In this paper, we consider a repair-cost limit replacement problem with imperfect repair and develop a graphical method to determine the optimal repair-cost limit which minimizes the expected cost per unit time in the steady-state, using the Lorenz transform of the underlying repair-cost distribution function. The method proposed can be applied to an estimation problem of the optimal repair-cost limit from empirical repair-cost data. Numerical examples are devoted to examine asymptotic properties of the non-parametric estimator for the optimal repair-cost limit.

The paper studies a model of repairable systems which is flexible enough to incorporate the standard imperfect repair and many other models from the literature. Palm stationarity of virtual ages, inter-failure times and degrees of repair is studied. A Loynes-type scheme and Harris recurrent Markov chains combined with coupling methods are used. Results on the weak total variation and moment convergences are obtained and illustrated by examples with IFR, DFR, heavy-tailed and light-tailed lifetime distributions. Some convergences obtained are monotone and/or at a geometric rate.

Stochastic comparison results for replacement policies are shown in this paper using the formalism of point processes theory. At each failure moment a repair is allowed. It is performed with a random degree of repair including (as special cases) perfect, minimal and imperfect repair models. Results for such repairable systems with schemes of planned replacements are also shown. The results are obtained by coupling methods for point processes.

A non-homogeneous Poisson shock model has a continuous mean function Λ(t). The kth shock Sk causes simultaneous failure of the components j ∊ J ∊ {1, ···, n} with probability pJ(Sk). If Tj is the lifetime of component j, it is shown that (T1, · ··, Tn) belongs to various multivariate non-parametric life classes depending on the life class of .

Several notions of stochastic convexity and concavity and their properties are studied in this paper. Efficient sample path approaches are developed in order to verify the occurrence of these notions in various applications. Numerous examples are given. The use of these notions in several areas of probability and statistics is demonstrated. In queueing theory, the convexity (as a function of c) of the steady-state waiting time in a GI/D/c queue, and (as a function of the arrival or service rates) in a GI/G/1 queue, is established. Also the convexity of the queue length in the M/M/c case (as a function of the arrival rate) is shown, thus strengthening previous results while simplifying their derivation. In reliability theory, the convexity of the payoff in the success rate of an imperfect repair is obtained and used to find an optimal repair probability. Also the convexity of the damage as a function of time in a cumulative damage shock model is shown. In branching processes, the convexity of the population size as a function of a parameter of the offspring distribution is proved. In non-parametric statistics, the stochastic concavity (or convexity) of the empirical distribution function is established. And, for applications in the theory of probability inequalities, we identify several families of distributions which are convexly parametrized.

Properties of the conditional hazard rates of X1, · ··, Xn and Y1, · ··, Yn, which imply (X1, · ··, Xn) (Y1, · ··, Yn), are found. These are used to find conditions on the hazard rates of T = (T1, · ··, Tn) which ensure that T has the MIHR | property of Arjas (1981a) and the ‘weakened by failure’ property of Arjas and Norros (1984). Applications for load-sharing model and multivariate imperfect repair are given.