Coherent systems, i.e., multicomponent systems where every component
monotonically affects the working state or failure of the whole system, are among the
main objects of study in reliability analysis. Consider a coherent system with possibly
dependent components having lifetime T , and assume we know that it failed before a
given time t > 0. Its inactivity time t − T can be evaluated under different conditional
events. In fact, one might just know that the system has failed and then consider the
inactivity time (t − T |T ≤ t), or one may also know which ones of the components
have failed before time t, and then consider the corresponding system’s inactivity time
under this condition. For all these cases, we obtain a representation of the reliability
function of system inactivity time based on the recently defined notion of distortion
functions. Making use of these representations, new stochastic comparison results for
inactivity times of systems under the different conditional events are provided. These
results can also be applied to order statistics which can be seen as particular cases
of coherent systems (k-out-of-n systems, i.e., systems which work when at least k of
their n components work).

EB00000004216K

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• Comparison results for inactivity times of k-out-of-n and general coherent systems with dependent components