Two models of reliability by imprecise parameters of lifetime distributions

By analyzing the reliability of a system, it is very often assumed that all probabilities are precise, that is, that every probability involved is perfectly determinable. However, the information about reliability of components may be supplied by experts and it is difficult to expect that all experts provide precise and true reliability assessments. One of the promising tools for dealing with such assessments and for computing the system reliability on their base is the imprecise probability theory (also called the theory of lower previsions [19], the theory of interval statistical models [14], the theory of interval probabilities [21, 22]), whose general framework is provided by upper and lower previsions. Some examples of the successful application of imprecise probabilities to reliability analysis can be found in [13, 16, 17], where it is assumed that probability distributions of the component times to failure are unknown and there exists only some partial information about the component reliability behavior in the form of interval probabilities, interval moments, etc. At the same time, there are cases when a type of probability distributions of the component times to failure is known, for example, from physical nature of components, but their parameters or a part of parameters are defined by experts. If experts provide possible intervals of parameters and these experts are absolutely reliable, i.e., they provide always true assessments, then the problem of computing the system reliability is reduced to the well known interval analysis [1]. However, in reality there is some degree of our belief to each expert’s judgement whose value is determined by experience and competence of the expert. Therefore, it is necessary to take into account the available information about experts to obtain more credible assessments of the system reliability. The uncertainty in parameters can be considered in a framework of hierarchical uncertainty models which are rather common in uncertainty theory. The different application examples of these models can be found in [6, 8]. A comprehensive review of hierarchical models is given in [5] where it is argued that the most common hierarchical model is the Bayesian one [2, 3, 10, 11, 12, 15, 24].