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Let $X = (X_1, X_2)$ be a 2-dimensional random variable and $X(n), n \in\mathbb{N}$ a sequence of i.i.d. copies of $X$. The associated random walk is$S(n)= X(1) + \cdots +X(n)$. The corresponding absorbed-reflected walk $W(n), n\in \mathbb{N}$ in the first quadrant is given by $W(0) = x \in \mathbb{R}_+^2$and $W(n) = \max \{ 0, W(n-1) - X(n) \}$, where the maximum is takencoordinate-wise. This is often called the Lindley process and models thewaiting times in a two-server queue. We characterize recurrence of thisprocess, assuming suitable, rather mild moment conditions on $X$. It turns outthat this is directly related with the tail asymptotics of the exit time of therandom walk $x + S(n)$ from the quadrant, so that the main part of this paperis devoted to an analysis of that exit time in relation with the drift vector,i.e., the expectation of $X$.

Recurrence of 2-dimensional queueing processes, and random walk exit times from the quadrant