## Maintenance Models with General Degree of Repair

Apart from Policy 5 and the cumulative repair cost limit replacement policies, all maintenance policies considered so far were based on maintenance actions, which only comprise replacements (“good as new”-concept) and minimal repairs (“bad as old”-concept). In many cases, this is a satisfactory approximation to real-life maintenance management. While it may makes sense, doing replacements of aging systems both in case of emergency (corrective) and preventive maintenance, preventive minimal repairs of an operating system do not make any sense. Replacements of an aging system by an equivalent new one reduce the failure rate to its smallest value min: In this chapter, we will assume that (t) is nondecreasing in [0;1) so that min = (0): After a minimal repair, (t) is on the same level as immediately before the repair (failure). Therefore, replacements and minimal repairs can be considered extreme maintenance measures. This motivates the de…nition of the degree of a repair as the ratio

dr(t) = (t) r(t) (t) (0) ; (12.1)

where (t) = (t0) is the value of the failure rate immediately before a failure at time t and r(t) = r(t + 0) is the value of the failure rate immediately after a repair done at time t, (0) r(t) (t): In this case, dr = 0 (dr = 1) refers to a minimal repair (replacement). r de…nes the virtual age v of the system after a repair at time t via r(t) = (v); 0 v t: Thus, from the reliability (survival) point of view, a repair at time t with degree dr(t) reduces the age of the system by t v time units, and, for any calender time point t; the value of the failure rate is actually determined by the virtual age of the system at time t. (Obviously, the virtual age of a technical system is the

the system to continue its work, but actually increases its failure rate, may happen in practice, but we will exclude this atypical situation. Hence, when pursuing a maintenance policy with general repairs, every repair reduces the virtual age of the system, which it had immediately before the failure (repair) so that the virtual age of the system at calender time t depends on all repairs up to time t: Assigning to a minimal repair the degree dr = 0 makes sense, since such a repair has zero in‡uence on the (virtual) age and, therefore, on the survival probability of the system. However, in the literature 1 dr is frequently de…ned as degree of repair. It is now widely used terminology to call a repair imperfect if 0 dr(t) < 1

and perfect if dr(t) = 1 (see e.g. Wang and Pham [303]). Thus, the imperfect repair concept includes our two-failure-type model applied in Chapter 10 and the maintenance policies 10 to 12 derived from it. In the literature, the two failure type model is now called “(p(t); q(t))-modeling”and started with the papers [26], [54], and [55]. The special case p(t) p; q(t) 1 p = q is used in [29], [20], and [72]. For …rst summaries see the monographs [28], [42]. For more recent summaries see [148],[124] , [125], [303], [238], [268]. The (p; q)- model had been applied to preventive repairs as well: A preventive repair is minimal with probability p (as mentioned before, in this case the preventive repair is useless) or renews the system with probability q = 1 p: Maybe the …rst papers along this line were [84], [235], [233]. The terms imperfect repair and perfect repair are likely to be a bit dis-

turbing to maintenance engineers/mechanics, since, from their layman point of view, they may believe to have carried out a perfect imperfect repair and a perfect repair not that perfect. Hence, in what follows we prefer to use the term general repair if the degree of a repair can assume any value between 0 and 1: Two simple analytical models for quantifying the development in time of the system failure rate, when following a general repair policy, will be considered. In both models, let ft1; t2; :::g be the sequence of repair (failure) time points, and fa1; a2; :::g be a sequence of real numbers with 0 an 1; n = 1; 2; ::: To avoid routine modi…cations to the models, all repairs are assumed to take negligibly small times.