We study the conditions for unimodality of the lifetime distribution of a coherent system when the ordered component lifetimes in the system are described by generalized order statistics. Results for systems with independent and identically distributed lifetimes of components are included in this setting. The findings are illustrated with some examples for different types of systems. In particular, coherent systems with strictly bimodal density functions are presented in the case of independent standard uniform distributed lifetimes of components. Furthermore, we use the results to derive a sharp upper bound on the expected system lifetime in terms of the mean and the standard deviation of the underlying distribution.

Suppose that a device is subjected to shocks and that denotes the probability of surviving k shocks. Then is the probability that the device will survive beyond t, where is the counting process which governs the arrival of shocks. A-Hameed and Proschan (1975) considered the survival function H(t) under what they called the pure birth shock model. In this paper we shall prove that is IFRA and DMRL under conditions which differ from those used by A-Hameed and Proschan (1975).

A device is subject to a sequence of shocks occurring randomly at times n = 1, 2, ⃛. At each point in time, shocks occur according to a Poisson distribution with parameter λ. Shocks cause damage and damage accumulates additively. They can cause the device to fail, and the probability of such a failure depends on the accumulated damage. Failure occurs because of shocks and can occur only at times n = 1, 2, ⃛. The device can be replaced before or at failure. If the device fails it is immediately replaced at a fixed cost. Replacement before failure can only occur at times n = 1, 2, ⃛, and is done at a lower cost depending on the amount of accumulated damage at replacement. In this paper we determine the optimal replacement policy that minimizes the expected cost per unit time.

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 kλ (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.