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A $$k$$-out-of-$$n$$ reliability system with an unreliable server and phase type repairs and services: The $$(N,T)$$ policy. (English) Zbl 1003.90013
Summary: We study a $$k$$-out-of-$$n$$ reliability system in which a single unreliable server maintains $$n$$ identical components. The reliability system is studied under the $$(N,T)$$ policy. An idle server takes a vacation for a random amount of time $$T$$ and then attends to any failed component waiting in line upon completion of the vacation. The vacationing server is recalled instantaneously upon the failure of the $$N$$-th component. The failure times of the components are assumed to follow an exponential distribution. The server is subject to failure with failure times exponentially distributed. Repair times of the component, fixing times of the server, and vacationing times of the server are assumed to be of phase type. Using matrix-analytic methods we perform steady state analysis of this model. Time spent by a failed component in service, total time in the repair facility, vacation time of the server, non-vacation time of the server, and time until failure of the system are all shown to be of phase type. Several performance measures are evaluated. Illustrative numerical examples are presented.

##### MSC:
 90B25 Reliability, availability, maintenance, inspection in operations research 60K25 Queueing theory (aspects of probability theory)
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