Extended repair model for a deteriorating system with its repairman having vacations
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Extended repair model for a deteriorating system with its repairman having vacations Junyuan Wang1
· Jimin Ye1
Received: 11 March 2019 / Accepted: 15 October 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract A new repairable system model with a single component and a repairman is proposed. It is assumed that the system has two types of failure and is replaced at the occurrence of the N th type I failure, or the first type II failure (catastrophic failure), whichever occurs first. The system is repaired after type I failure. The repairman has multiple vacations when the system is working. And the system is detected by repairman after his vacation. The system will work again when it repaired. Successive working time and the successive repair time are extended geometric process. The explicit expression of the long-run average cost rate function of the system is derived. Numerical cases are provided to verify the effectiveness of the theoretical approaches. Finally, sensitive analysis of parameters is carried out. Keywords Stochastic deterioration · Extended geometric process · Vacation · Replacement policy · Average cost rate
1 Introduction In reliability and maintenance area, it is usually assumed that a failed system after repair would be returned as good as new state. Since most repairable system is deteriorating due to accumulative wear. This assumption is not always reasonable in application. Barlow and Hunter [1], Barlow and Proschan [2] proposed two types of preventive maintenance policies. Brown and Proschan [4] studied the imperfect repair model, a failed system is either perfectly repaired or minimally repaired with probability p and 1 − p, respectively. In real life, the successive operating time after repair may become shorter and shorter. However, the repair time of the system will become longer
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Junyuan Wang [email protected] Jimin Ye [email protected]
1
School of Mathematics and Statistics, Xidian University, Xi’an 710071, China
123
J. Wang, J. Ye
and longer with the increasing of failure number for deteriorating system. Lam [17] studied a machine maintenance model, in which it’s working time becomes shorter stochastically. On the other hand, repair time becomes longer stochastically. Lam [17] and Zhang [37] introduced geometric process (GP) repairable model, which could better describes the working time and repair time of the machine. And Lam studied two kinds of replacement policies of this model, i.e., one is the failure number N of the system and the other is the working time T of the system. Meanwhile, Lam [17] proved that optimal policy N ∗ is easier to carry out than optimal policy T ∗ under some conditions. Stadje and Zuckerman [29] proved that optimal replacement policy N ∗ is as good as the optimal replacement policy T ∗ of the system from the perspective of cost. Finkelstein [9] proposed a more general deteriorating renewal process model. Zhang [37] generalized Lam’s work and proposed the bivariate replacement policy (T , N ), and proved that the bi
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