Timing Characteristics of Hierarchical Discrete Event Systems

A discrete event system (DES) is a dynamical system whose evolution in time develops as the result of the occurrence of physical events at possibly irregular time intervals. Although many DES´s operation is asynchronous, others have dynamics which depend on a clock or some other complex timing schedule. Here we provide a formal representation of the advancement of time for logical DES via interpretations of time. We show that the interpretations of time along with a timing structure provide a framework to study principles of the advancement of time for hierarchical DES (HDES). In particular, it is shown that for a wide class of HDES the event rate is higher for DES at the lower levels of the hierarchy than at the higher levels of the hierarchy. Relationships between event rate and event aggregation are shown. We define a measure for event aggregation and show that there exists an inverse relationship between the amount of event aggregation and the event rate at any two successive levels in a class of HDES. Next, we study how to design the timing structure to ensure that there will be a decrease in the event rate (by some constant factor) between any two levels of a wide class of HDES. It is shown that if the communications between the various DES in the HDES satisfy a certain admissibility condition then there will be a decrease in the event rate.