In this paper we consider aperiodic checkpoint placement problems with equality constraints over an infinite time horizon and develop both exact and approximate algorithms to determine the optimal checkpoint sequences minimizing the relevant expected costs. More precisely, the problem is to minimize the expected recovery cost (expected checkpointing cost) subject to a given level of the expected checkpointing cost (expected recovery cost). First, we develop exact computation algorithms to derive the optimal aperiodic checkpoint sequence by applying the Lagrange multiplier. Second, we propose approximate algorithms based on the variational calculus approach. Numerical examples are devoted to compare two computation algorithms in terms of both accuracy of the resulting checkpoint sequences and their computation efficiency.