  This paper shows how the steady-state availability and failure frequency can
be calculated in a single pass for very large systems, when the availability is
expressed as a product of matrices. We apply the general procedure to
$k$-out-of-$n$:G and linear consecutive $k$-out-of-$n$:F systems, and to a
simple ladder network in which each edge and node may fail. We also give the
associated generating functions when the components have identical
availabilities and failure rates. For large systems, the failure rate of the
whole system is asymptotically proportional to its size. This paves the way to
ready-to-use formulae for various architectures, as well as proof that the
differential operator approach to failure frequency calculations is very useful
and straightforward.
