J. Appl. Prob. 47, 601–607 (2010) Printed in England  Applied Probability Trust 2010 A NOTE ON CONVERGING GEOMETRIC-TYPE PROCESSES MAXIM FINKELSTEIN, ∗ University of the Free State and Max Planck Institute for Demographic Research Abstract The process of deterioration of repairable systems with each repair is modeled using converging geometric-type processes. It is proved that the expectation of the number of repairs in each interval of time is infinite. A new regularization procedure is suggested and the corresponding optimization problem is discussed. Keywords: Geometric process; renewal process; repairable system; renewal function; minimal repair; overhaul 2010 Mathematics Subject Classification: Primary 60K10 Secondary 60N05 1. Introduction Deterioration in performance of imperfectly repaired items is often modeled using a stochas- tically decreasing sequence of lifetimes (cycles). This can be done in various ways. One popular approach is based on the virtual age concept, which states that after the repair action the age of a deteriorating item that started operation at t = 0 and failed at t = t f is equal to some ‘younger’, intermediate value t ν , 0 ≤ t ν ≤ t f . The case in which t ν = 0 corresponds to the perfect repair, whereas t ν = t f means that a minimal repair has been performed. Different models exist for defining this intermediate value. The most popular method employs the linear reduction of the age at failure (see Kijima (1989), Doyen and Gadoin (2004), and Finkelstein (2007), to name a few). Note that the corresponding renewal-type process in this case is the process with dependent interarrival times that are governed by a generic cumulative distribution function (CDF) F(t). On the other hand, a sequence of stochastically decreasing, independent lifetimes can also constitute a useful model for the deterioration of repairable items and the corresponding example will be considered in this paper. Denote the duration of the i th cycle by X i ≥ 0,i = 1, 2,... , and the corresponding CDF by F i (t). Assume that the mean μ i = E[X i ] exists, and denote the variance by σ 2 i . Let μ i +1 <μ i , which is obviously weaker than the (usual) stochastic ordering: X i +1 < st X i , i = 1, 2,..., (1) meaning that F i (t) > F i +1 (t), t > 0 (see Ross (1996)). The geometric process introduced in Lam (1988) and thoroughly investigated in Lam (2007) is a meaningful example of (1). Let Y 1 ,Y 2 ,... be the independent and identically distributed sequence of continuous lifetime random variables with CDF F(t), F(0) = 0, mean μ, and Received 14 October 2009; revision received 15 March 2010. ∗ Postal address: Department of Mathematical Statistics, University of the Free State, 339 Bloemfontein 9300, South Africa. Email address: finkelm@ufs.ac.za 601 https://www.cambridge.org/core/terms. https://doi.org/10.1239/jap/1276784913 Downloaded from https://www.cambridge.org/core. IP address: 18.206.13.133, on 11 Jun 2020 at 11:47:41, subject to the Cambridge Core terms of use, available at