  Let ${Z_n}_{n\ge 0}$ be a random walk with a negative drift and i.i.d.
increments with heavy-tailed distribution and let $M=\sup_{n\ge 0}Z_n$ be its
supremum. Asmussen & Kl{\"u}ppelberg (1996) considered the behavior of the
random walk given that $M>x$, for $x$ large, and obtained a limit theorem, as
$x\to\infty$, for the distribution of the quadruple that includes the time
$\rtreg=\rtreg(x)$ to exceed level $x$, position $Z_{\rtreg}$ at this time,
position $Z_{\rtreg-1}$ at the prior time, and the trajectory up to it (similar
results were obtained for the Cram\'er-Lundberg insurance risk process). We
obtain here several extensions of this result to various regenerative-type
models and, in particular, to the case of a random walk with dependent
increments. Particular attention is given to describing the limiting
conditional behavior of $\tau$. The class of models include Markov-modulated
models as particular cases. We also study fluid models, the Bj{\"o}rk-Grandell
risk process, give examples where the order of $\tau$ is genuinely different
from the random walk case, and discuss which growth rates are possible. Our
proofs are purely probabilistic and are based on results and ideas from
Asmussen, Schmidli & Schmidt (1999), Foss & Zachary (2002), and Foss,
Konstantopoulos & Zachary (2007).
