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Tail Asymptotics of the Stationary Distribution of a Two-Dimensional Reflecting Random Walk with Unbounded Upward Jumps

Part of: Stochastic processes Limit theorems Special processes

Published online by Cambridge University Press:  22 February 2016

Masahiro Kobayashi  and
Masakiyo Miyazawa
Masahiro Kobayashi*
Affiliation:
Tokyo University of Science
Masakiyo Miyazawa*
Affiliation:
Tokyo University of Science
*
∗ Postal address: Tokyo University of Science, 2641 Yamazaki, Noda, Chiba, 278-8510, Japan.
∗ Postal address: Tokyo University of Science, 2641 Yamazaki, Noda, Chiba, 278-8510, Japan.
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Abstract

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We consider a two-dimensional reflecting random walk on the nonnegative integer quadrant. This random walk is assumed to be skip free in the direction to the boundary of the quadrant, but may have unbounded jumps in the opposite direction, which are referred to as upward jumps. We are interested in the tail asymptotic behavior of its stationary distribution, provided it exists. Assuming that the upward jump size distributions have light tails, we find the rough tail asymptotics of the marginal stationary distributions in all directions. This generalizes the corresponding results for the skip-free reflecting random walk in Miyazawa (2009). We exemplify these results for a two-node queueing network with exogenous batch arrivals.

Keywords

Two-dimensional reflecting random walkstationary distributionlinear convex ordertail asymptoticsunbounded jumptwo-node queueing networkbatch arrival60K25

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60F10: Large deviations 60G50: Sums of independent random variables; random walks
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 46 , Issue 2 , June 2014 , pp. 365 - 399
Copyright
© Applied Probability Trust 

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