Quasi-stationary distributions for queueing and other models

TL;DR

This paper discusses conjectures on the convergence behaviors of transition probabilities in certain Markov chains, with implications for understanding their limiting distributions, especially in queueing and related models.

Contribution

It introduces conjectures on polynomial, algebraic, and sub-exponential convergence rates for specific Markov chains and explores their implications for limiting conditional distributions.

Findings

Conjectures on convergence rates for Markov chains.

Implications for existence of limiting conditional distributions.

Examples mainly in queueing and branching models.

Abstract

This note presents conjectures on polynomial/algebraic/sub-exponential convergence of transition probabilities for $\lambda$-null recurrent and $\lambda$-transient Markov chains in continuous time. The only known positive examples are in queueing, branching, and related models. Implications for the existence of limiting conditional distributions for $\lambda$-transient chains are discussed.