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

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