Recurrence of two-dimensional queueing processes, and random walk exit times from the quadrant

Abstract

Let $\mathit{X}=({\mathit{X}}_1,{\mathit{X}}_2)$ be a two-dimensional random variable and $\mathit{X}(\mathit{n})$, $\mathit{n}\in \mathbb{N}$, a sequence of i.i.d. copies of X. The associated random walk is $\mathit{S}(\mathit{n})=\mathit{X}(1)+\cdots +\mathit{X}(\mathit{n})$. The corresponding absorbed-reflected walk $\mathit{W}(\mathit{n})$, $\mathit{n}\in \mathbb{N}$, in the first quadrant is given by $\mathit{W}(0)=\mathit{x}\in {\mathbb{R}}_{+}^2$ and $\mathit{W}(\mathit{n})=max\{0,\mathit{W}(\mathit{n}-1)-\mathit{X}(\mathit{n})\}$, 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 $\mathit{x}+\mathit{S}(\mathit{n})$ 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.