An Overview of Transience Bounds in Max-Plus Algebra

TL;DR

This paper surveys upper bounds on the transient phase length in max-plus linear systems, introduces an extension of Nachtigall's result for proving such bounds, and discusses asymptotic tightness with examples.

Contribution

It presents a new approach for establishing transience bounds in max-plus algebra by extending Nachtigall's result and demonstrates asymptotic tightness with specific examples.

Findings

Extended Nachtigall's result for transience bounds

Provided an asymptotic tightness example

Discussed implications for max-plus linear systems

Abstract

We survey and discuss upper bounds on the length of the transient phase of max-plus linear systems and sequences of max-plus matrix powers. In particular, we explain how to extend a result by Nachtigall to yield a new approach for proving such bounds and we state an asymptotic tightness result by using an example given by Hartmann and Arguelles.