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: http://www.emse.fr/~xie/master

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

System basics 3

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

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

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)

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

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

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

System dynamics : sample path 10

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

A queueing system 12

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

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

Discrete event systems 15

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

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

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

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

A manufacturing system 1 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

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

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