Σύγκριση συστημάτων αξιοπιστίας με ανταλλάξιμες μονάδες
Comparing reliability structures with exchangeable components
Μπαράκη, Μαρία - Ελένη Κ.
The subject of the present MSc Dissertation is the comparison of reliability structures with exchangeable components. Even though most of the published work in this area deals with reliability systems with independent and identically distributed components, the assumption of dependence between the lifetimes of system’s components is more realistic. This is the reason why more and more published work over the last years deals with components with dependent lifetimes. Principal concept for this study is the systems’ signature, which is very helpful for the calculation of their reliability function. In the first chapter the reader is introduced in the concept of system’s reliability. With a flashback over the past decades, the evolution of this concept and the significance of statistics in calculating system’s reliability are illustrated. More specifically, we introduce order statistics, which is a tool with an important role in reliability system theory, quality control and other applied areas. Finally, we present the concept of signature, which is a very useful tool for studying the reliability function of a system with exchangeable components. In the second chapter, the concept of exchangeable random variables is introduced. Firstly, the finite/infinite exchangeable sequence and De Finetti’s Theorem, according to which infinite exchangeable sequences are conditionally independent and identically distributed, are reviewed. Then, several results are given regarding the distribution of order statistics in a sample of exchangeable random variables. Finally, additional results regarding the distribution of runs in the case of an exchangeable sequence of binary trials are provided. The third chapter introduces the reader to the concept of the signature of a coherent reliability structure. The reliability function of the system is expressed in terms of the system’s signature for the case where the system’s components are independent and identically distributed or the components are exchangeable. In addition, various results pertaining to the stochastic comparison of certain systems, based on the comparison of their signatures, are provided. For this purpose, four different kinds of stochastic order are considered. The fourth chapter refers to the subject of aging preservation properties under the formation of coherent reliability systems. More specifically, various results regarding whether a coherent system is 𝐼𝐹𝑅/𝐷𝐹𝑅 are provided, along with several conditions based on the system’s signature, examining the same question. Finally, some new concepts of multivariate aging for exchangeable random variables and their properties are presented, focusing our attention only on the bivariate case.