Fixed-delay Events in Generalized Semi-Markov Processes Revisited

TL;DR

This paper investigates the long-term behavior of generalized semi-Markov processes with fixed and variable delays, revealing conditions for stability and providing algorithms for frequency approximation.

Contribution

It demonstrates that multiple fixed-delay events can cause instability in GSMPs and proves that at most one fixed-delay event ensures the existence of an invariant measure.

Findings

Multiple fixed-delay events can lead to unstable GSMP behavior.

GSMPs with at most one fixed-delay event always have an invariant measure.

Algorithms are provided for approximating state frequencies in stable GSMPs.

Abstract

We study long run average behavior of generalized semi-Markov processes with both fixed-delay events as well as variable-delay events. We show that allowing two fixed-delay events and one variable-delay event may cause an unstable behavior of a GSMP. In particular, we show that a frequency of a given state may not be defined for almost all runs (or more generally, an invariant measure may not exist). We use this observation to disprove several results from literature. Next we study GSMP with at most one fixed-delay event combined with an arbitrary number of variable-delay events. We prove that such a GSMP always possesses an invariant measure which means that the frequencies of states are always well defined and we provide algorithms for approximation of these frequencies. Additionally, we show that the positive results remain valid even if we allow an arbitrary number of reasonably restricted fixed-delay events.