New First Passage Times and Their Distributions

The First Passage Time (FPT) is a very important index in the real world, which is ubiquitous in physics, chemistry, and biology, with numerous examples of applications, especially in reliability field, the lifetime is a kind of FPTs. A huge literature has been given on the FPTs, some of them have been appeared in Nature, Science and other famous journals. However, the current FPT cannot cover the increasing needs in scientific research and engineering. Therefore, some new FPTs are needed from a point-view of theory and applications. In this note, two new FPT are introduced for a given finite state and continuous time Markov process. After giving the definitions of FPTs, the detailed contents such as distributions of FPTs and the distributions of the number of sojourn times until the realizations of the FPTs are presented. The Laplace transform technique is used in the derivation of formulae. Finally, the future researches on FPTs shall be discussed.