A Matrix-Multiplicative Solution for Multi-Dimensional QBD Processes

Valeriy Naumov ()
Additional contact information
Valeriy Naumov: Service Innovation Research Institute, Annankatu 8 A, 00120 Helsinki, Finland

Mathematics, 2024, vol. 12, issue 3, 1-15

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.

Keywords: discrete-time Markov chain; multi-dimensional QBD; matrix-geometric solution; matrix-multiplicative solution (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/12/3/444/pdf (application/pdf)
https://www.mdpi.com/2227-7390/12/3/444/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:12:y:2024:i:3:p:444-:d:1329781

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().