Initial Provisioning of a Standby System with Deteriorating and Repairable Spares

A technique is developed for finding the time dependent operating probabilities used by reliability systems designers for provisioning a system with N + k identical units, k of which are called spares and N called operating units, and s repair facilities. System failure occurs when less than N units are operational. Units fail with exponential interfailure times and are repaired with exponential service time. Idle spares fail due to deterioration at a rate possibly different from that of the operating units. Graphs are presented which show the minimum numbers of spares needed to achieve system reliabilities of 0.90 and 0.99 as a function of time. The technique is applicable for finding, numerically, the first passage time distribution for any system modeled by birth and death processes.