Queueing Systems

, Volume 90, Issue 3–4, pp 351–403 | Cite as

Exact asymptotic formulae of the stationary distribution of a discrete-time two-dimensional QBD process

  • Toshihisa OzawaEmail author
  • Masahiro Kobayashi


We consider a discrete-time two-dimensional process \(\{(X_{1,n},X_{2,n})\}\) on \(\mathbb {Z}_+^2\) with a supplemental process \(\{J_n\}\) on a finite set, where the individual processes \(\{X_{1,n}\}\) and \(\{X_{2,n}\}\) are both skip-free. We assume that the joint process \(\{\varvec{Y}_n\}=\{(X_{1,n},X_{2,n},J_n)\}\) is Markovian and that the transition probabilities of the two-dimensional process \(\{(X_{1,n},X_{2,n})\}\) are modulated depending on the state of the supplemental process \(\{J_n\}\). This modulation is space homogeneous except for the boundaries of \(\mathbb {Z}_+^2\). We call this process a discrete-time two-dimensional quasi-birth-and-death process. Under several conditions, we obtain the exact asymptotic formulae of the stationary distribution in the coordinate directions.


Quasi-birth-and-death process Stationary distribution Asymptotic property Matrix analytic method Two-dimensional reflecting random walk 

Mathematics Subject Classification

60J10 60K25 



We are grateful to Professor Masakiyo Miyazawa for valuable discussions with him about the convergence domain of the generating function \(\varvec{\varphi }(z,w)\). Also, the authors would like to thank the referees for their valuable comments and suggestions. This work was supported by JSPS KAKENHI Grant Number JP17K18126.

Supplementary material

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Supplementary material 1 (pdf 633 KB)


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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of Business AdministrationKomazawa UniversityTokyoJapan
  2. 2.Department of Mathematical ScienceTokai UniversityKanagawaJapan

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