Computing optimal replacement time and mean residual life in reliability shock models

•Particular class of reliability shock models is studied.•Phase-type modeling is proposed.•Optimal replacement time and mean residual life are computed.•Matrix-based efficient formulae are obtained.

In this paper, matrix-based methods are presented to compute the optimal replacement time and mean residual lifetime of a system under particular class of reliability shock models. The times between successive shocks are assumed to have a common continuous phase-type distribution. The system's lifetime is represented as a compound random variable and some properties of phase-type distributions are utilized. Extreme shock model, run shock model, and generalized extreme shock model are shown to be the members of this class. Graphical illustrations and numerical examples are presented for the run shock model when the interarrival times between shocks follow Erlang distribution.