In assessing reliability, two types of uncertainties exist: (1) aleatory (irreducible) variability and (2) epistemic (reducible) uncertainty. Reliability is a function of the aleatory variability and the epistemic uncertainty hinders us from accurately assessing reliability. The simulation-based methods can predict accurate reliability of the physical system provided (1) accurate input distribution models for variabilities; and (2) accurate simulation and surrogate models. However, first, only limited numbers of test data are used in practical applications for modeling input distributions. Secondly, the simulation and surrogate models could be biased. Finally, for accurate target output distribution for validation of simulation and surrogate models, a large number of output test data is required, which is prohibitively expensive. The insufficient input and output test data induce epistemic uncertainties in input distribution models and target output distribution, respectively. Thus, reliability becomes uncertain and follows certain distribution. In this paper, surrogate modeling and computational methods are developed to obtain confidence-based reliability assessment and uncertainty quantification with limited numbers of input and output physical test data. As the surrogate model induces epistemic uncertainty, it is desirable to reduce the uncertainty by developing accurate surrogate models. In this paper, the dynamic Kriging (DKG) modeling method with a local window concept is introduced. To combine uncertainties induced by limited input and output test data as well as biased simulation and surrogate models, a hierarchical Bayesian model is formulated. Once the epistemic uncertainty distribution of the reliability is obtained, the user can select a target confidence level. At the target confidence level, the confidence-based target output PDF and reliability can be obtained, which are confidence-based estimations of the true output PDF and reliability.