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A Matrix-Multiplicative Solution for Multi-Dimensional QBD Processes

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  • Valeriy Naumov

    (Service Innovation Research Institute, Annankatu 8 A, 00120 Helsinki, Finland)

Abstract

We consider an irreducible positive-recurrent discrete-time Markov process on the state space X = ℤ + M × J , where ℤ + is the set of non-negative integers and J = { 1 , 2 , … , n } . The number of states in J may be either finite or infinite. We assume that the process is a homogeneous quasi-birth-and-death process (QBD). It means that the one-step transition probability between non-boundary states ( k , i ) and ( n , j ) may depend on i , j , and n − k but not on the specific values of k and n . It is shown that the stationary probability vector of the process is expressed through square matrices of order n , which are the minimal non-negative solutions to nonlinear matrix equations.

Suggested Citation

  • Valeriy Naumov, 2024. "A Matrix-Multiplicative Solution for Multi-Dimensional QBD Processes," Mathematics, MDPI, vol. 12(3), pages 1-15, January.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:3:p:444-:d:1329781
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    References listed on IDEAS

    as
    1. Blanc, J.P.C., 1990. "Performance evaluation of polling systems by means of the power-series algorithm," Other publications TiSEM a5f5fb56-c17c-4c46-8d5e-b, Tilburg University, School of Economics and Management.
    2. 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.
    3. Frédéric Lavancier & Ronan Le Guével, 2021. "Spatial birth–death–move processes: Basic properties and estimation of their intensity functions," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 83(4), pages 798-825, September.
    4. 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.
    Full references (including those not matched with items on IDEAS)

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