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

TL;DR

This paper investigates the dynamics of implicit Max-Min-Plus-Scaling (MMPS) systems, providing conditions for solutions, analyzing global behavior, and introducing stability concepts, with a case study on urban railway networks.

Contribution

It offers the first comprehensive analysis of implicit MMPS system dynamics, including solution existence, growth rates, fixed points, and stability, with practical case validation.

Findings

Sufficient conditions for solution existence in implicit MMPS systems

Characterization of growth rates and fixed points

Introduction of stability concepts for normalized 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.