Periodical replacement model for a multi-unit system subject to failure rate interaction

A periodical replacement model for a multi-unit system subject to failure rate interaction between units is presented. In a N-unit system, one is dominant unit and the others are secondary units. When a secondary unit fails, it increases the failure rate of the dominant unit by some amount, while the failure of dominant unit make the system into total failure. Without failure rate interaction, the failure rates of all units also increase with age. All single failures of secondary units are assumed to be corrected by minimal repairs. The system is replaced at age T, or on total failure, whichever occurs first. The aim of this paper is to derive the expected cost rate per unit time as a criterion of optimality and to seek the optimal period T* to minimize that cost rate.