Event segment

A segment or trajectory is a relation between an element of an arbitrary set ${\displaystyle Z}$ and a time of time base ${\displaystyle \mathbb {T} }$ [Zeigler76] and [ZPK00]. As timed sequences of events, event segments are a special class of the general segment. Event segments are used to define Timed Event Systems such as DEVS, timed automata, and timed petri nets.

Event Segments

Event and Null Event

An event is a label that abstracts a change. Given an event set ${\displaystyle Z}$ , the null event denoted by ${\displaystyle \epsilon \not \in Z}$  stands for nothing change.

Time Base

The time base of our concerning systems is denoted by ${\displaystyle \mathbb {T} }$ , and defined

${\displaystyle \mathbb {T} =[0,\infty )}$ 

as the set of non-negative real numbers.

Timed Event

A timed event ${\displaystyle (z,t)}$  over an event set ${\displaystyle Z}$  and the time base ${\displaystyle \mathbb {T} }$  denotes that an event ${\displaystyle z\in Z}$  occurs at time ${\displaystyle t\in \mathbb {T} }$ .

Null Event Segment

The null event segment over time interval ${\displaystyle [t_{l},t_{u}]\subset \mathbb {T} }$  is denoted by ${\displaystyle \epsilon _{[t_{l},t_{u}]}}$  which means that there is no event over ${\displaystyle [t_{l},t_{u}]}$ .

Unit Event Segment

An unit event segment is either a null event segment or a timed event.

Concatenation of Event Segments

Given an event set ${\displaystyle Z}$ , concatenation of two unit event segments ${\displaystyle \omega }$  over ${\displaystyle [t_{1},t_{2}]}$  and ${\displaystyle \omega '}$  over ${\displaystyle [t_{3},t_{4}]}$  is denoted by ${\displaystyle \omega \omega '}$  whose time interval is ${\displaystyle [t_{1},t_{4}]}$ , and implies ${\displaystyle t_{2}=t_{3}}$ .

Multi-Event Segment

A multi-event segment ${\displaystyle (z_{1},t_{1})(z_{2},t_{2})\cdots (z_{n},t_{n})}$  over an event set ${\displaystyle Z}$  and a time interval ${\displaystyle [t_{l},t_{u}]\subset \mathbb {T} }$  is concatenations of unit event segments ${\displaystyle \epsilon _{[t_{l},t_{1}]},(z_{1},t_{1}),\epsilon _{[t_{1},t_{2}]},(z_{2},t_{2}),\ldots ,(z_{n},t_{n}),}$  and ${\displaystyle \epsilon _{[t_{n},t_{u}]}}$  where ${\displaystyle t_{l}\leq t_{1}\leq t_{2}\leq \cdots \leq t_{n-1}\leq t_{n}\leq t_{u}}$ .

Timed Language

The universal timed language over an event set ${\displaystyle Z}$  and a time interval ${\displaystyle [t_{l},t_{u}]\subseteq \mathbb {T} }$ , is denoted by ${\displaystyle \Omega _{Z,[t_{l},t_{u}]}}$ , and is defined as the set of all possible event segments. Formally,

${\displaystyle \Omega _{Z,[t_{l},t_{u}]}=\{(z,t)^{*}|z\in Z\cup \{\epsilon \},t\in [t_{l},t_{u}]\}}$ 

where ${\displaystyle ^{*}}$  denotes a none or multiple concatenation(s) of timed events. Notice that the number of events in a string ${\displaystyle \omega \in \Omega _{Z,[t_{l},t_{u}]}}$  can be either of zero, finite or infinite. Infinite many events in a string ${\displaystyle \omega \in \Omega _{Z,[t_{l},t_{u}]}}$  implies that ${\displaystyle t_{u}-t_{l}\rightarrow \infty }$ , however ${\displaystyle t_{u}-t_{l}\rightarrow \infty }$  does not imply infinite many events in it.

A timed language over an event set ${\displaystyle Z}$  and a timed interval ${\displaystyle [t_{l},t_{u}]}$  is a set of event segments over ${\displaystyle Z}$  and ${\displaystyle [t_{l},t_{u}]}$ . If ${\displaystyle L}$  is a language over ${\displaystyle Z}$  and ${\displaystyle [t_{l},t_{u}]}$ , then ${\displaystyle L\subseteq \Omega _{Z,[t_{l},t_{u}]}}$ .

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.