Allocation Of Spares And Stochastic Comparisons Based On K-out-of-n Systems

In the reliability theory text, a k-out-of-n system is a very popular structure of redundancy in fault-tolerant systems, which has widely applications in electronic en-gineering, aerospace, weapons and equipment, and water conservancy and hydropower and other related areas of system design. In this thesis, we have a thorough study on allocation of active redundancies and carry out stochastic comparisons based on k-out-of-n systems with independent and heterogenous components.Firstly, we consider the problem of optimal allocation of m active redundancies to k-out-of-n systems consisting of independent components. If the components and redundancies are independent and identical, we showed that through balancing the allocation of redundancies one can optimize the hazard rate of the resulting system, consequently, the optimal allocation is presented in the sense of hazard rate order-ing. Further, we investigate the problem of allocation of active redundancies in the other situation where the components and redundancies are not necessary identical, wherein we develop two models based on assumptions of system with heterogeneous components and of system with identical components, respectively. For the former, a relatively simple sufficient condition is provided so as to stochastically optimize the lifetime of resulting systems through balancing the allocation of active redundancies. For the latter, with the help of no conditions, the optimal policy is proved to bal-ance the allocation of active redundancies. In addition, as an application, we carry out stochastic comparisons of order statistics based on the heterogenous samples having the proportional hazard rate, which strengthen the corresponding results in the literature.Secondly, we pay our attention to stochastic comparisons of k-out-of-n systems with heterogenous components. Inspired by Misra and Misra (2012), we draw support from m spares and proceed the study to consider the following problems:(1) the spares stochastically perform better than all components do; (2) the spares stochastically per-form worse than all components do. In both situations we employ these m spares to replace components of the system, and carry out stochastic comparisons between the original k-c-out-of-n system and the resulting one with respect to collection of stochastic orders, which involve hazard rate ordering, reversed hazard rate ordering, likelihood ration ordering and mean residual life ordering. Our main results serve as nice supple-ments to corresponding ones of Boland and Proschan (1994) and Nanda and Shaked (2001), and further generalize the corresponding ones by Misra and Misra (2012) to k-out-of-n systems or series and parallel systems consisting of n components.Lastly, we focus on the topic about residual life of parallel system and inactive time of series system, wherein the involving systems have independent and heterogenous components. We begin with the problem of stochastic comparisons of residual lifetime and inactive time. Up to now all the results in the literature are established with respect to usual stochastic ordering. Inspired by this fact, on the basis of different structures of residual life and inactive time developed in the literature, we carry out stochastic comparisons with respect to reversed hazard rate ordering. In addition, we also examine the monotonicity properties of residual life and inactive time with respect to elapsed time.