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December 2021 Recurrence of two-dimensional queueing processes, and random walk exit times from the quadrant
Marc Peigné, Wolfgang Woess
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Ann. Appl. Probab. 31(6): 2519-2537 (December 2021). DOI: 10.1214/20-AAP1654

Abstract

Let X=(X1,X2) be a two-dimensional random variable and X(n), nN, a sequence of i.i.d. copies of X. The associated random walk is S(n)=X(1)++X(n). The corresponding absorbed-reflected walk W(n), nN, in the first quadrant is given by W(0)=xR+2 and W(n)=max{0,W(n1)X(n)}, where the maximum is taken coordinate-wise. This is often called the Lindley process and models the waiting times in a two-server queue. We characterize recurrence of this process, assuming suitable, rather mild moment conditions on X. It turns out that this is directly related with the tail asymptotics of the exit time of the random walk x+S(n) from the quadrant, so that the main part of this paper is devoted to an analysis of that exit time in relation with the drift vector, that is, the expectation of X.

Funding Statement

The first author acknowledges support by a visiting professorship at TU Graz.
The second author was supported by Austrian Science Fund projects FWF P31889 and W1230 as well as from the European Research Council (ERC) under Kilian Raschel’s Starting Grant Agreement No759702.

Acknowledgements

The authors thank the referee for her/his efforts to improve the paper.

Citation

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Marc Peigné. Wolfgang Woess. "Recurrence of two-dimensional queueing processes, and random walk exit times from the quadrant." Ann. Appl. Probab. 31 (6) 2519 - 2537, December 2021. https://doi.org/10.1214/20-AAP1654

Information

Received: 1 September 2019; Revised: 1 June 2020; Published: December 2021
First available in Project Euclid: 13 December 2021

Digital Object Identifier: 10.1214/20-AAP1654

Subjects:
Primary: 60G50
Secondary: 37H05 , 60K25

Keywords: exit times , Lindley process , Queueing theory , random walk in the quadrant , recurrence

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.31 • No. 6 • December 2021
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