbooks Required text book:
K. S. Trivedi, Probability and Statistics with Reliability, Queuing and
Computer Science Applications, Second Edition, John Wiley.

8. 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 , , 2.5.5,2.5.7, ):
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

9. 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.

10. 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.

11. 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 distributions
Week #5 (Feb. 18):
9. Feb. 18: Exponential distribution, reliability and failure rate
10. Feb. 20: Expectations of random variables, moments

12. 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 performance
Week #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

13. 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 models
Week #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.

14. 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.

15. 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.

16. 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.

17. Grading system Homeworks – 0% - Ungraded homeworks. Midterm - 20%
Project – 25%
- Two to three problems.
Final %
- Heavy emphasis on the final

18. 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

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

20. Introduction and motivation
Why study probability theory?
Answer questions such as:

21. Probability model Examples of random/chance phenomenon:
What is a probability model?

22. Sample space Definition: Example: Status of a computer system
Example: Status of two components: CPU, Memory
Example: Outcomes of three coin tosses

23. Types of sample space
Based on the number of elements in the sample space:
Example: Coin toss
Countably finite/infinite
Countably infinite

24. Events Definition of an event: Example: Sequence of three coin tosses:
Example: System up.

25. Events (contd..)
Universal event
Null event
Elementary event

26. 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

27. 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

28. Example (contd..) Collectively exhaustive events:
Defining each sample point to be an event