# Event segment

(Redirected from 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 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 ${\displaystyle Z}$ [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.

## Event segments

### 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.

### 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.

### Timed event

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

### Null segment

The null segment over time interval ${\displaystyle [t_{l},t_{u}]\subset \mathbb {T} }$ is denoted by ${\displaystyle \epsilon _{[t_{l},t_{u}]}}$ which means nothing in ${\displaystyle Z}$ occurs 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 ${\displaystyle t_{2}=t_{3}}$.

### Event trajectory

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

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

${\displaystyle \omega :[t_{l},t_{u}]\rightarrow Z^{*}.}$

## Timed language

The universal timed language ${\displaystyle \Omega _{Z,[t_{l},t_{u}]}}$ over an event set ${\displaystyle Z}$ and a time interval ${\displaystyle [t_{l},t_{u}]\subset \mathbb {T} }$, is the set of all event trajectories over ${\displaystyle Z}$ and ${\displaystyle [t_{l},t_{u}]}$.

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

## 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.
• [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