A note on the passage time of finite state Markov chains

TL;DR

This paper derives generating functions for passage times in finite state Markov chains, providing a more convenient method for calculating expectations of absorption and passage times in specific scenarios.

Contribution

It introduces new formulas for passage time distributions in finite Markov chains, simplifying calculations especially for expectations.

Findings

Derived generating functions for absorption times at state d from any starting state.

Provided formulas for passage times starting from the stationary distribution in reversible, ergodic chains.

Showed the method's advantage over existing techniques in ease of computation.

Abstract

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.