Analysis for series–parallel systems under Marshall–Olkin run shock model

This paper develops a reliability framework for series–parallel systems subject to run-dependent Marshall–Olkin (MO) shocks, in which each parallel block may contain an arbitrary number of components sharing a block-common shock source. Within a parallel block, the component lifetimes are dependent, since they share the block-common run-shock time. This dependence prevents a direct application of the phase-type maximum closure. We resolve this obstacle through a structural decomposition lemma that expresses each block lifetime in terms of mutually independent random variables. This lemma allows exact phase-type representations at all system levels through nested Kronecker product constructions. From these representations, we obtain survival functions, mean time to failure, variance, hazard rate, and mean residual life in closed matrix-exponential form. We also show that the conditional mean residual life of any component, given the survival of all other components in its block, depends on their observation times only through their maximum. This is a structural invariance property of the MO parallel-block model. Closed-form dimension formulas describe how the state-space size grows with the system size, the phase dimensions, and the run-length threshold k $k$ k . The methodology is illustrated on a three-block series–parallel system with Erlang-2 interarrival times.