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New results for the Brown-Proschan model of imperfect repair. (English) Zbl 0623.62089

Let us consider the model of imperfect repair (i.e. at each failure the repair is ”perfect” with the probability \(p\in (0,1)\) or ”minimal” with the probability 1-p). It is well known that the distribution \(F_ p\) of the sojourn time between renewals is such that \(\bar F{}_ p(t)=\bar F^ p(t)\), where F denotes the life- time distribution function of a new equipment. In the paper an alternative proof of this fact is given via ”shock model representation of the sojourn time”.
Moreover following statements are true: (i) \(F_ p\in A\) for some \(p\neq 0\Rightarrow F\in A\), when \(A=IFR\), IFRA, NBU (DFR, DFRA, NWU); (ii) \(F_ p\in A\) for all sufficiently large \(p<1\Rightarrow F\in A\), when \(A=DMRL\), NBUE, HNBUE, \({\mathcal L}\); (iii) \(F_ p\in A\) for all sufficiently large \(p<1\) and the mean residual sojourn time \(\mu\) (p)\(\to \mu (1)\) as \(p\to 1\Rightarrow F\in A\), when \(A=IMRL\), NWUE, HNWUE, \({\mathcal L}.\)
Inequalities for the mean residual sojourn time and limit distributions of the sojourn time are included.
Reviewer: D.Bobrowski

MSC:

62N05 Reliability and life testing
60G35 Signal detection and filtering (aspects of stochastic processes)
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References:

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