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 $Z$ [Zeigler76] and [ZPK00]. A trajectory of a system variable is a concatenation of segments. We call it a trajectory of constant-segments (respectively linear-segments) if its concatenating segments are constant (respectively linear).

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 $Z$, the null event denoted by $\epsilon \not \in Z$ stands for nothing change.

Time base

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

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

as the set of non-negative real numbers.

Timed event

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

Null event segment

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

Unit event segment

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

Concatenation

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

Multi-event segment

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

Timed language

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

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

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

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

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.{{cite book}}: CS1 maint: multiple names: authors list (link)
• [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