A non-regenerative model of a redundant repairable system: bounds for the unavailability and asymptotical insensitivity to the lifetime distribution

In this paper we investigate steady state reliability parameters of an F : r -out-of- N redundant repairable system with m ( 1 ≤ m ≤ r − 1 ) repair channels in light traffic conditions. Such a system can also be treated as a closed queueing network of a simple kind. It includes two nodes, with infinite number of channels and m channels, respectively. Each of the N customers pass cyclically from one node to the other; the service time distributions are of a general form for both the nodes. It is an N -component system with a general distribution A ( t ) of free-of-failure periods of the components is considered. Failed components are repaired by an m -channel queueing system with a general distribution B ( t ) of repair times. The system is assumed to be failed if and only if the number of failed components is at least r . (Only the rather difficult case r ≥ m + 1 is considered.) Let μ be the intensity of the stationary point process of the occurrences of (partial) busy periods within which systems failures happen at least once, and let Q be the steady-state unavailability of the system. Two-sided bounds are established for Q and μ based on the behavior of the renewal rate of an auxiliary renewal process. The bounds are used for deriving some asymptotical insensitivity properties in light traffic conditions.