On some applications of a discrete-time model of failure and repair

Research on a discrete-time model of failure and repair studied by Rocha-Martinez and Shaked (1995) is continued in this paper. Among various related results, we prove that if for one point x∈]0,1 the probability generating function of a non-negative integer valued random variable S satisfies Φ_S(x)⩽x^m for some integer m⩾0, then E(S)⩾m. We use these results to show that for any M (the ‘input’ lifetime of a unit in the model) the R_m's (the allowed number of repairs on the unit at time m, m⩾0) can be chosen such that M_u (the ‘output’ lifetime of the unit through the model) is in hazard rate ordering (therefore in stochastic ordering) arbitrarily large and such that E(R_m) is a minimum in some sense. As a first application, we see how a low-quality item (car, computer, washing machine, etc.) might fulfil strict durability regulations under an appropriate imperfect repair strategy (and be able to compete against the existing leading brand in the market) in such a way that the mean number of repairs be a minimum in some sense. As a second application we show how it can be easily proven that if M is of class: NBU, NWE, DMRL, IMRL, NBUE or NWUE, then M_u is not necessarily of the same class. © 1998 John Wiley & Sons, Ltd.