timed event system

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The General System has been described by Zeigler{{cite book |last=Zeigler |first=Bernard |title=Theory of Modeling and Simulation |year=1976 |publisher=Wiley Interscience |location=New York |edition=first}}{{cite book |last1=Zeigler |first1=Bernard |last2=Kim |first2=Tag Gon |last3=Praehofer |first3=Herbert |title=Theory of Modeling and Simulation |year=2000 |publisher=Academic Press |location=New York |isbn=978-0-12-778455-7 |edition=second |author1-link=Bernard P. Zeigler}} with the standpoints 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 came from the class of General System to allows non-deterministic behaviors in it.{{cite conference |last=Hwang |first=Moon H. |title=Qualitative Verification of Finite and Real-Time DEVS Networks |conference=Proceedings of 2012 TMS/DEVS |location=Orlando, FL, USA |year=2012 |pages=43:1–43:8 |isbn=978-1-61839-786-7}} Since the behaviors of DEVS can be described by Timed Event System, DEVS and RTDEVS is a sub-class or an equivalent class of Timed Event System.

Timed Event Systems

A timed event system is a structure

\mathcal{G}=

where

$\,Z$ is the set of events;
$\,Q$ is the set of states;
$\,Q_0 \subseteq Q$ is the set of initial states;
$Q_A \subseteq Q$ is the set of accepting states;
$\Delta \subseteq Q \times \Omega_{Z,[t_l,t_u]} \times Q$ is the set of state trajectories in which $(q,\omega,q') \in \Delta$ indicates that a state $q \in Q$ can change into $q' \in Q$ along with an event segment $\omega \in \Omega_{Z,[t_l, t_u]}$ . If two state trajectories $(q_1,\omega_1,q_2)$ and $(q_3, \omega_2, q_4) \in \Delta$ are called contiguous if $q_2 = q_3$ , and two event trajectories $\omega_1$ and $\omega_2$ are contiguous. Two contiguous state trajectories $(q,\omega_1,p)$ and $(p,\omega_2, q') \in \Delta$ implies $(q,\omega_1\omega_2,q') \in \Delta$ .

Behaviors and Languages of Timed Event System

Given a timed event system $\mathcal{G}=$ , 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 (q_0, \omega, q) \in
\Delta, q_0 \in Q_0, q \in Q_A\}.

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

By sending the observation time length $t$ to infinity, we define infinite length observation language of $\mathcal{G}$

is denoted by $L(\mathcal{G}, \infty)$ , and defined as

L(\mathcal{G},\infty)= \{\omega \in \underset{t \rightarrow \infty} \lim
\Omega_{Z,[0,t]}: \exists \{q: (q_0, \omega, q) \in
\Delta, q_0 \in Q_0 \} \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 $\omega \in L(\mathcal{G},\infty)$ .

References

Category:Automata (computation)

Category:Formal specification languages

timed event system

Timed Event Systems

Behaviors and Languages of Timed Event System

See also

References