Copula Function-based Reliability Analysis And Stochastic Comparison Of Simple System With A Standby Unit

In most real life situations, the components working in the common environment are dependent and this dependence should be considered in system design and management from a practical point of view. However, in the study of system reliability under standby redundancy, the components are usually assumed to be independent previously. The appearance of Copula function provides an efficient tool for the study of reliability of system consisted of the dependent components.There are only two main ways considered in this paper to allocate redundancies to a system, the cold standby and the hot standby. The former is put into operation when one of the working components fails and it does not fail while in standby. The latter is put in parallel to components of systems and it does fail while in standby, especially these spares start functioning simultaneously as original components.First of all, this paper studied a series system and a parallel system with a single cold standby unit when the components are dependent. The dependence between the components has been modeled by Copulas. The explicit expression and the boundary have presented in this paper for the MTTF of the series system and the parallel system in term of the Copula function and marginal lifetime distribution in two different cases. As an application, the numerical calculations of MTTF are obtained corresponding to two different two-dimensional Copula functions.After that, this paper also studied a series system with a single hot standby unit when the components are dependent. The hot standby unit is put in parallel with one of components in two unit series system and the problem is how to allocate the hot standby unit so that the resulting system performs better. The first innovation in this paper is to extend the stochastic comparison result for the problem of allocating one redundancy to the series system with two components under exponential framework or weibull framework. The second innovation is obtaining an explicit expression and the numerical calculations for the MTTF of the two unit series system with a single hot standby in term of the three different three-dimensional Copula functions and marginal lifetime distributions.