Geometric and Subexponential Asymptotics of Markov Chains of M/G/1 Type

Abstract

This paper considers the steady-state solution of Markov chains of M/G/1 type. We first derive the matrix product-form solution of the steady-state probability. This formula is considered as a natural generalization of the matrix-geometric solution of quasi birth-and-death processes to Markov chains of M/G/1 type. Based on this formula, we study the asymptotics of the tail distribution. For the light-tailed case, we show a sufficient condition for the geometric asymptotics of the tail distribution, employing the Markov key renewal theorem. Contrary to the previous works, some periodic characteristics of transitions in upper levels are explicitly taken into account and a new geometric asymptotic formula is established. Furthermore, for the heavy-tailed case, we show a subexponential asymptotics formula for the tail distribution under a mild condition.