A cold system with two different components and a single vacation of the repairman

In this paper, we model a cold standby repairable system with one repairman. Suppose that the system consists of two dissimilar components and one repairman, in which component 1 is the major working unit and component 2 is the cold stand-by unit, component 1 has priority in use and repair; then the repairman takes a single vacation. Under these assumptions , we derive a model of partial differential equations. Furthermore, using the functional analysis method, we discuss the dynamic asymptotical behavior of the system and get some reliability indices. Finally we analyze a variety for indices of reliability and profit of the system with change of other parameters of the system by numerical simulation. With the development of modern technology and the world economy, the reliability problem of a system attracts more and more attentions. More and more people are working on the reliability of systems including the economic system, in particular , on the reliability of repairable systems. In the repairable system aspect, there are two points being noted: one is the deteriorating system, for instance, see [1–5]; the other is that the system can be repaired ''as good as new'' but the repairman takes the vacation, here we refer to [6–10]. In fact, the former considers a profile of long term running of the system while the latter is based on consideration of the present economic case. It is well known that with increasing time, the system fails easily. In order to enhance the reliability of the system, some people propose a strategy consisting of increasing standby components for example, see [11–15]. Obviously, it has a larger cost. So it is more economical to use a cold standby system that consists of a main component, a standby component and a repairman on vacation. We note that the cold standby component is in the waiting state for a long time, it need not to be the same as the main component from an economic point of view. On the other hand, the vacation rate of the repairman in [10] is assumed to be a constant; in order to retain the profits of the system the vacation rate should be changed. Based on the above reasons , we consider in the present paper a cold standby system with two different components and the repairman under a single vacation. A single vacation means that the repairman returns to the system after …