Inverting block Toeplitz matrices in block Hessenberg form by means of displacement operators: Application to queueing problems

The concept of displacement rank is used to devise an algorithm for the inversion of an n × n block Toeplitz matrix in block Hessenberg form Hn having m × m block entries. This kind of matrices arises in many important problems in queueing theory. We explicitly relate the first and last block rows and block columns of H−1n with the corresponding ones of H−1n/2. These block vectors fully define all the entries of H−1n by means of a Gohberg-Semencul-like formula. In this way we obtain a doubling algorithm for the computation of H−12i, i = 0, 1,…, q, n = 2q, where at each stage of the doubling procedure only a few convolutions of block vectors must be computed. The overall cost of this computation is O(m2n log n + m3n) arithmetic operations with a moderate overhead constant. The same technique can be used for solving the linear system Hnx = b within the same computational cost. The case where Hn is in addition to a scalar Toeplitz matrix is analyzed as well. An application to queueing problems is presented, and comparisons with existing algorithms are performed showing the higher efficiency and reliability of this approach