Event segment
Encyclopedia
A segment or trajectory is a relation between an element of an arbitrary set 
and a time of time base 
[Zeigler76] and [ZPK00]. As timed sequences of events, event segments are a special class of the general segment. Event segments are used to define Timed Event Systems
such as DEVS
, timed automata, and timed petri nets.
, the null event denoted by 
stands for nothing change.
, and defined
as the set of non-negative real numbers.
over an event set 
and the time base 
denotes that an event 
occurs at time 
.
is denoted by 
which means that there is no event over 
.
, concatenation of two unit event segments 
over 
and 
over 
is denoted by 
whose time interval is 
, and implies 
.
over an event set 
and a time interval 
is concatenation of unit event segments 
and 
where
.
and a time interval 
, is denoted by
, and is defined as the set of all possible event segments. Formally,
where
denotes a none or multiple concatenation(s) of timed events. Notice that the number of events in an event segment 
can be one of zero, finite or infinite.
Infinitely many events in an event segment
implies that 
, however 
does not imply infinite many events in it.
A timed language over an event set
and a timed interval
is a set of event segments over 
and 
. If 
is a language over 
and 
, then 
.
and a time of time base [Zeigler76] and [ZPK00]. As timed sequences of events, event segments are a special class of the general segment. Event segments are used to define Timed Event SystemsTimed event system
The General System has been described in [Zeigler76] and [ZPK00] with the stand points to define the time base, the admissible input segments, the system states, the state trajectory with an admissible input segment, the output for a given state....
such as DEVS
DEVS
DEVS abbreviating Discrete Event System Specification is a modular and hierarchical formalism for modeling and analyzing general systems that can be discrete event systems which might be described by state transition tables, and continuous state systems which might be described by differential...
, timed automata, and timed petri nets.
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.Time base
The time base of the concerning systems is denoted by, and defined as the set of non-negative real numbers.
Timed event
A timed event over an event set and the time base denotes that an event occurs at time .Null event segment
The null event segment over time interval is denoted by which means that there is no event over .Unit event segment
A unit event segment is either a null event segment or a timed event.Concatenation
Given an event set, concatenation of two unit event segments over and over is denoted by whose time interval is , and implies .Multi-event segment
A multi-event segment over an event set and a time interval is concatenation of unit event segments and where.Timed language
The universal timed language over an event set and a time interval , is denoted by, and is defined as the set of all possible event segments. Formally, where
denotes a none or multiple concatenation(s) of timed events. Notice that the number of events in an event segment can be one of zero, finite or infinite.Infinitely many events in an event segment
implies that , however does not imply infinite many events in it.A timed language over an event set
and a timed interval is a set of event segments over and . If is a language over and , then .