Only the High-Fidelity (HF) simulation is employed for the development of the adaptive Kriging model, making it difficult to further improve the estimation efficiency while ensuring prediction accuracy. To solve this challenge, a novel Adaptive Multiple-fidelity Kriging model based on the Acceptance Rejection Importance Sampling (AMK-ARIS) is developed, where the adaptive Multiple-fidelity Kriging (MK) model is employed to efficiently integrates data obtained from various fidelity sources. The proposed AMK-ARIS is composed of the following strategies: (1) the acceptance rejection importance sampling (ARIS) algorithm is proposed to select importance candidate samples; (2) a new active learning function is presented to obtain the best point for MK model in a more accurate way; (3) a novel active learning strategy is proposed to efficiently and accurately conduct reliability analysis for time-dependent problems. The validity of the proposed method is subsequently verified through both a numerical example and a practical engineering application.