 
Summary: A Novel Computational Method for Solving
Finite QBD Processes \Lambda
Nail Akar Nihat C. O–guz and Khosrow Sohraby y
Sprint Corporation Computer Science Telecommunications
9300 Metcalf Avenue University of MissouriKansas City
Overland Park, KS 66212 5100 Rockhill Road, Kansas City, MO 64110
akar@sprintcorp.com fncoguz,sohrabyg@cstp.umkc.edu
Abstract: We present a novel numerical method that exploits invariant subspace computa
tions 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 ß k for level k, 0 Ÿ k Ÿ K, is known to be expressible in the mixed matrix
geometric form ß k = v 1 R k
1 +v 2 R K \Gammak
2 , where R 1 and R 2 are certain solutions to two quadratic
matrix equations, and v 1 and v 2 are vectors to be determined using the boundary condi
tions. We show that the matrixgeometric factors R 1 and R 2 can be simultaneously obtained
irrespective of K via finding arbitrary bases for the left and rightinvariant subspaces of a
certain real matrix of size 2m. To find these bases, we employ either Schur decomposition or
a matrixsign function iteration with quadratic convergence rate. The vectors v 1 and v 2 are
obtained by solving a linear matrix equation, which is constructed with a time complexity
