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A system is subject to shocks that arrive according to a non-homogeneous Poisson process. As these shocks occur, the system experiences one of two types of failures: a type-I failure (minor), rectified by a minimal repair; or a type-II failure (catastrophic) that calls for a replacement. In this paper, we consider a periodic replacement model with minimal repair based on a cumulative repair-cost limit. Under such a policy, the system is anticipatively replaced at the n -th type-I failure, or at the k-th type-I failure (k <; n) at which the accumulated repair cost exceeds the pre-determined limit, or at any type-II failure, whichever occurs first. The minimum-cost replacement policy is studied by showing its existence, uniqueness, and structural properties. Our model is a generalization of several classical models in maintenance literature. Some numerical analyses are also presented.