Tree Structured QBD Markov Chains and Tree‐Like QBD Processes

Abstract

In this paper, we show that an arbitrary tree structured quasi‐birth–death (QBD) Markov chain can be embedded in a tree‐like QBD process with a special structure. Moreover, we present an algebraic proof that applying the natural fixed point iteration (FPI) to the nonlinear matrix equation V = B + ∑ _s=1 ^d U _s (I ? V)^?1 D _s that solves the tree‐like QBD process, is equivalent to the more complicated iterative algorithm presented by Yeung and Alfa (1996).