Discrete Event Systems
- Corso del 1° Anno di Magistrale (1° Semestre).
- Docente: Simone Paoletti.
- Link to Drive with Video Lectures
Perquisites:
Contents and Program:
Introduction
Untimed models of discrete event systems
- Types of DESs:
- ‘State Automata’: Logical (or untimed) Discrete Event-Driven System (with or without outputs) Logical: because the system doesn’t have a concept of time. Discrete Event-Driven: the system evolves only when an event occurs. NOTE: Because the system doesn’t have time it’s neither Discrete nor Dynamical.
- Other types of DESs are: Timed Automata, Stochastic Timed Automata, and Stochastic Timed Automata with Poisson Clock Structure
Timed models of discrete event systems
- ‘Timed Automata’: Same as State Automata but added the concept of time, for when events happen.
Usually these system are Dynamical, though the Event are always discrete so they are still non-differentiable.
Stochastic timed models of discrete event systems
- ‘Stochastic Timed Automata’: Adding also stochastic probabilities to timed automata; an event can happen at time with probability
- ‘Stochastic Timed Automata with Poisson Clock Structure’: Particular case of Stochastic Timed Automata, where all probability distributions are exponential.
Markov chains
Queuing theory
Queuing theory is the subject dedicated to the specific study of queuing systems/networks.
Markovian queuing networks
Benefits of having a Continuous Time Homogeneous Markov Chain representation of the model.
Index - All Notes:
Lecture Notes:
Introduction
- DES - Differences Between Time-Driven Dynamics and Event-Driven Dynamics
- DES - Definition of ‘Dynamical’
- DES - Definition of ‘Discrete’
- DES - What are Discrete Systems
Untimed models of discrete event systems
- DES - Model of a State Automata
- DES - Definition of ‘State of a System’
- DES - Definition of ‘Transition Function’
- DES - Definition of ‘Domain of a Transition Function’
- DES - Definition of ‘Feasible Event Sequence’
- DES - Definition of ‘State Automata’
- DES - Definition of ‘State Automata with Output’
Timed models of discrete event systems
- DES - Model of a Timed Automata
- DES - Definition of ‘Timed Automata’
- DES ~ Example of a FIFO Clock Structure
- DES ~ Example of a RR Clock Structure
- DES - Definition of an ‘Time-Driven System’
- DES - Definition of ‘Score’
Stochastic timed models of discrete event systems
- DES - Model of a Stochastic Timed Automata
- DES - Definition of ‘Expected Value’
- DES - Definition of ‘Variance’
- DES - Uniform Distribution
- DES - Exponential Distribution
- DES - Poisson Distribution
- DES - Theorem ‘Total Probability Rule’
- DES - Definition of ‘Stochastic Timed Automata’
Timed automata with Poisson clock structures
- DES - Model of a Stochastic Timed Automata with Poisson Clock Structure
- DES - Definition of ‘Memory-Less Property’
- DES - Definition of ‘Extended Memory-Less Property’
- DES - Definition of ‘Poisson Clock Structure’
- DES - Definition of ‘Superposition’
- DES - Definition of ‘State Holding Time’
- DES - Theorem ‘Distribution of the State Holding Time for a Poisson Clock Structure’
- DES - Definition of a ‘Poisson Process’
- DES - Definition of ‘Stochastic Timed Automata with Poisson Clock Structure’
Markov chains
- DES - Model of a CTHMC
- DES - Definition of ‘Stochastic Process’
- DES - Definition of ‘Independent Stochastic Process’
- DES - Definition of ‘Homogeneity’
- DES - Definition of ‘Markov Process’
- DES - Definition of ‘Continuous Time Homogeneous Markov Chain (CTHMC)’
- DES - Definition of ‘Transition Function Matrix for CTHMC’
- DES - Classification of the States of a CTHMC
- DES - Definition of ‘Reachable State’
- DES - Definition of ‘Closed Subset’
- DES - Definition of ‘Irreducible Subset’
- DES - Definition of ‘Irreducible CTHMC’
- DES - Definition of ‘Recurrent State or Transient State’
- DES - Chapman-Kolmogorov Equation
- DES - Definition of ‘Transition Rate Matrix Q’
- DES - Graphical Representation of CTHMC
- DES - Graphical Transition from Stochastic State Automata with Poisson Clock Structure to CTHMC
- DES - Transient Analysis
- DES - Steady-State Analysis
- DES - Definition of ‘Limit State Probability Vector’
- DES - Theorem of ‘Existence, Uniqueness and Consistency of the Limit State Probability Vector’
Queuing Theory:
- DES - Definition of ‘Queuing System’
- DES - Definition of ‘Queuing Network’
- DES - Specification of Queuing Systems
- DES - Kendal’s Notation
- DES - Analysis of Queuing Networks
- DES - Steady-State Conditions for Queuing Systems
- DES - Characterization of Steady-State
- DES - Little’s Law
- DES - Pasta Property
- DES - Performance Indicators
Markovian Queuing Networks:
Discrete Systems
- DES - Law of Large Numbers
- DES - Central Limit Theorem
- DES - Empirical CDF
- DES - Histograms and Empirical PDF
- DES - Estimation of Models of Discrete Event Systems
- DES - Analysis of Discrete Event Systems
- DES - Definition of ‘Stochastic Equivalence of Discrete-Event Models’
- DES - Stochastically Equivalent CTHMC of a Discrete-Event Model
- DES - Discrete Event Simulation
- DES - Random Number Generation
- DES - As-is and What-if Analysis
- DES - Model of a DTHMC
- DES - Definition of ‘Discrete Time Homogeneous Markov Chain (DTHMC)’
- DES - Chapman Kolmogorov Equation in Discrete Time
- DES - Transient Analysis in Discrete Time
- DES - Graphical Representation of DTHMC
- DES - Definition of ‘State-Holding Time’ in Discrete Time
- DES - Geometric Distribution
- DES - Classification of States of a DTHMC
- DES - Definition of ‘Reachable State’ in Discrete Time
- DES - Definition of ‘Closed Subset’ in Discrete Time
- DES - Definition of ‘Absorbing State’
- DES - Definition of ‘Irreducible Subset’ in Discrete Time
- DES - Definition of ‘Irreducible DTHMC’
- DES - Definition of ‘Recurrent State or Transient State’ in Discrete Time
- DES - Definition of ‘Periodic and Aperiodic States’ in Discrete Time
- DES - Definition of ‘Period of a DTHCM’
- DES - Steady States Analysis of DTHMCs
- DES - Theorem of ‘Existence, Uniqueness and Consistency of the Limit State Probability Vector’ in Discrete Time
- DES ~ Example ‘How to Calculate the Limit Probability Vector of a DTHMC’
Exercises:
Discrete Event Systems - All Exercises
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All types of exercises:
MATLAB Scripts:
All My Notes
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