  In this paper we consider first-passage percolation on certain 1-dimensional
periodic graphs, such as the $\Z\times\{0,1,\ldots,K-1\}^{d-1}$ nearest
neighbour graph for $d,K\geq1$. We find that both length and weight of
minimal-weight paths present a typical 1-dimensional asymptotic behaviour.
Apart from a strong law of large numbers, we derive a central limit theorem, a
law of the iterated logarithm, and a Donsker theorem for these quantities. In
addition, we prove that the mean and variance of the length and weight of the
minimizing path between two points are monotone in the distance between the
points.
  The main idea used to deduce the mentioned properties is the exposure of a
regenerative structure within the process. We describe this structure carefully
and show how it can be used to obtain a detailed description of the process
based on classical theory for i.i.d.\ sequences. In addition, we show how the
regenerative idea can be used to couple two first-passage processes to
eventually coincide. Using this coupling we derive a 0-1 law.
