Assessment of a multi-state system under a shock model

A system is subject to random shocks over time. Let c1 and c2 be two critical levels such that c1 < c2. A shock with a magnitude between c1 and c2 has a partial damage on the system, and the system transits into a lower partially working state upon the occurrence of each shock in (c1, c2). A shock with a magnitude above c2 has a catastrophic affect on the system and it causes a complete failure. Such a shock model creates a multi-state system having random number of states. The lifetime, the time spent by the system in a perfect functioning state, and the total time spent by the system in partially working states are defined and their survival functions are derived when the interarrival times between successive shocks follow phase-type distribution.