Timed event system
The General System has been described in [Zeigler76] and [ZPK00] 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 [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
where
- is the set of events;
- is the set of states;
- is the set of initial states;
- is the set of accepting states;
- is the set of state trajectories in which indicates that a state can change into along with an event segment . If two state trajectories and are called contiguous if , and two event trajectories and are contiguous. Two contiguous state trajectories and implies .
![{\displaystyle \Delta \subseteq Q\times \Omega _{Z,[t_{l},t_{u}]}\times Q}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd4aa1a8249e4c2515cbf417a658ae967a8c0215)
![{\displaystyle \omega \in \Omega _{Z,[t_{l},t_{u}]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b7cfe075628a7736029e86ae72332f6f2c53bfb9)
Behaviors and Languages of Timed Event System
Given a timed event system , the set of its behaviors is called its language depending on the observation time length. Let be the observation time length. If , -length observation language of is denoted by , 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}\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/208d64904cee9a5882695934e19baef37159d226)
We call an event segment a -length behavior of , if .
![{\displaystyle \omega \in \Omega _{Z,[0,t]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bfb0d5bc3fdd4876f2b1031c6b0eec316c497056)
By sending the observation time length to infinity, we define infinite length observation language of is denoted by , 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}\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1074fd4431b43b11336aeefc545b6e32ef8c9ad6)
We call an event segment an infinite-length behavior of , if .
![{\displaystyle \omega \in {\underset {t\rightarrow \infty }{\lim }}\Omega _{Z,[0,t]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3d49a62bff81d741b329204e536d50753433377)
See also
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 978-0-12-778455-7.
- [Hwang2012] Moon H. Hwang. "Qualitative Verification of Finite and Real-Time DEVS Networks". Proceedings of 2012 TMS/DEVS. Orlando, FL, USA. pp. 43:1–43:8. ISBN 978-1-61839-786-7.