Computing the Matrix G of Multi-Dimensional Markov Chains of M/G/1 Type

We consider Md-M/G/1 processes, which are irreducible discrete-time Markov chains consisting of two components. The first component is a nonnegative integer vector, while the second component indicates the state (or phase) of the external environment. The level of a state is defined by the minimum value in its first component. The matrix G of the process represents the conditional probabilities that, starting from a given state of a certain level, the Markov chain will first reach a lower level in a specific state. This study aims to develop an effective algorithm for computing matrices G for Md-M/G/1 processes.