Imperfect repair

TLDR

A device is repaired each time it fails. Each repair either restores the device to a perfect state with probability p or returns it to a functioning state equivalent to its age at failure with probability 1–p, and the repair takes negligible time. The authors derive the distribution Fp of intervals between successive perfect‑repair states from the underlying life distribution F, show that Fp preserves the same reliability class as F, and establish several monotonicity properties, making the results useful for stochastic process analysis and the imperfect‑repair model.

Abstract

A device is repaired at failure. With probability p , it is returned to the ‘good-as-new' state (perfect repair), with probability 1 – p , it is returned to the functioning state, but it is only as good as a device of age equal to its age at failure (imperfect repair). Repair takes negligible time. We obtain the distribution F p of the interval between successive good-as-new states in terms of the underlying life distribution F . We show that if F is in any of the life distribution classes IFR, DFR, IFRA, DFRA, NBU, NWU, DMRL, or IMRL, then F p is in the same class. Finally, we obtain a number of monotonicity properties for various parameters and random variables of the stochastic process. The results obtained are of interest in the context of stochastic processes in general, as well as being useful in the particular imperfect repair model studied.