Grassmann [1993] established a UL-type two-matrix factorization for the matrix I-P, where P is an irreducible transition probability matrix of finite size. Grassmann and Heyman [1993] gave the same factorization for an irreducible Markov chain of GI/G/1 type, while Heyman [1995] extended the result to an irreducible and positive recurrent Markov chain with infinitely-many states. Based on the Wiener-Hopf equations for the transition probability matrix of GI/G/1 type, Zhao [2000] obtained a UL-type RG-factorization for the matrix I 􀀐P for the first time. For a level dependent QBD process, Li and Cao [2004] first provided two types: UL- and LU-types of RG-factorizations. Li and Zhao [2002, 2003, 2004] generalized the UL-type RG-factorization to an irreducible block-structured Markov renewal process with
infinitely-many states, which can immediately lead to the UL-type RG-factorization for any irreducible block-structured Markov chain. Li and Liu [2004] constructed the LU-type RG-factorization for any irreducible block-structured Markov chain, which is parallel to but different from that of Li and Zhao [2004]. It is worthwhile to note that the UL-type RG-factorization is very useful for computing the stationary probability vector and more generally, analyzing the stationary performance measures; while the LU-type RG-factorization is a key for calculating the transient performance measures such as the first passage time and the sojourn time, as illustrated in Chapters 6 to 11 of the book by Q.L. Li [2010].