Toeplitz-Circulant Preconditioners for Toeplitz Systems and their Applications to Queueing Networks with Batch Arrivals

The preconditioned conjugate gradient method is employed to solve Toeplitz systems $T_n {\bf x} = {\bf b}$ where the generating functions of the n-by-n Toeplitz matrices $T_n $ are functions with zeros. In this case, circulant preconditioners are known to give poor convergence, whereas band-Toeplitz preconditioners offer only linear convergence and can handle only real-valued functions with zeros of even orders. We propose here preconditioners hich are products of band-Toeplitz matrices and circulant matrices. The band-Toeplitz matrices are used to cope with the zeros of the given generating function and the circulant matrices are used to speed up the convergence rate of the algorithm. Our preconditioner can handle complex-valued functions with zeros of arbitrary orders. We prove that the preconditioned Toeplitz matrices have singular values clustered around 1 for large n. We apply our preconditioners to solve the stationary probability distribution vectors of Markovian queueing models with batch arrivals. We show that if the number of servers is fixed independent of the queue size n, then the preconditioners are invertible and the preconditioned matrices have singular values clustered around 1 for large n. Numerical results are given to illustrate the fast convergence of our methods.