Exercise 5.11
Prof. Dr.-Ing. Jo ̈rg Raisch
Germano Schafaschek
Soraia Moradi
Behrang Nejad
Fachgebiet Regelungssysteme
Fakulta ̈t IV Elektrotechnik und Informatik Technische Universita ̈t Berlin Lehrveranstaltung ”Ereignisdiskrete Systeme“ Wintersemester 2020/2021
Exercise sheet 5
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Consider a simple process consisting of three computer operations. The first operation is divided into two sequential tasks, T1 and T2, and is performed in parallel with the second operation, consisting only of task T3. The outputs of T2 and T3 must then be combined as an input for a subsequent operation consisting of a single task, T4. The duration of each task is given by a constant wi, i = 1, 2, 3, 4. As soon as the last operation is completed, it is possible (although not mandatory) for the first two to immediately start again, which they can do independently of each other. Initially, no task is being performed.
a) Model the system as a timed event graph with holding times.
b) Determine recursive equations for the firing times of the transitions.
Exercise 5.22
Consider a circular railway track with three stations, S1, S2, and S3. Two trains run along this track in the same direction, going from S1 to S2, from S2 to S3, from S3 to S1, and so on. The time a train needs to travel between two consecutive stations Si and Sj is denoted by δij . Station S1 is also visited by trains that travel on a separate, external track; once such an external train leaves S1, a new one arrives after δe time units.
In order to facilitate connections between the lines, trains from the two tracks (circular and external) must wait for each other at S1 and must remain at the station together for at least δc time units. Assume that initially in the circular track there is one train stopped at S1 and one at S3, and that an external train is also stopped at S1.
a) Model the system as a timed event graph with holding times. Represent explicitly the arrival and departure of the trains at each station.
b) Derive a set of recursive equations describing the firing times of the transitions.
c) Suppose that, as a safety measure, a train in the circular track is not allowed to leave its current station before the platform at the next station is free. Modify the model from item a) so as to incorporate this restriction.
1Inspired by an example from the book “Introduction to Discrete Event Systems”, Cassandras and Lafortune. 2Inspired by an example from the book “Max Plus at Work”, Heidergott, Olsder, and van der Woude.
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