Let be a two-dimensional random variable and , , a sequence of i.i.d. copies of X. The associated random walk is . The corresponding absorbed-reflected walk , , in the first quadrant is given by and , 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 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.
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.
The authors thank the referee for her/his efforts to improve the paper.
"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