A novel computational method for solving finite qbd processes

Abstract

We present a novel numerical method that exploits invariant subspace computations for finding the stationary probability distribution of a finite QBD process. Assuming that the QBD state space is defined in two dimensions with m phases and K+1 levels, the solution vector π k for level , is known to be expressible in the mixed matrix-geometric form , where R 1and R 2are certain solutions to two quadratic matrix equations, and v 1and v 2are vectors to be determined using the boundary conditions. We show that the matrix-geometric factors R 1and R 2can be simultaneously obtained irrespective of K via finding arbitrary bases for the left-and right-invariant subspaces of a certain real matrix of size 2m To find these bases, we employ either Schur decomposition or a matrix-sign function iteration with quadratic convergence rate. The vectors v 1and v 2are obtained by solving a linear matrix equation, which is constructed with a time complexity of . Therefore, the effect of the number of levels on the overall complexity is minimal. Besides its numerical efficiency, the proposed method is numerically stable, and in the limiting case of , it is shown to yield the well-known matrix-geometric solution for infinite QBD processes. We also extend the method to the cases of level-dependent and non-canonical transitions, and provide a numerical example to demonstrate its computational features in comparison with other well-known methods