Event segment

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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 $Z$ [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

{{center|1= $\mathbb{T}=[0,\infty)$ }}

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 is 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 :

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] {{cite book|author = Bernard Zeigler | year = 1976| title = Theory of Modeling and Simulation| publisher = Wiley Interscience, New York |edition=first}}
[ZKP00] {{cite book|author1=Bernard Zeigler |author2=Tag Gon Kim |author3=Herbert Praehofer | year = 2000| title = Theory of Modeling and Simulation| publisher = Academic Press, New York | isbn= 978-0-12-778455-7 |edition=second}}
[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

Category:Automata (computation)

Category:Formal specification languages