Finite and In nite QBD Chains : A Simple and UnifyingAlgorithmic

In this paper, we present a novel algorithmic approach , the hybrid matrix geometric/invariant subspace method, for nding the stationary probability distribution of the nite QBD process which arises in performance analysis of computer and communication systems. Assuming that the QBD state space is deened in two dimensions with m phases and K + 1 levels , the solution vector for level k, k ; 0 k K is shown to be in a modiied matrix geometric form where R 1 and R 2 are certain solutions to two nonlinear matrix equations and v 1 and v 2 are vectors to be determined using the boundary conditions. We show that the matrix geometric factors R 1 and R 2 can simultaneously be obtained independently of K via nding the sign function of a real matrix by an iterative algorithm with quadratic convergence rates. The time complexity of obtaining the coeecient vectors v 1 and v 2 is shown to be O(m 3 log 2 K) which indicates that the contribution of the number of levels on the overall algorithm is minimal. Besides the numerical ef-ciency, the proposed method is numerically stable and in the limiting case of K ! 1, it is shown to yield the well-known matrix geometric solution k = 0 R k 1 for innnite QBD chain.