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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 [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.
The time base of the concerning systems is denoted by , and defined
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 , the null event denoted by stands for nothing change.
A timed event is a pair where and denotes that an event occurs at time .
The null segment over time interval is denoted by which means nothing in occurs over .
Unit event segment
Given an event set , concatenation of two unit event segments over and over is denoted by whose time interval is , and implies .
An event trajectory over an event set and a time interval is concatenation of unit event segments and where .
Mathematically, an event trajectory is a mapping a time period to an event set . So we can write it in a function form :
The universal timed language over an event set and a time interval , is the set of all event trajectories over and .
A timed language over an event set and a timed interval is a set of event trajectories over and if .
- [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