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

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.

💡 Research Summary

The paper investigates the dynamics of autonomous, time‑invariant implicit Max‑Min‑Plus‑Scaling (MMPS) systems, a recent generalization of max‑plus, min‑plus, and max‑min‑plus models that also incorporates scaling operations. While explicit MMPS models can be rewritten using a Kleene‑star construction, implicit models—where the current state depends on itself within the same event cycle—cannot be easily transformed into an explicit form, making their analysis challenging. The authors address this gap by developing a comprehensive theoretical framework that covers existence of solutions, computation of growth rates and fixed points, and stability analysis without converting the system to an explicit representation.

Key contributions

1. Canonical representation – The authors extend the ABC canonical form for explicit MMPS systems to an ABCD form that captures implicit dependencies:
   \