Ergodic properties of concurrent systems

TL;DR

This paper investigates the ergodic properties of concurrent systems modeled as monoid actions, establishing a unique Markov measure, analyzing the M"obius matrix, and introducing a speedup metric to quantify concurrency.

Contribution

It introduces a spectral approach to analyze ergodic properties of concurrent systems, proving the existence of a unique Markov measure and defining a new concurrency metric.

Findings

Existence and uniqueness of a Markov measure on infinite trajectories.

The kernel of the M"obius matrix has dimension 1.

Introduction of the speedup metric for concurrency measurement.

Abstract

A concurrent system is defined as a monoid action of a trace monoid on a finite set of states. Concurrent systems represent state models where the state is distributed and where state changes are local. Starting from a spectral property on the combinatorics of concurrent systems, we prove the existence and uniqueness of a Markov measure on the space of infinite trajectories relatively to any weight distributions. In turn, we obtain a combinatorial result by proving that the kernel of the associated M\"obius matrix has dimension 1; the M\"obius matrix extends in this context the M\"obius polynomial of a trace monoid. We study ergodic properties of irreducible concurrent systems and we prove a Strong law of large numbers. It allows us to introduce the speedup as a measurement of the average amount of concurrency within infinite trajectories. Examples are studied.