In 1985, Grassmann, Taksar, and Heyman [2] published their celebrated paper introducing a numerically stable algorithm for computing the stationary probabilities of a finite-state Markov chain. This algorithm later became known as the GTH-algorithm (or the state-reduction method) and has since become one of the standard tools in applied probability.

In 1990, the algorithm was extended by Grassmann and Heyman [1] to handle stationary distributions of block-structured Markov chains with repeating rows. This extension allowed for a broader class of Markov chains to be analyzed effectively using the same fundamental principles.

In this talk, we first provide probabilistic interpretations for all components of the GTH-algorithm in the context of block-structured Markov chains. We then demonstrate how, through the concept of censoring, the GTH-algorithm can be generalized to infinite-state Markov chains.

Finally, we show that the RG-factorization is analogous to LU-decomposition in Gaussian elimination and is mathematically equivalent to the GTH-algorithm. It is also noteworthy that the censored Markov chain offers a minimal error approximation to the stationary distribution of the original chain under the

ℓ1ℓ 1 -norm.