| The paper deals with a one-unit reliability model with a good state and K failure states. We can't find it at once and make minimal repair at the kth failure and renewal repair at the Kth failure after detecting it. When all the system life and repair time distributions are arbitrary, using the supplementary variable method. We obtain system availability.There are three parts in this paper1. The section of preparative knowledge. In this part, some correlative conceptions methods are given. Such as, the definition of Markov process; the basic conceptions of system: life-time distribution, reliability, failure-frequency function; a primary reliable index in reliable theory research: availability; a main distribution of life-time distributions: exponential distribution; the basic steps of analysing the Morkov precess; the supplementary variable method and Laplace transformation used in this paper.2. The section of system analysis. The part is the core of this paper. In this part, detail analysis of system is done. First, we give the conception of system which generalizes the past models. In this paper, a one-unit detection model with a good state and K failure states is given. In this model, all time distributions involved by system are arbitrary distribution and the hypothesis that all failure states can't transform each other is given. Next, we analysis the system above.we can't consider the system process as Markov process simply because the distributions are arbitrary. In order to consider the process as generalized Markov process, We use the supplementary variable method. Then We give the differential equations of state transformation and get the solution to the equations. Finally, two propositions are given, which are also the main object of this paper: instant availability and average stead availability.3. The section of Result. In this part, we give the conduction of the result in part two and discuss the work which needs to be done in the future.
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