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Some partial orderings which compare probability distributions with the exponential distribution, are found to be very useful to understand the phenomenon of ageing. Here, we introduce some new generalized partial orderings which describe the same kind of characterization of some generalized ageing classes. We give some equivalent conditions for each of the orderings. Inter-relations among the generalized orderings have also been discussed.

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Stochastic orders are useful to compare the lifetimes of two systems. We discuss both active redundancy as well as standby redundancy. We show that redundancy at the component level is superior to that at the system level with respect to different stochastic orders, for different types of systems.Index Terms-Coherent system, -out-of-system, reliability, stochastic orders. ACRONYMSAND ABBREVIATIONS st usual stochastic sp stochastic precedence hr hazard rate rhr reversed hazard rate lr likelihood ratio up shifted hazard rate down shifted hazard rate up shifted reversed hazard rate up shifted likelihood ratio down shifted likelihood ratio NOTATION underlying nonnegative random variable probability density function of random variable cumulative distribution function of random variable survival (reliability) function of random variable hazard rate function of random variable reversed hazard rate function of random variable lifetime of a coherent system Manuscript

Relative ageing describes how one system ages with respect to another. The ageing faster orders are used to compare the relative ageing of two systems. Here, we study ageing faster orders in the hazard and reversed hazard rates. We provide some sufficient conditions for one coherent system to dominate another with respect to ageing faster orders. Further, we investigate whether the active redundancy at the component level is more effective than that at the system level with respect to ageing faster orders, for a coherent system. Furthermore, a used coherent system and a coherent system made out of used components are compared with respect to ageing faster orders.

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We consider an open problem of obtaining the optimal operational sequence for the 1-out-of-n system with warm standby. Using the virtual age concept and the cumulative exposure model, we show that the components should be activated in accordance with the increasing sequence of their lifetimes. Lifetimes of the components and the system are compared with respect to the stochastic precedence order and its generalization. Only specific cases of this optimal problem were considered in the literature previously.