Bounding the Equilibrium Distribution of Markov Population Models

TL;DR

This paper introduces a novel method for estimating bounds on the equilibrium distribution of infinite-state Markov models, aiding analysis of complex biological systems where traditional methods fail.

Contribution

It proposes a new approach combining geometric bounds with detailed state-wise analysis to approximate steady state distributions in infinite Markov models.

Findings

Provides a way to bound the steady state distribution in infinite Markov models

Enables analysis of biological systems with complex state spaces

Offers a framework for state-wise probability bounds

Abstract

Arguing about the equilibrium distribution of continuous-time Markov chains can be vital for showing properties about the underlying systems. For example in biological systems, bistability of a chemical reaction network can hint at its function as a biological switch. Unfortunately, the state space of these systems is infinite in most cases, preventing the use of traditional steady state solution techniques. In this paper we develop a new approach to tackle this problem by first retrieving geometric bounds enclosing a major part of the steady state probability mass, followed by a more detailed analysis revealing state-wise bounds.