## 1 Introduction  
Nowadays, availability theory plays a more and more important role in many fields, such as information network [1], military field [2], traffic field [3] and some other basic fields [4; 5]. Steady-state availability as an important aspect of reliability theory, which representing the proportion of time in normal state after a long run, has been studied for many years [6; 7; 8]. However, with the rapid development of science and technology, equipment updating and out becomes faster and the equipment service period may now be only a few years or less. Moreover, the complicated mechatronic systems nowadays always consist of a lot of mechanical elements. The interaction of subsystems often leads to undulations of the availability in the early use of new equipment. Therefore, the availability of these complicated systems usually fluctuates heavily in the early stage, but this fluctuation is quite far from the steady-state availability to be adopted for a long period by system developers and users. This factor seriously affects their understanding of the system availability level and the evaluation of the system performance of these complicated mechatronic systems.  
Under this circumstance, the research on the instantaneous availability (IA) of complicated mechatronic system starts to receive wide attention [9; 10]. IA represents the probability that the system is working at any point of time. With IA, we can easily know the fluctuation of availability of a system and evaluate the performance of a system. Currently, the mathematical analysis on the IA fluctuation of complicated mechatronic systems is unfolded from two main aspects. The first one is a study of how to set up an IA model, solve the IA and analyze the IA fluctuation for one-unit system with multiple states (working, delay-repair, repair, etc.) [11; 12; 13]. Traditionally, the most popular method to assess IA has been the 2-state (working-repair) model [11]. In such case that the system failure time and repair time follow exponential distribution, Markov process is widely used [14]. If they follow general distribution (e.g., uniform, gamma, and Weibull distributions), the Markov renewal process and simulation analysis are often used [11]. [PERSON] et al. studied the fluctuation mechanism of 3-state IA model through simulation analysis and concluded that when parameters, including the failure time, the maintenance time and the support delay time, follow exponential distribution, IA of system has no fluctuation [15; 16]. The second aspect is a study of how to set up an IA model, solve the IA and analyze the IA fluctuation for multi-unit systems (series system and parallel system). Based on the renewal model, the IA fluctuation of the two-unit series repairable system is analyzed in [17]. It turns out that there exists no fluctuation of IA if the working time and repair time are under exponential distribution.  
However, among all the literatures which we can get hold of, none of them combines these two aspects together, i.e., studies how to set up an IA model, solve the IA and analyze the IA fluctuation for multi-unit series repairable systems or parallel repairable systems with multiple states. So, in this paper, we will analyze the mathematical model and IA fluctuations for the two-unit series repairable system and parallel repairable system with three states. Firstly we will give the mathematical models of the IAs of the series systems and the parallel systems with three states. Then the numerical solutions of IAs will be given to analyze the fluctuations of IAs. Finally, some numerical examples will be given to show the effectiveness of the methods in this paper.  
This paper is organized as follows. In Section 2, we study the numerical solution of IA for two-unit series system with three states. Section 3 shows the numerical solution of IA for two-unit parallel system with three states. Section 4 concludes this paper.