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

*(English)*Zbl 0858.60076Summary: We investigate steady state reliability parameters of an \(F:r\)-out-of-\(N\) redundant repairable system with \(m\) \((1\leq m\leq r-1)\) 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. 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 \(r\geq m+1\) is considered.) Let \(\mu\) 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 \(\mu\) 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.

##### MSC:

60K10 | Applications of renewal theory (reliability, demand theory, etc.) |

60K15 | Markov renewal processes, semi-Markov processes |

60K25 | Queueing theory (aspects of probability theory) |

90B22 | Queues and service in operations research |

90B25 | Reliability, availability, maintenance, inspection in operations research |