Two Types of RG-Factorizations of Quasi-birth-and-death Processes and Their Applications to Stochastic Integral Functionals

In this paper, we provide UL-type and LU-type RG-factorizations for an irreducible continuous-time level-dependent quasi-birth-and-death (QBD) process with either finitely-many levels or infinitely-many levels, and then apply the RG-factorizations to solve a class of linear QBD-equations, which is always crucial for analyzing a stochastic model described as a QBD process. Based on the results obtained for the linear QBD-equations, we analyze up-, down- and return-integral functionals. We explicitly express the Laplace transforms of the conditional distributions of the three types of stochastic integral functionals and their conditional moments.