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

Valeriy Naumov () and Konstantin Samouylov
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Valeriy Naumov: Service Innovation Research Institute, Annankatu 8 A, 00120 Helsinki, Finland
Konstantin Samouylov: Institute of Computer Science and Telecommunications, RUDN University, 6 Miklukho-Maklaya St., Moscow 117198, Russia

Mathematics, 2025, vol. 13, issue 2, 1-14

Abstract: 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.

Keywords: discrete-time Markov chain; Markov chain of M/G/1 type; matrix G; system of nonlinear matrix equations (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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