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 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],[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 $\mathbb{T}$ , and defined

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

Timed event

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

Null segment

The null segment over time interval $[t_l, t_u] \subset \mathbb{T}$ is denoted by $\epsilon_{[t_l, t_u]}$ which means nothing in $Z$ occurs 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 $t_2 = t_3$ .

Event trajectory

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

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

\omega:[t_l,t_u] \rightarrow Z^* .

Timed language

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

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

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