Event segment

A segment of a system variable shows a homogenous status of system dynamics over a time period. Here, a homogenous status of a variable is a state which can be described by one coefficient of a simple formula. For example of homogenous statuses, we can bring status of constant ('ON' of a switch) and linear (60 miles or 96km per hour for speed). Mathematically, a segment is a function mapping from a set of times which can be defined by an real interval, to the set ${\displaystyle Z}$ [Zeigler76] and [ZPK00]. A trajectory of a system variable is a concatenation of segments. If its concatenating segments are constant (respectively linear), we call it a trajectory of constant-segments (respectively linear-segments). An event segment is a special class of the constant segment that is a timed event or a null-segment. The 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 the 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

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

Concatenation

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 concatenation 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 ${\displaystyle [t_{l},t_{u}]\subset \mathbb {T} }$ , 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 an event segment ${\displaystyle \omega \in \Omega _{Z,[t_{l},t_{u}]}}$  can be one of zero, finite or infinite. Infinitely many events in an event segment ${\displaystyle \omega \in \Omega _{Z,[t_{l},t_{u}]}}$  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 978-0-12-778455-7.
• [Hwang13] M.H. Hwang, ``Revisit of system variable trajectories``, Proceedings of the Symposium on Theory of Modeling & Simulation - DEVS Integrative M&S Symposium , San Diego, CA, USA, April 7 - 10, 2013