Statistics : Theory of Stochastic Orders and Aging and Applications

The manner in which a component (or system) improves or deteriorates with time can be described by concepts of aging. Various aging notions have been proposed in the literature. Similarly lifetimes of two different systems can be compared using the concepts of stochastic orders between the probability distributions of corresponding (random) lifetimes. Various stochastic orders between probability distributions have been defined in the literature. We study the concepts of aging and stochastic orders for various coherent systems. In many situations, the performance of a system can be improved by introducing some kind of redundancy into the system. The problem of allocating redundant components to the components of a coherent system, in order to optimize its reliability or some other system performance characteristic, is of considerable interest in reliability engineering. These problems often lead to interesting theoretical results in Probability Theory. We study the problem of optimally allocating spares to the components of various coherent systems, in order to optimize their reliability or some other system performance characteristic. Performances of systems arising out of different allocations are studied using concepts of aging and stochastic orders.