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Based on some practical problems in industrial engineering, a species of two-unit parallel repairable system model is introduced in this paper. The prominent feature of the model is described as follows: if a repair time is sufficiently short (less than some threshold value) that does not affect the system operation, i.e. the effect could be ignored, the system can be considered as being operating during this repair time. Two scenarios are discussed where the threshold value is regarded as a constant and non-negative random variable, respectively. Reliability indices such as instantaneous availability are formulated.