The safety performance of operating nuclear power plants is strongly affected by the unavailability of safety systems that are designed to mitigate accident conditions. The unavailability of these safety systems during plant operation is controlled by the plant's technical specifications which prescribes limits on the downtime duration (outage time) of the individual safety equipment. In this study, risk- and reliability-based methodologies for the determination of allowable cumulative downtime for safety components and safety systems are developed. The limits on the cumulative downtime durations are determined by taking into account the statistical variations expected from a stochastic process which models both the downtime occurrences as well as the downtime durations. This stochastic process is also expected to track with time, the distribution of the cumulative downtime (or the cumulative residence time while in the failed state). Such processes are referred to as memory processes in the context of this study. Three mathematical models are developed for evaluating the types of processes was developed which can generate thousands of simulations and therefore allows the construction of the cumulative downtime distribution. Various solution techniques for these mathematical models were also developed and applied. The question of partial information was addressed, and the use of the maximum entropy principle in this area was detailed. A solution method for estimating the parameters of maximum entropy distribution using the Laguerre polynomials are also discussed. Finally, these theoretical models were applied to a typical auxiliary feed water system in a pressurized water reactor and a pilot trial for determination of the allowable cumulative downtime for a component and the system was performed.