Multi-Dimensional Markov Chains of M/G/1 Type

We consider an irreducible discrete-time Markov process with states represented as ( k , i ) where k is an M -dimensional vector with non-negative integer entries, and i indicates the state (phase) of the external environment. The number n of phases may be either finite or infinite. One-step transitions of the process from a state ( k , i ) are limited to states ( n , j ) such that n ≥ k − 1 , where 1 represents the vector of all 1s. We assume that for a vector k ≥ 1 , the one-step transition probability from a state ( k , i ) to a state ( n , j ) may depend on i, j , and n − k , but not on the specific values of k and n . This process can be classified as a Markov chain of M/G/1 type, where the minimum entry of the vector n defines the level of a state ( n , j ). It is shown that the first passage distribution matrix of such a process, also known as the matrix G , can be expressed through a family of nonnegative square matrices of order n , which is a solution to a system of nonlinear matrix equations.