Chapter 1 - Reliability Theory

Abstract

In this chapter, we provide a brief account of the background materials that are required for the discussions in the ensuing chapters. The meaning, objectives and importance of reliability theory, as a discipline, is discussed. We then point out why reliability modelling and analysis in discrete time is separately viewed in this work. Life distributions play an important role in reliability modelling. Accordingly, various properties of life distributions that are of interest in reliability modelling are briefly reviewed. We also discuss the role of mixture distributions and weighted distributions. Among the weighted distributions, special attention is paid to equilibrium distributions and their higher orders. Some basic topics like convolutions and shock models are reviewed along with some essential results in this connection. This is followed by describing some geometric concepts like convexity, concavity, and star-shapedness used in the derivation of certain properties of reliability concepts. In order to model complex devices or systems with several components, multivariate distributions become necessary and for this reason some basic tools and techniques in multivariate analysis are presented. Multivariate equilibrium distributions of different forms are defined in the discrete case. An important aspect to be considered in modelling and analyzing multivariate data is the dependence relation that exists between the components. So, we review several dependence measures like correlation, Kendall's tau, Spearman's rho and Blomquist's β and also dependence concepts that indicate positive or negative association like tail monotonicity, stochastic monotonicity, and total positivity. Moreover, the multivariate extensions of these notions are also presented. Following this, discrete versions of time-dependent measures of Clayton, Anderson, Louis, Holm and Harvald, Bairamov, Kotz and Kozubowski, and Nair and Sankaran are discussed. This chapter finally concludes with the definitions of Schur-convexity, concavity and constancy that form an essential tool in Bayesian reliability analysis.