In this paper we consider two kinds of sequential checkpoint placement problems with infinite/finite time horizon. For these problems, we apply the approximation methods based on the variational principle and develop the computation algorithms to derive the optimal checkpoint sequence approximately. Next, we focus on the situation where the knowledge on system failure is incomplete, i.e. the system failure time distribution is unknown. We develop the so-called min-max checkpoint placement methods to determine the optimal checkpoint sequence under the uncertain circumstance in terms of the system failure time distribution. In numerical examples, we investigate quantitatively the min-max checkpoint placement methods, and refer to their potential applicability in practice.