A Fixed Point Approach to the Classification of Markov Chains with a Tree Structure

In this paper, we study the classification problem of discrete time and continuous time Markov processes with a tree structure. We first show some useful properties associated with the fixed points of a nondecreasing mapping. Mainly we find the conditions for a fixed point to be the minimal fixed point by using fixed point theory and degree theory. We then use these results to identify conditions for Markov chains of M/G/1 type or GI/M/1 type with a tree structure to be positive recurrent, null recurrent, or transient. The results are generalized to Markov chains of matrix M/G/1 type with a tree structure. For all these cases, a relationship between a certain fixed point, the matrix of partial differentiation (Jacobian) associated with the fixed point, and the classification of the Markov chain with a tree structure is established. More specifically, we show that the Perron-Frobenius eigenvalue of the matrix of partial differentiation associated with a certain fixed point provides information for a complete classification of the Markov chains of interest.