## Lifetime of Coherent Systems

In case of independent element lifetimes and no repair of failed elements, the survival (availability) function of the system is fully determined by the vector of the survival functions of the elements p(t) = (F 1(t); F 2(t); :::; Fn(t)) and the availability function of the system h():

F s(t) = 1 Fs(t) = h(p(t)): An already classic problem related to Fs(t) is the IFR lifetime closure problem for coherent systems: If the Fi(t), i = 1; 2; :::; n; are IFR (De…nition 1.1), is Fs(t) IFR as well? Esary and Proschan [117] gave a counter example: The lifetime distribution of a parallel system consisting of two independent elements with di¤erent exponential lifetime distributions is not IFR. (The exponential distribution is both IFR and DFR.) Actually, the failure rate of such a system increases up to a certain time point t0, and after t0 it decreases. Later Birnbaum et al. [50] showed that the lifetime closure problem for coherent systems with independent elements has a positive answer for the IFRA-distribution (De…nition 1.2): If the Fi(t), i = 1; 2; :::; n; are IFRA, then Fs(t) is IFRA. However, there are cases where Fi(t); i = 1; 2; :::; n; being IFR implies

Fs(t) being IFR, too. Samaniego [265] gave a “simple” condition: Fs(t) is IFR if the elements have iid IFR lifetimes and the function

g(x) =

Pn1 i=0 (n i)i+1

xiPn1

xi

is increasing in x; x > 0; where (1; 2; :::; n) is the signature of the system (Section 2.2.4).