Modelling of a deteriorating system with repair satisfying general distribution

In this paper, we model a deteriorating system that cannot be repaired ''as good as new'' after failures, the model comes from [19, Applied Mathematics and Computation, 217(2011), 4980–4989]. Suppose that the system has one repairman who can have multiple vacations, and if the system fails when the repairman is on vacation, it will wait for repair until the repairman is available. Herein the repair time is taken into account and supposes that the repair satisfies the general distribution. Under these assumptions, by means of the geometric process and the supplementary variable techniques, we derive a complete model of the partial differential equations, which will correct an error of mathematical model in [19]. Moreover, we deduce some important reliability indices of the system such as the availability of system, the probability of the repairman working and the rate of occurrence of failures. In particular, we prove that the rate of occurrence of failures m f is not equal to zero. With development of the modern technology and extensive use of electronic products, the reliability problem of systems involving electronic products has become a hot topic. In particular, the problems of the maintenance and repair of the electronic products have attracted more attention in engineering. Many researchers have worked in this field, including system modelling (see [1–3]), model analysis and system optimization (see [4–7] and the references therein). The repairability of system is an important factor to ensure the reliability of system. Since repairability problems arise not only in engineering but also other subjects, for instance, the study of medicament and human health (see [8]), electronic-commerce, etc. Therefore , much more attention concentrates on the study of repairable system. In most research works mentioned above, a common character is that the system will be repaired as soon as it fails and be as good as new after repair. In practice, it is not always the case. On one hand, the repairman cannot repair it in time, or the system after failure cannot be repaired immediately because of absence of the repairman. On the other hand, the system cannot be repaired as good as new after failures, and it is true particularly for the electronic products. It is well known that an electronic product has longer lifespan if it is in use, but it deteriorates if it is not used. Therefore, how to characterize reliability of electronic products becomes an important issue …