Timed event system

The General System has been described in [Zeigler76] and [ZPK00] with the stand points to define (1) the time base, (2) the admissible input segments, (3) the system states, (4) the state trajectory with an admissible input segment, (5) the output for a given state.

A Timed Event System defining the state trajectory associated with the current and event segments is a sub-class of the class of General System. Since the behaviors of DEVS can be described by Timed Event System, DEVS is a sub-class or a equivalent class of Timed Event System.

Timed Event Systems

A timed event system is a structure

$\mathcal{G}=<Z, Q, q_0, Q_A,\Delta>$

where

• $\,Z$ is the set of events;
• $\,Q$ is the set of states;
• $\,q_0 \in Q$ is the initial state;
• $Q_A \subseteq Q$ is the set of acceptance states;
• $\Delta: Q \times \Omega_{Z,[t_l,t_u]} \rightarrow Q$ is the state trajectory function that defines how a state $q \in Q$ changes into $q' \in Q$ along with an event segment $\omega \in \Omega_{Z,[t_l, t_u]}$.

If $\, \omega$ is concatenation of two unit event segments, i.e., $\,\omega=\omega_1 \omega_2$ then

$\, \Delta(q,\omega) = \Delta(\Delta(q,\omega_1),\omega_2).$

In general, if $\, \omega$ is concatenation of n unit event segments, i.e., $\,\omega=\omega_1 \omega_2 \cdots \omega_n$ where $n \ge 2$ then

$\Delta(q,\omega) = \Delta(\ldots \Delta(\Delta(q,\omega_1),\omega_2)\ldots), \omega_n).$

Deterministic System and Non-deterministic System

Let A and B be two arbitrary sets. Then function $f:A \rightarrow B$ is called deterministic if for $a \in A$, f(a) is identical any time. Otherwise, f is called non-deterministic.

A Timed Event System $\mathcal{G}=<Z, Q, q_0, Q_A,\Delta>$ is deterministic if

1. selecting $\,q_0$ is deterministic, and
2. $\,\Delta$ is deterministic.

Otherwise, $\mathcal{G}$ is non-deterministic.

Behaviors and Languages of Timed Event System

Given a timed event system $\mathcal{G}=<Z,Q,q_0,Q_A,\Delta>$, the set of its behaviors is called its language depending on the observation time length. Let t be the observation time length. If $0 \le t <\infty$, t-length observation language of $\mathcal{G}$ is denoted by $L(\mathcal{G}, t)$, and defined as

$L(\mathcal{G},t)=\{\omega \in \Omega_{Z,[0,t]}: \exists \text{ the case } \Delta(q_0,\omega) \in Q_A\}.$

Notice that the reason why we need "there exists the case" is that we allow $\mathcal{G}$ can be nondeterministic so the number of possible results Δ(q₀,ω) can be many. Finally, we call an event segment $\omega \in \Omega_{Z,[0,t]}$ a t-length behavior of $\mathcal{G}$, if $\omega \in L(\mathcal{G},t)$.

We can define behaviors with the infinite time length. Given an infinite-observation event segment $\omega \in \underset{t \rightarrow \infty} \lim \Omega_{Z,[0,t]}$ and a timed event system $\mathcal{G}$, let inf(Δ(q₀,ω)) denote the set of $\mathcal{G}$'s states that are infinitely many or long visited by ω and $\mathcal{G}$ . Then infinite length observation language of $\mathcal{G}$ is denoted by $L(\mathcal{G})$, and defined as

$L(\mathcal{G})= \{\omega \in \underset{t \rightarrow \infty} \lim \Omega_{Z,[0,t]}: \exists \text{ the case } inf(\Delta(q_0,\omega)) \subseteq Q_A \}.$

We call an event segment $\omega \in \underset{t \rightarrow \infty} \lim \Omega_{Z,[0,t]}$ an infinite-length behavior of $\mathcal{G}$, if $\omega \in L(\mathcal{G})$.

References

• [Zeigler76] Bernard Zeigler (1976). Theory of Modeling and Simulation (first ed.). Wiley Interscience, New York. 
• [ZKP00] Bernard Zeigler, Tag Gon Kim, Herbert Praehofer (2000). Theory of Modeling and Simulation (second ed.). Academic Press, New York. ISBN 978-0127784557.