Let τ(x)τ(x) be the epoch of first entry into the interval (x,∞)(x,∞), x>0x>0, of the reflected process YY of a Lévy process XX, and define the overshoot Z(x)=Y(τ(x))−xZ(x)=Y(τ(x))−x and undershoot z(x)=x−Y(τ(x)−)z(x)=x−Y(τ(x)−) of YY at the first-passage time over the level xx. In this paper we establish, separately under the Cramér and positive drift assumptions, the existence of the weak limit of (z(x),Z(x))(z(x),Z(x)) as xx tends to infinity and provide explicit formulas for their joint CDFs in terms of the Lévy measure of XX and the renewal measure of the dual of XX. Furthermore we identify explicit stochastic representations for the limit laws. We apply our results to analyse the behaviour of the classical M/G/1 queueing system at buffer-overflow, both in a stable and unstable case.