System availability with non-exponentially distributed outages

This paper studies the steady-state availability of systems with times to outages and recoveries that are generally distributed. Availability bounds are derived for systems with limited information about the distributions. Also investigated are the applicability of convenient exponential models in evaluating availability for systems that have two-sided bounded distributions of times to planned outages. A general closed-form formula is derived for the steady-state availability of a system with multiple outage types of arbitrary distributions. The formula shows that only the mean values of times to repair (TTRi, i = 1, 2, . . . , n) affect the steady-state availability; i.e., distributions of TTRi with the same mean value have the same effect in determining the steady-state system availability. However, the distributions of times to outages, (TTOi, i = 1, 2, . . . , n), have an important impact on the steady-state system availability. Bounds are provided for the steady-state availability for a system subject to unplanned outages, for which times-to-outages are exponentially distributed and planned outages for which times-to-outages have bounded distributions. In practice, the distribution of time to planned outages is generally bounded due to economic constraints and industrial competition. The bounds derived here are good estimates of the system's steady-state availability, if the only known information of time-to-planned-outage is its two-sided bounds. Popular all-exponential models that assume that all times to outages and recoveries are exponentially distributed can under-estimate or over-estimate system availability if used for a system with generally distributed times to outages, of which limited information is known. Therefore explicit criteria are presented for determining when an all-exponential model, if applied to systems with outages of two-sided bounded general distributions, is a good approximation.