Dynamics of Implicit Time-Invariant Max-Min-Plus-Scaling Discrete-Event Systems

Abstract

Max-min-plus-scaling (MMPS) systems generalize max-plus, min-plus and max-min-plus models with more flexibility in modelling discrete-event dynamics. Especially, implicit MMPS models capture a wide range of real world discrete-event applications. This article analyzes the dynamics of an autonomous, time-invariant implicit MMPS system in a discrete-event framework. First, we provide sufficient conditions under which an implicit MMPS system admits at least one solution to its state-space representation. Then, we analyze its global behavior by determining the key parameters; the growth rates and fixed points. For a solvable MMPS system, we assess the local behavior of the system around its set of fixed points via a normalization procedure. Further, we present the notion of stability for the normalized system. A case study of the urban railway network substantiates the theoretical results.