A Matrix-Multiplicative Solution for Multi-Dimensional QBD Processes

Abstract

We consider an irreducible positive-recurrent discrete-time Markov process on the state space X=ℤ+M×J, where ℤ+ is the set of non-negative integers and J={1,2,…,n}. The number of states in J may be either finite or infinite. We assume that the process is a homogeneous quasi-birth-and-death process (QBD). It means that the one-step transition probability between non-boundary states (k,i) and (n,j) may depend on i,j, and n−k but not on the specific values of k and n. It is shown that the stationary probability vector of the process is expressed through square matrices of order n, which are the minimal non-negative solutions to nonlinear matrix equations.

Keywords

• discrete-time Markov chain
• multi-dimensional QBD
• matrix-geometric solution
• matrix-multiplicative solution