On Cyclic Reduction Applied to a Class of Toeplitz-Like Matrices Arising in Queueing Problems
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.
KeywordsStochastic Matrice Block Column Cyclic Reduction Triangular Block Block Toeplitz Matrix
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