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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 Ozawa
  • Masahiro Kobayashi
Article
  • 65 Downloads

Abstract

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.

Keywords

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

Mathematics Subject Classification

60J10 60K25 

Notes

Acknowledgements

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)

References

  1. 1.
    Fayolle, G., Malyshev, V.A., Menshikov, M.V.: Topics in the Constructive Theory of Countable Markov Chains. Cambridge University Press, Cambridge (1995)CrossRefGoogle Scholar
  2. 2.
    Flajolet, P., Sedgewick, R.: Analytic Combinatorics. Cambridge University Press, Cambridge (2009)CrossRefGoogle Scholar
  3. 3.
    Foley, R.D., McDonald, D.R.: Bridges and networks: exact asymptotics. Ann. Appl. Probab. 15(1B), 542–586 (2005)CrossRefGoogle Scholar
  4. 4.
    Gail, H.R., Hantler, S.L., Taylor, B.A.: Spectral analysis of M/G/1 and G/M/1 type Markov chain. Adv. Appl. Probab. 28, 114–165 (1996)CrossRefGoogle Scholar
  5. 5.
    Gohberg, I., Lancaster, P., Rodman, L.: Matrix Polynomials. SIAM, Philadelphia (2009)CrossRefGoogle Scholar
  6. 6.
    Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991)CrossRefGoogle Scholar
  7. 7.
    Kingman, J.F.C.: A convexity property of positive matrices. Q. J. Math. Oxf. 2(12), 283–284 (1961)CrossRefGoogle Scholar
  8. 8.
    Kobayashi, M., Miyazawa, M.: Revisit to the tail asymptotics of the double QBD process: refinement and complete solutions for the coordinate and diagonal directions. In: Matrix-Analytic Methods in Stochastic Models, pp. 145–185 (2013)Google Scholar
  9. 9.
    Latouche, G., Ramaswami, V.: Introduction to Matrix Analytic Methods in Stochastic Modeling. SIAM, Philadelphia (1999)CrossRefGoogle Scholar
  10. 10.
    Li, Q.-L., Zhao, Y.Q.: \(\beta \)-invariant measures for transition matrices of GI/M/1 type. Stoch. Models 19(2), 201–233 (2003)CrossRefGoogle Scholar
  11. 11.
    Markushevich, A.I.: Theory of Functions of a Complex Variable. AMS Chelsea Publishing, Providence (2005)Google Scholar
  12. 12.
    Miyazawa, M., Zhao, Y.Q.: The stationary tail asymptotics in the GI/G/1 type queue with countably many background states. Adv. Appl. Probab. 36(4), 1231–1251 (2004)CrossRefGoogle Scholar
  13. 13.
    Miyazawa, M.: Tail decay rates in double QBD processes and related reflected random walks. Math. Oper. Res. 34(3), 547–575 (2009)CrossRefGoogle Scholar
  14. 14.
    Miyazawa, M.: Light tail asymptotics in multidimensional reflecting processes for queueing networks. TOP 19(2), 233–299 (2011)CrossRefGoogle Scholar
  15. 15.
    Miyazawa, M.: Superharmonic vector for a nonnegative matrix with QBD block structure and its application to a Markov modulated two dimensional reflecting process. Queueing Syst. 81, 1–48 (2015)CrossRefGoogle Scholar
  16. 16.
    Neuts, M.F.: Matrix-Geometric Solutions in Stochastic Models. Dover Publications, New York (1994)Google Scholar
  17. 17.
    Ozawa, T.: Asymptotics for the stationary distribution in a discrete-time two-dimensional quasi-birth-and-death process. Queueing Syst. 74, 109–149 (2013)CrossRefGoogle Scholar
  18. 18.
    Ozawa, T.: Stability of multidimensional skip-free Markov modulated reflecting random walks: revisit to Malyshev and Menshikov’s results and application to queueing networks. Submitted (2015). arXiv:1208.3043
  19. 19.
    Ozawa, T., Kobayashi, M.: Exact asymptotic formulae of the stationary distribution of a discrete-time 2d-QBD process: an example and additional proofs (2018). arXiv:1805.04802
  20. 20.
    Seneta, E.: Non-negative Matrices and Markov Chains, Revised Printing. Springer, New York (2006)Google Scholar
  21. 21.
    Tweedie, R.L.: Operator-geometric stationary distributions for Markov chains with applications to queueing models. Adv. Appl. Probab. 14, 368–391 (1982)CrossRefGoogle Scholar

Copyright information

© 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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