Computations with Markov Chains pp 21-38 | Cite as

# On Cyclic Reduction Applied to a Class of Toeplitz-Like Matrices Arising in Queueing Problems

## Abstract

We observe that the cyclic reduction algorithm leaves unchanged the structure of a block Toeplitz matrix in block Hessenberg form. Based on this fact, we devise a fast algorithm for computing the probability invariant vector of stochastic matrices of a wide class of Toeplitz-like matrices arising in queueing problems. We prove that for any block Toeplitz matrix *H* in block Hessenberg form it is possible to carry out the cyclic reduction algorithm with *O*(*k* ^{3} *n* + *k* ^{2} *n* log *n*) arithmetic operations, where *k* is the size of the blocks and n is the number of blocks in each row and column of *H.* The probability invariant vector is computed within the same cost. This substantially improves the *O*(*k* ^{3} *n* ^{2}) arithmetic cost of the known methods based on Gaussian elimination. The algorithm, based on the FFT, is numerically weakly stable. In the case of semi-infinite matrices the cyclic reduction algorithm is rephrased in functional form by means of the concept of generating function and a convergence result is proved.

## Keywords

Stochastic Matrice Block Column Cyclic Reduction Triangular Block Block Toeplitz Matrix## Preview

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