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

Author

Listed:
  • 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)

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.

Suggested Citation

  • Valeriy Naumov & Konstantin Samouylov, 2025. "Multi-Dimensional Markov Chains of M/G/1 Type," Mathematics, MDPI, vol. 13(2), pages 1-14, January.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:2:p:209-:d:1563710
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    References listed on IDEAS

    as
    1. Toshihisa Ozawa & Masahiro Kobayashi, 2018. "Exact asymptotic formulae of the stationary distribution of a discrete-time two-dimensional QBD process," Queueing Systems: Theory and Applications, Springer, vol. 90(3), pages 351-403, December.
    2. Toshihisa Ozawa, 2022. "Tail asymptotics in any direction of the stationary distribution in a two-dimensional discrete-time QBD process," Queueing Systems: Theory and Applications, Springer, vol. 102(1), pages 227-267, October.
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