Large Deviations Analysis for Distributed Algorithms in an Ergodic Markovian Environment

TL;DR

This paper analyzes the probability and timing of deadlocks in distributed resource-sharing systems modeled by ergodic Markov processes, using large deviations theory to understand rare events and system stability.

Contribution

It introduces a large deviations framework for deadlock analysis in Markovian environments with boundary reflections, providing new estimates and solving a Hamilton-Jacobi equation.

Findings

Freidlin-Wentzell estimates for deadlock times

Quasi-potential as viscosity solution of Hamilton-Jacobi equation

Identification of stable attractors and limit cycles

Abstract

We provide a large deviations analysis of deadlock phenomena occurring in distributed systems sharing common resources. In our model transition probabilities of resource allocation and deallocation are time and space dependent. The process is driven by an ergodic Markov chain and is reflected on the boundary of the d-dimensional cube. In the large resource limit, we prove Freidlin-Wentzell estimates, we study the asymptotic of the deadlock time and we show that the quasi-potential is a viscosity solution of a Hamilton-Jacobi equation with a Neumann boundary condition. We give a complete analysis of the colliding 2-stacks problem and show an example where the system has a stable attractor which is a limit cycle.