Bathtub Model Research Articles

Mixed hazard models and their applications

Abstract Life density functions were often taken to be uni-modal for mathematical simplicity and because failure data are observed in a relatively (or extremely) short period compared with the product true life. In this way, (1) over-estimation of the product mean life (2) improper burn-in action and (3) improper preventive maintenance/replacement policy will result. However, multi-mode or mixture models and also the bathtub model can reflect real situations better. But the former will appear in general only by accelerating the test and the latter lacks both theoretical and experimental evidence. A so-called S-shape model which is in fact a mixed Weibull has recently been proposed as a substitute for the bathtub model. In this paper, real data for a sample of certain CMOS (Complementary Metal-Oxide-Semiconductor) components for an accelerated test at 125°C and 5V are illustrated so that they are well fitted by a mixture of two well-separated Weibull pdfs (probability density functions) whose shape parameters are both larger than 1. Such a result further contradicts and challenges the claims from the bathtub model that the hazard rate must be decreasing in an earlier period. If the cost is not under consideration, how long the burn-in is going to take (for such a CMOS batch) depends on the requirement of mission reliability and can be better calculated in terms of such a mixed model. MLE (Maximum Likelihood Estimation) and the Weibull plotting method aided by a Bayesian approach are presented to estimate parameters of the mixed Weibull in such a case of good separation. If two such groups of data are separated more and more widely, estimation through such methods will become more and more reliable.