Finite and Infinite QBD Chains: A Simple and Unifying Algorithmic Approach

I n this paper, we present a novel algorithmic approach, the hybrid matrix geometric/invariant subspace method, f o r finding the stationary probability distribut ion of the f inite QBD process which arises in performance analysis of computer and communication systems. Assuming that the QBD state space i s defined in two dimensions with m phases and K f 1 levels, the solution Vector for level k , T k , O < k < K is shown to be in a modified matrix geometric f o r m "k = VIR! + v ~ R F ~ where RI and Rz are certain solutions to two nonlinear matrix equations and u l and v2 are vectors to be determined using the boundary conditions. We show that the matrix geometric factors R.1 and R2 can simultaneously be obtained independently of K via finding the sign function of a real matrix b y an iterative algorithm with quadratic convergence rates. The time complexity of obtaining the coeficient VeCtOTS ul and v2 is shown to be O(m3 log, K ) which indicates that the contribution of the number of levels o n the overall algorithm is manamal. Besides the numerical efficiency, the proposed method is numerically stable and in the limiting case of K --t 03, i t is shown to yield the well-known matr ix geometric solution x k = TOR: for infinite QBD chain.