  The paper studies the expectation of the inspection time in complex aging
systems. Under reasonable assumptions, this problem is reduced to studying the
expectation of the length of the shortest path in the directed degradation
graph of the systems where the parameters are given by a pool of experts. The
expectation itself being sometimes out of reach, in closed form or even through
Monte Carlo simulations in the case of large systems, we propose an easily
computable lower bound. The proposed bound applies to a rather general class of
linear programs with random nonnegative costs and is directly inspired from the
upper bound of Dyer, Frieze and McDiarmid [Math.Programming {\bf 35} (1986),
no.1,3--16].
