The core content in this paper discusses the Bayesian approach, which essentially estimates and predicts the reliability of a repairable system during the actual testing process. As specified in previous literature, the nonhomogeneous Poisson process (NHPP) with a power-law intensity function has often been employed as a model for describing the failure process of repairable systems since it provides a reasonable way of presenting the rate of change for the reliability of a system as a function of time. This research proposed the same time-truncated sampling assumption to mainly ensure that the joint posterior distribution of the power-law failure process has a considerably closed-form representation. This proposed approach facilitates the analysis of the actual performance, with which analysts can provide the prior information about the authentic means and variations of the shape and scale parameters of the power-law intensity. The corresponding posterior means and variations can then be significantly and easily obtained without any complex integral arithmetic. Using Monte Carlo simulation for comparison, the Bayesian estimation with the proposed prior outperforms the maximum likelihood estimation, particularly when the prior aging of the repairable is underestimated.