  A set of general formulas is proposed for the probability of failure on
demand (PFD) assessment of MooN architecture (i.e. k-out-of-n) systems subject
to proof tests. The proof tests can be partial or full. The partial tests (e.g.
visual inspections, partial stroke testing) are able to detect only some system
failures and leave the others latent, whereas the full tests refer to overhauls
which restore the system to an as good as new condition. Partial tests may
occur at different time instants (periodic or not), up to the full test. The
system performances which are investigated are the system availability
according to time, the PFD average in each partial test time interval, and the
total PFD average calculated on the full test time interval. Following the
given expressions, parameter estimations are proposed to assess the system
failure rates and the partial test effectiveness according to feedback data
from previous test policies. Subsequently, an optimization of the partial test
strategy is presented. In the 2oo6 system given as example, an improvement of
about 10% of the total PFD average has been obtained, just by a better
(non-periodic) distribution of the same number of partial tests, in the full
test time interval.
