Classification of Markov processes of M/G/1 type with a tree structure and its applications to queueing models

This paper studies the classification problem of Markov processes of M/G/1 type with a tree structure. It is shown that the classification of positive recurrence, null recurrence, and transcience of the Markov processes of interest is determined completely by the Perron–Frobenius eigenvalue of a nonnegative matrix. The results are used to find classification criteria for a number of discrete time or continuous time queueing systems with multiple types of customers.