Abstract
This paper considers a simple repairable system with a warning device and a repairman who can have delayed-multiple vacations. By Markov renewal process theory and the probability analysis method, the system is first transformed into a group of integrodifferential equations. Then, the existence and uniqueness as well as regularity of the system dynamic solution are discussed with the functional analysis method. Further, the asymptotic stability, especially the exponential stability of the system dynamic solution, is studied by using the strongly continuous semigroup theory orC0semigroup theory. The reliability indices and some applications (such as the comparisons of indices and profit of systems with and without warning device), as well as numerical examples, are presented at the end of the paper.
Highlights
A repairable system is a system which, after failing to perform one or more of its functions satisfactorily, can be restored to fully satisfactory performance by any method, rather than the replacement of the entire system
A perfect repair can restore a system to an “as good as new” state, a normal repair is assumed to bring the system to any condition, and a minimal repair, or imperfect repair, can restore the system to the exact state it was before failure
We proposed a simple repairable system with a warning device and a repairman who can have delayedmultiple vacations
Summary
A repairable system is a system which, after failing to perform one or more of its functions satisfactorily, can be restored to fully satisfactory performance by any method, rather than the replacement of the entire system. The available references concerning repairman vacation in repairable systems can be classified into two categories: one is focused on the system indices and the other is the optimization problems. Hu et al [13] studied the steady-state availability and the mean up-time of a seriesparallel repairable system consisting of one master control unit, two slave units, and a single repairman who operates single vacation by using the supplementary variable method and the vector Markov process theory. We are devoted to studying the asymptotic behavior of the system by strongly continuous semigroup theory and make comparisons of indices (such as reliability, availability, and the probability of the repairman’s vacation) and profit of the two systems with and without warning device. A brief conclusion is offered in the last section
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