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

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## 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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