NBU Processes with General State Space

Waiting times for many kinds of events can be represented as first passage times for appropriate stochastic processes. In some applications the most natural stochastic process is multidimensional. In other applications it may be more natural to consider a stochastic process which takes on values in a more general space. In this paper we consider new better than used (NBU) stochastic processes on partially ordered separable Banach spaces, that is processes which have NBU first passage times to closed upper sets. Various equivalent conditions for a process to be NBU are derived. Some of the results can be used also to characterize increasing hazard rate average (IHRA) processes. Examples of such processes include processes with shocks and recovery and processes with random repair times. Some applications in reliability theory and in other areas of probability are given.