PROBABILITY AND STATISTICAL METHODS

Course Objectives:
1.Develop a solid foundation in probability theory and random processes.
2. Learn fundamental modeling and analysis techniques for stochastic systems so you can use them in applications found in various Engineering disciplines, Operations Research, and Computer Science.

Course Outline:

1.  FOUNDATIONS OF PROBABILITY THEORY.
1.1.  Basic concepts (sample space, event space, probability space)
1.2.  Probability measures and probability functions
1.3.  Discrete and continuous probability spaces
1.4.  Dependent and independent events, conditional probability

2.  RANDOM VARIABLES.
2.1.  Definitions
2.2.  Probability distribution and density functions
2.3.  Functions of random variables
2.4.  Expectation, moments, characteristic functions
2.5.  Sequences of random variables, convergence, laws of large numbers and central limit theorem

3.  RANDOM PROCESSES.
3.1.  Definitions
3.2.  Random process properties (stationarity, ergodicity, correlation)
3.3.  Spectral analysis, random process transformations
3.4.  Special random processes used in modeling: Gaussian, Poisson, Markov; applications
3.5.  Introduction to Estimation

DISCRETE EVENT AND HYBRID SYSTEMS

Course Objectives:

1. Learn about Discrete Event Systems (DES) and their applications, as well as recently emerging Hybrid Systems (HS) that combine both continuous (time-driven) and discrete (event-driven) dynamics.
2. Develop the ability to conceptualize cutting-edge issues in the DES and HS domain, and formulate problems for potential research purposes.

Course Outline:

1. REVIEW OF SYSTEM THEORY FUNDAMENTALS

1.1.  Basic concepts
1.2.  Time-driven vs. event-driven systems
1.3.  Examples of Discrete Event Systems (DES): computer systems;  communication networks; automated manufacturing;  traffic systems
1.4.  The queueing system model.

2.  UNTIMED MODELS OF DISCRETE-EVENT SYSTEMS.
2.1.  State Automata
2.2.  Petri Nets

3.  TIMED MODELS OF DISCRETE-EVENT SYSTEMS.
3.1.  Timed State Automata
3.2.  Timed Petri Nets
3.3.  Review of probability theory and stochastic processes
3.4.  Stochastic Timed State Automata
3.5.  The Poisson counting process and Markov chain models

4.  INTRODUCTION TO DISCRETE EVENT (MONTE-CARLO) SIMULATION
4.1.  Basic concepts in discrete event simulation
4.2.  Model construction and applications
4.3.  Introduction to estimation theory

5.  MARKOV DECISION PROCESSES
5.1.  Dynamic Programming
5.2.  Solving resource contention problems: admission control,

routing,  scheduling

6. PERTURBATION ANALYSIS AND CONCURRENT ESTIMATION

7. INTRODUCTION TO HYBRID SYSTEMS 