On the Solution of a Nonlinear Matrix Equation arising in Queueing Problems

• Authors: Dario Bini;Beatrice Meini
• Venue: SIAM Journal on Matrix Analysis and Applications
• Year: 1996

By extending the cyclic reduction technique to infinite block matrices we devise a new algorithm for computing the solution $G_0$ of the matrix equation $G=\sum_{i=0}^{+\infty}G^iA_i$ arising in a wide class of queueing problems. Here $A_i$, $i=0,1,\ldots,$ are $k\times k$ nonnegative matrices such that $\sum_{i=0}^{+\infty}A_i$ is column stochastic. Our algorithm, which under mild conditions generates a sequence of matrices converging quadratically to $G_0$, can be fully described in terms of simple operations between matrix power series, i.e., power series in $z$ having matrix coefficients. Such operations, like multiplication and reciprocation modulo $z^m$, can be quickly computed by means of FFT-based fast polynomial arithmetic; here $m$ is the degree where the power series are numerically cut off in order to reduce them to polynomials. These facts lead to a dramatic reduction of the complexity of solving the given matrix equation; in fact, $O(k^3m+k^2 m \log m)$ arithmetic operations are sufficient to carry out each iteration of the algorithm. Numerical experiments and comparisons performed with the customary techniques show the effectiveness of our algorithm. For a problem arising from the modelling of metropolitan networks, our algorithm was about 30 times faster than the algorithms customarily used in the applications. Cyclic reduction applied to quasi-birth--death (QBD) problems, i.e., problems where $A_i= O$ for $i2$, leads to an algorithm similar to the one of [Latouche and Ramaswami, J. Appl. Probab., 30 (1993), pp. 650--674], but which has a lower computational cost.