## A non-regenerative model of a redundant repairable system: Bounds for the unavailability and asymptotical insensitivity to the lifetime distribution.(English)Zbl 0858.60076

Summary: 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
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