Event segment

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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 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[edit]

Time base[edit]

The time base of the concerning systems is denoted by  \mathbb{T} , and defined


as the set of non-negative real numbers.

Event and null event[edit]

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[edit]

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[edit]

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[edit]

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


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[edit]

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[edit]

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]}.


  • [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