Fundamental matrix of transient QBD generator with finite states and level dependent transitions

Fundamental matrix plays an important role in a finite-state Markov chain to find many characteristic values such as stationary distribution, expected amount of time spent in the transient state, absorption probabilities. In this paper, the fundamental matrix of the finite-state quasi-birth-and-death (QBD) process with absorbing state and level dependent transitions is considered. We show that each block component of the fundamental matrix can be expressed as a matrix product form and present an algorithm for computing the fundamental matrix. Some applications with numerical results are also presented.