Imprecise Reliability: an Introductory Review

The main aim of the paper is to define what the imprecise reliability is, what problems can be solved by means of a framework of the imprecise reliability. From this point of view, various branches of reliability analysis are considered, including analysis of monotone systems, repairable systems, multistate systems, structural reliability, software reliability, human reliability, fault tree analysis. Various types of initial information used in the imprecise reliability are overviewed. Some open problems are given in conclusion. This is a contribution to the Documentation Section on the website of the Society for Imprecise Probability Theory and Applications (SIPTA): http://www.sipta.org c © 2003 by Lev V. Utkin and SIPTA 1. Why imprecise probabilities in reliability analysis A lot of methods and models of the classical reliability theory assume that all probabilities are precise, that is, that every probability involved is perfectly determinable. Moreover, it is usually assumed that there exists some complete probabilistic information about the system and component reliability behavior. The completeness of the probabilistic information means that two conditions must be fulfilled: (1) all probabilities or probability distributions are known or perfectly determinable; (2) the system components are independent, i.e., all random variables, describing the component reliability behavior, are independent. The precise system reliability measures can be always (at least theoretically) computed if both conditions are satisfied (it is assumed here that the system structure is defined precisely and there exists a function linking the system time to failure (TTF) and TTFs of components). If at least one of the conditions is violated, then only the interval reliability measures can be obtained. In reality, it is difficult to expect that the first condition is fulfilled. If the information we have about the functioning of components and systems is based on a statistical analysis, then a probabilistic uncertainty model should be used in order to mathematically represent and manipulate that information. However, the reliability assessments that are combined to describe systems and components may come from various sources. Some of them may be objective measures based on relative frequencies or on well-established statistical models. A part of the reliability assessments may be supplied by experts. If a system is new or exists only as a project, then there are not sufficient statistical data in many cases. Even if such data exist, we do not always observe their stability from the statistical point of view. Moreover, the failure time may be not accurately observed or even missed. Sometimes, failure does not occur or occurs partially and we get a censored observation of the failure time.