1. Introduction to discrete event systems
Chapter I
Introduction to discrete event systems
Learning objectives :
Introduce fundamental concepts of system theory
Understand features of event-driven dynamic systems
Textbook :
C. Cassandras and S. Lafortune, Introduction to Discrete Event Systems, Springer, 2007
Support en Français et en Anglais:

2. Discrete-event system by an example of a queueing system
Plan
System basics
Discrete-event system by an example of a queueing system
Discrete event systems

3. System basics 3

4. Interacting components Function the system is supposed to perform
The concept of system
•System:
A combination of components that act together to perform a function not possible with any of the individual parts (IEEE)
•Salient features :
Interacting components
Function the system is supposed to perform

5. The Input-Output Modeling process
Define a set of measurable variables
Select a subset of variables that can be changed over time (Input variables)
Select another set of variables directly measurable (Output variables, responses, stimulus)
Derive the Input-Output relation

6. The Input-Output Modeling process
Example 1 : An electric circuit with two resistances r and R
y(t)/u(t)= R/(r+R)
Example 2 : An electric circuit with a resistance R and a capacitor (condensateur) C
u(t) = vR(t) + y(t)
vR(t) = iR
i=C.dy(t)/dt
Y(s)/U(s) = 1/(1+CRs)

7. Static and dynamic systems
Static systems :
Output y(t) independent of the past values of the input u(t), for t < t.
The IO relation is a function : y(t) = g(u(t))
Dynamic systems :
Output y(t) depends on past values of the input u(t), for t < t.
Memory of the input history is needed to determine y(t)
The IO relation is a differential equation. 7

8. The concept of state Definition :
The state of a system at time t0 is the information required at t0 such that the output y(t), for all t ≥ t0 is uniquely determined from this information and from u(t), t ≥ t0.
The state is generally a vector of state variables x(t). 8

9. System dynamics State equation :
The set of equations required to specify the state x(t) for all t≥ t0, given x(t0) and the function u(t), t≥ t0.
State space :
The state space of a system is a set of all possible values that the state may take.
Output equation : 9

10. System dynamics : sample path10

11. Discrete system
The system is observed at regular intervals at time t = nD for all constant elementary period D. 11

12. A queueing system 12

13. x(t) = number of customers in the system Random customer arrivals
State of the system :
x(t) = number of customers in the system
Random customer arrivals
Random service times
FIFO service 13

14. arrival times t of customers where x(t+0) = x(t0) +1
System dynamic
The state of the system remains unchanged except at the following instants (events)
arrival times t of customers where
x(t+0) = x(t0) +1
departure times t of customers where
x(t+0) = x(t0)  1
Sample path 14

15. Discrete event systems15

16. The concept of event
An event occurs instantaneously and causes transitions from one discrete state to another
An event can be
a specific action taken (press a button)
a spontaneous occurrence dictated by nature (failures)
sudden fulfillment of some conditions (buffer full).
Notation : e = event, E = set of event.
Queueing system: E = {a, d} with a = arrival, d = departure 16

17. Time-driven and event-driven systems
Time-driven systems
Continuous time systems
Discrete systems (driven by regular clock ticks)
State transitions are synchronized by the clock
Event-driven systems
State changes at various time instants (may not known in advance) with some event e announcing that it is occurring
State transitions as a result of combining asynchronous and concurrent event processes. 17

18. Characteristics of discrete event systems
Definition.
A Discrete Event Systems (DES) is a discrete-state, event-driven system, that is, its state evolution depends entirely on the occurrence of asyncrhonuous discrete events over time.
Essential defining elements:
E : a discrete-event set
X : a discrete state space 18

19. Two Points of Views Untimed models (logical behavior)
Input : event sequence {e1, e2, ...} without information about the occurrence times.
Sample path: sequence of states resulting from {s1, s2, ...}
Timed models (quantitative behavior)
Input : timed event sequence {(e1, t1), (e2, t2), ...}.
Sample path : the entire sample path over time. Also called a realization. e1 e2 e3 e4 e5 e1 e2 e3 e4 e5 s1 s2 s3 s4 s5 s6 t1 t2 t3 t4 t5 19

20. A manufacturing system1 2
part departures
part arrivals
A two-machine transfer line with an intermediate buffer of capacity 3.
Essential defining elements:
E = {a, d1, d2}
X = {(x1, x2) : x1 ≥ 0, x2  {0, 1, 2, 3, B}} 20

21. System classifications
Static vs dynamic systems
Time-varying vs time-invariant systems
Linear vs nonlinear systems
continuous-state vs discrete state systems
time-drived vs event-driven systems
deterministic vs stochastic systems
discrete-time vs continuous-time systems 21

22. Goals of system theory Modeling and analysis Design and synthesis
Control
Performance evaluation
Optimization 22