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

Abstract

Max-min-plus-scaling (MMPS) systems extend max-plus and max-min-plus models by incorporating a scaling operation, making them suitable for modeling discrete-event systems involving timing, synchronization, and resource constraints. This paper focuses on implicit MMPS systems, where the current state depends on itself, which complicates analysis because solutions may not exist or may be non-unique. We investigate conditions under which such systems are solvable. We prove that the implicit MMPS system is solvable if the corresponding implicit interconnection graph does not contain any circuits. When there exist a circuit, we develop necessary and sufficient conditions for the system to be solvable. The paper also discusses solvable regions and proposes an iterative method for computing system states. We demonstrate all the theoretical results with a numerical example.