Matrix-Analytic Methods in Stochastic Models pp 145-185 | Cite as

# Revisiting the Tail Asymptotics of the Double QBD Process: Refinement and Complete Solutions for the Coordinate and Diagonal Directions

## Abstract

We consider a two-dimensional skip-free reflecting random walk on a nonnegative integer quadrant. We are interested in the tail asymptotics of its stationary distribution, provided its existence is assumed. We derive exact tail asymptotics for the stationary probabilities on the coordinate axis. This refines the asymptotic results in the literature and completely solves the tail asymptotic problem on the stationary marginal distributions in the coordinate and diagonal directions. For this, we use the so-called analytic function method in such a way that either generating functions or moment-generating functions are suitably chosen. The results are exemplified by a two-node network with simultaneous arrivals.

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Sorting## Notes

### Acknowledgements

We are grateful to Mark S. Squillante for encouraging us to complete this work. We are also thankful to the three anonymous referees. This research was supported in part by the Japan Society for the Promotion of Science under Grant No. 21510165.

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