Event segment

A segment of a system variable in computing 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 a set of coefficients of a formula. For example, of homogenous statuses, we can bring status of constant ('ON' of a switch) and linear (60 miles or 96 km per hour for speed). Mathematically, a segment is a function mapping from a set of times which can be defined by a real interval, to the set [math]\displaystyle{ Z }[/math] [Zeigler76],[ZPK00], [Hwang13]. A trajectory of a system variable is a sequence of segments concatenated. We call a trajectory constant (respectively linear) if its concatenating segments are constant (respectively linear). An event segment is a special class of the constant segment with a constraint in which the constant segment is either one of 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.

## Contents

* 1 Event segments
* 1.1 Time base
* 1.2 Event and null event
* 1.3 Timed event
* 1.4 Null segment
* 1.5 Unit event segment
* 1.6 Concatenation
* 1.7 Event trajectory
* 2 Timed language
* 3 See also
* 4 References

## Event segments

### Time base

The time base of the concerning systems is denoted by [math]\displaystyle{ \mathbb{T} }[/math], and defined

[math]\displaystyle{ \mathbb{T}=[0,\infty) }[/math]

as the set of non-negative real numbers.

### Event and null event

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

### Timed event

A timed event is a pair [math]\displaystyle{ (t,z) }[/math] where [math]\displaystyle{ t \in \mathbb{T} }[/math] and [math]\displaystyle{ z \in Z }[/math] denotes that an event [math]\displaystyle{ z \in Z }[/math] occurs at time [math]\displaystyle{ t \in \mathbb{T} }[/math].

### Null segment

The null segment over time interval [math]\displaystyle{ [t_l, t_u] \subset \mathbb{T} }[/math] is denoted by [math]\displaystyle{ \epsilon_{[t_l, t_u]} }[/math] which means nothing in [math]\displaystyle{ Z }[/math] occurs over [math]\displaystyle{ [t_l, t_u] }[/math].

### Unit event segment

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

### Concatenation

Given an event set [math]\displaystyle{ Z }[/math], concatenation of two unit event segments [math]\displaystyle{ \omega }[/math] over [math]\displaystyle{ [t_1, t_2] }[/math] and [math]\displaystyle{ \omega' }[/math] over [math]\displaystyle{ [t_3, t_4] }[/math] is denoted by [math]\displaystyle{ \omega\omega' }[/math] whose time interval is [math]\displaystyle{ [t_1, t_4] }[/math], and implies [math]\displaystyle{ t_2 = t_3 }[/math].

### Event trajectory

An event trajectory [math]\displaystyle{ (t_1,z_1)(t_2,z_2) \cdots (t_n,z_n) }[/math] over an event set [math]\displaystyle{ Z }[/math] and a time interval [math]\displaystyle{ [t_l, t_u] \subset \mathbb{T} }[/math] is concatenation of unit event segments [math]\displaystyle{ \epsilon_{[t_l,t_1]},(t_1,z_1), \epsilon_{[t_1,t_2]},(t_2,z_2),\ldots, (t_n,z_n), }[/math] and [math]\displaystyle{ \epsilon_{[t_n,t_u]} }[/math] where [math]\displaystyle{ t_l\le t_1 \le t_2 \le \cdots \le t_{n-1} \le t_n \le t_u }[/math].

Mathematically, an event trajectory is a mapping [math]\displaystyle{ \omega }[/math] a time period [math]\displaystyle{ [t_l,t_u] \subseteq \mathbb{T} }[/math] to an event set [math]\displaystyle{ Z }[/math]. So we can write it in a function form :

[math]\displaystyle{ \omega:[t_l,t_u] \rightarrow Z^* . }[/math]

## Timed language

The universal timed language [math]\displaystyle{ \Omega_{Z,[t_l, t_u]} }[/math] over an event set [math]\displaystyle{ Z }[/math] and a time interval [math]\displaystyle{ [t_l, t_u] \subset \mathbb{T} }[/math], is the set of all event trajectories over [math]\displaystyle{ Z }[/math] and [math]\displaystyle{ [t_l,t_u] }[/math].

A timed language [math]\displaystyle{ L }[/math] over an event set [math]\displaystyle{ Z }[/math] and a timed interval [math]\displaystyle{ [t_l, t_u] }[/math] is a set of event trajectories over [math]\displaystyle{ Z }[/math] and [math]\displaystyle{ [t_l, t_u] }[/math] if [math]\displaystyle{ L \subseteq \Omega_{Z, [t_l, t_u]} }[/math].

## See also

* Outline of computing

## 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.
* [Giambiasi01] Giambiasi N., Escude B. Ghosh S. “Generalized Discrete Event Simulation of Dynamic Systems”, in: Issue 4 of SCS Transactions: Recent Advances in DEVS Methodology-part II, Vol. 18, pp. 216–229, dec 2001
* [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

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