Studies On The Distribution Structures And Stochastic Comparisons Of Conditional Lifetimes Of Coherent Systems

Coherent systems are very important structures in reliability theory and sur-vival analysis. Some simple coherent systems, such as k-out-of-n systems, have widely applications in electronic engineering, aerospace and other related areas. Signature-based representations of the reliability function of coherent systems with independent and identical components have been proved to be very useful in study-ing the aging characteristics of such systems and in comparing the performance of different coherent systems. In this thesis, we have a thorough study on the distri-bution structures and stochastic comparisons of conditional lifetimes of coherent systems.Firstly, we study the conditional residual lifetime of a coherent system with in-dependent and identical distributed component lifetimes(i.e. the residual lifetime of the coherent system when the system is working and at least k components of the system have failed at time t). We build a new mixture representation of the reliability function of the conditional residual lifetime in terms of the reliabil-ity functions of the conditional residual lifetimes of k-out-of-n system, and then we carry out stochastic comparisons between the conditional residual lifetimes of two coherent systems having either different structures or different component lifetimes.Secondly, we consider the conditional inactive time of a coherent system (i.e. the inactive time of the coherent system when the system has failed and at least n-k+1components of the system are still alive at time i), wherein the involv-ing systems have independent and identical distributed component lifetimes. A mixture representation of the reliability function of the conditional inactivity time of a coherent system in terms of the reliability functions of conditional inactivity times of k-out-of-n system is presented, at the same time, some stochastic ordering properties for the conditional inactivity times of different systems are obtained, based on the stochastically ordered coefficient vectors.Thirdly, we pay our attention to the conditional residual lifetime and the con-ditional inactive time of coherent system with exchangeable components. We also provide mixture representation of the reliability function of the conditional residual lifetime (the conditional inactivity time) of a coherent system, and then we obtain stochastic order properties of the conditional residual lifetime (the conditional i-nactive time) of coherent system, based on the stochastically ordered coefficient vectors.Lastly, we focus on the topic about the conditional residual lifetime and the conditional inactive time of coherent system(i.e. the residual lifetime and the inac-tivity time of coherent system when exactly i components of the system have failed at time t), wherein the involving systems have independent and heterogeneous com-ponents. We construct a group of independent and identical random variables, to get help from mixture representation of reliability function of conditional resid-ual lifetime (conditional inactive time) of coherent system with independent and identical components, we establish their mixture representations for the case of independent and heterogeneous components, respectively. In addition, based on these mixture representations, we compare the performance of competing systems with independent and heterogeneous components.