Subexponential Asymptotics of the Stationary Distributions of GI/G/1-Type Markov Chains

TL;DR

This paper investigates the subexponential asymptotics of stationary distributions in GI/G/1-type Markov chains, providing new conditions and exploring cases with stochastic and substochastic phase transition matrices.

Contribution

It introduces a weaker sufficient condition for subexponential asymptotics in the stochastic case and studies the less-explored substochastic case, highlighting differences between the two.

Findings

Subexponential asymptotics differ between stochastic and substochastic cases.

A weaker condition for subexponential asymptotics is established in the stochastic case.

Locally subexponential asymptotics are analyzed for both cases.

Abstract

This paper considers the subexponential asymptotics of the stationary distributions of GI/G/1-type Markov chains in two cases: (i) the phase transition matrix in non-boundary levels is stochastic; and (ii) it is strictly substochastic. For the case (i), we present a weaker sufficient condition for the sub exponential asymptotics than those given in the literature. As for the case (ii), the subexponential asymptotics has not been studied, as far as we know. We show that the subexponential asymptotics in the case (ii) is different from that in the case (i). We also study the locally subexponential asymptotics of the stationary distributions in both cases (i) and (ii).