Shock models with underlying birth process

Abstract

This paper extends results of Esary, Marshall and Proschan (1973) and A-Hameed and Proschan (1973). We consider the life distribution of a device subject to a sequence of shocks occurring randomly in time according to a nonstationary pure birth process: given k shocks have occurred in [0, t ], the probability of a shock occurring in ( t, t + Δ] is λ k λ ( t )Δ + o (Δ). We show that various fundamental classes of life distributions (such as those with increasing failure rate, or those with the ‘new better than used' property, etc.) are obtained under appropriate assumptions on λ k , λ ( t ), and on the probability of surviving a given number of shocks.