# 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 an 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 [Hwang2012]. 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

${\displaystyle {\mathcal {G}}=}$

where

• ${\displaystyle \,Z}$ is the set of events;
• ${\displaystyle \,Q}$ is the set of states;
• ${\displaystyle \,Q_{0}\subseteq Q}$ is the set of initial states;
• ${\displaystyle Q_{A}\subseteq Q}$ is the set of accepting states;
• ${\displaystyle \Delta \subseteq Q\times \Omega _{Z,[t_{l},t_{u}]}\times Q}$ is the set of state trajectories in which ${\displaystyle (q,\omega ,q')\in \Delta }$ indicates that a state ${\displaystyle q\in Q}$ can change into ${\displaystyle q'\in Q}$ along with an event segment ${\displaystyle \omega \in \Omega _{Z,[t_{l},t_{u}]}}$. If two state trajectories ${\displaystyle (q_{1},\omega _{1},q_{2})}$ and ${\displaystyle (q_{3},\omega _{2},q_{4})\in \Delta }$ are called contiguous if ${\displaystyle q_{2}=q_{3}}$, and two event trajectories ${\displaystyle \omega _{1}}$ and ${\displaystyle \omega _{2}}$ are contiguous. Two contiguous state trajectories ${\displaystyle (q,\omega _{1},p)}$ and ${\displaystyle (p,\omega _{2},q')\in \Delta }$ implies ${\displaystyle (q,\omega _{1}\omega _{2},q')\in \Delta }$.

## Behaviors and Languages of Timed Event System

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

${\displaystyle L({\mathcal {G}},t)=\{\omega \in \Omega _{Z,[0,t]}:\exists (q_{0},\omega ,q)\in \Delta ,q_{0}\in Q_{0},q\in Q_{A}\}.}$

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

By sending the observation time length ${\displaystyle t}$ to infinity, we define infinite length observation language of ${\displaystyle {\mathcal {G}}}$ is denoted by ${\displaystyle L({\mathcal {G}},\infty )}$, and defined as

${\displaystyle L({\mathcal {G}},\infty )=\{\omega \in {\underset {t\rightarrow \infty }{\lim }}\Omega _{Z,[0,t]}:\exists \{q:(q_{0},\omega ,q)\in \Delta ,q_{0}\in Q_{0}\}\subseteq Q_{A}\}.}$

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