On Computational Aspects of the Invariant Subspace Approach to Teletraac Problems and Comparisons

A large class of teletraac problems are based on Markov chains of M/G/1 and G/M/1 type, the study of which requires numerically eecient and reliable algorithms to solve the nonlinear matrix equations arising in such chains. Given A i ; i 0, a sequence of square m-by-m nonnegative matrices and A(z) = P 1 i=0 A i z i with A(1) being an irreducible stochastic matrix, these equations for M/G/1 type Markov chains are of the form G = P 1 i=0 A i G i. The traditional transform approach to solve these chains which requires root nding are known to cause problems when some roots are close or identical. The alternative iterative schemes based on matrix-analytical methods have in general low linear convergence rates. Assuming that A(z) is a rational matrix, by which we cover a rich variety of teletraac models, a new theory that is suitable for eecient and reliable computation has recently been reported in 4]. Based on these results, we basically develop two class of algorithms and compare these with the existing algorithms in terms of execution time, accuracy, and numerical robustness via numerical examples. The rst class of algorithms is based on a reduction of the nonlinear equation above of possibly innnite terms to a matrix polynomial equation of the form G = D ?1 (G)N(G) via writing A(z) in terms of a fraction of two polynomial matrices, i.e., A(z) = D ?1 (z)N(z). By this reduction, signiicant gains in space requirement and execution times can be achieved since truncation is no more a requirement for computational purposes. The key to the second class of algorithms is the numerically eecient computation of a certain invariant subspace of an mf mf matrix with high convergence rates, which nds its roots in the solution of the well-known Riccati equations of control theory. Here, f = max(d + 1; n) where d and n are the degrees of the polynomial matrices D and N, respectively. This approach is especially eeective when f is not very large.