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**Age-dependent minimal repair.**
*(English)*
Zbl 0564.60084

A stochastic model is developed to describe the operation in time of the following maintained system setting. A piece of equipment is put in operation at time 0. Each time it fails, a maintenance action is taken which, with probability p(t), is a complete repair or, with probability \(q(t)=1-p(t)\), is a minimal repair, where t is the age of the equipment in use at the failure time. It is assumed that complete repair restores the equipment to its good as new condition, that minimal repair restores the equipment to its condition just prior to failure and that both maintenance actions take negligible time.

If the equipment’s life distribution F is a continuous function, the successive complete repair times are shown to be a renewal process with interarrival distribution \[ F_ p(t)=1-\exp \{-\int^{t}_{0}p(x)\bar F^{-1}(x)F(dx)\} \] for \(t\geq 0\). Preservation and monotone properties of the model extending the results of M. Brown and F. Proschan [ibid. 20, 851-859 (1983; Zbl 0526.60080)] are obtained.

If the equipment’s life distribution F is a continuous function, the successive complete repair times are shown to be a renewal process with interarrival distribution \[ F_ p(t)=1-\exp \{-\int^{t}_{0}p(x)\bar F^{-1}(x)F(dx)\} \] for \(t\geq 0\). Preservation and monotone properties of the model extending the results of M. Brown and F. Proschan [ibid. 20, 851-859 (1983; Zbl 0526.60080)] are obtained.

### MSC:

60K10 | Applications of renewal theory (reliability, demand theory, etc.) |

60K05 | Renewal theory |

90B25 | Reliability, availability, maintenance, inspection in operations research |