A note on the passage time of finite-state Markov chains

Consider a Markov chain with finite state {0, 1, …, d}. We give the generation functions (or Laplace transforms) of absorbing (passage) time in the following two situations: (1) the absorbing time of state d when the chain starts from any state i and absorbing at state d; (2) the passage time of any state i when the chain starts from the stationary distribution supposed the chain is time reversible and ergodic. Example shows that it is more convenient compared with the existing methods, especially we can calculate the expectation of the absorbing time directly.