Componentwise accurate fluid queue computations using doubling algorithms
Markov-modulated fluid queues are popular stochastic processes frequently used for modelling real-life applications. An important performance measure to evaluate in these applications is their steady-state behaviour, which is determined by the stationary density. Computing it requires solving a (nonsymmetric) M-matrix algebraic Riccati equation, and indeed computing the stationary density is the most important application of this class of equations. Xue et al. (Numer Math 120:671–700, 2012) provided a componentwise first-order perturbation analysis of this equation, proving that the solution can be computed to high relative accuracy even in the smallest entries, and suggested several algorithms for computing it. An important step in all proposed algorithms is using so-called triplet representations, which are special representations for M-matrices that allow for a high-accuracy variant of Gaussian elimination, the GTH-like algorithm. However, triplet representations for all the M-matrices needed in the algorithm were not found explicitly. This can lead to an accuracy loss that prevents the algorithms from converging in the componentwise sense. In this paper, we focus on the structured doubling algorithm, the most efficient among the proposed methods in Xue et al., and build upon their results, providing (i) explicit and cancellation-free expressions for the needed triplet representations, allowing the algorithm to be performed in a really cancellation-free fashion; (ii) an algorithm to evaluate the final part of the computation to obtain the stationary density; and (iii) a componentwise error analysis for the resulting algorithm, the first explicit one for this class of algorithms. We also present numerical results to illustrate the accuracy advantage of our method over standard (normwise-accurate) algorithms.
Mathematics Subject ClassificationPrimary 65F30 60K25
The first author would like to acknowledge the financial support of the Australian Research Council through the Discovery Grant DP110101663. The second author would like to acknowledge the financial support of Istituto Nazionale di Alta Matematica, and thank N. Higham and M. Shao for useful discussions on the accuracy of various algorithms for the computation of the exponential of a \(-M\)-matrix; in particular, N. Higham pointed us to the paper . Both authors are grateful to helpful comments of N. Bean.
- 4.Berman, A., Plemmons, R.J.: Nonnegative matrices in the mathematical sciences, Classics in Applied Mathematics, vol. 9. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1994, revised reprint of the 1979 original)Google Scholar
- 6.Bini, D.A., Iannazzo, B., Meini, B.: Numerical solution of algebraic Riccati equations, Fundamentals of Algorithms, vol. 9. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2012)Google Scholar
- 7.Bini, D.A., Iannazzo, B., Meini, B., Poloni, F.: Nonsymmetric algebraic Riccati equations associated with an M-matrix: recent advances and algorithms. In: Olshevsky, V., Tyrtyshnikov, E. (eds.) Matrix Methods: Theory, Algorithms and Applications, chapter 10, pp. 176–209. World Scientific Publishing (2010)Google Scholar
- 11.da Silva Soares, A.: Fluid queues—building upon the analogy with QBD processes. PhD thesis, Université libre de Bruxelles (2005)Google Scholar
- 12.Golub, G.H., Van Loan, C.F.: Matrix Computations. Johns Hopkins Studies in the Mathematical Sciences, 3rd edn. Johns Hopkins University Press, Baltimore (1996)Google Scholar
- 15.Higham, N.J.: The Matrix Function Toolbox. http://www.ma.man.ac.uk/higham/mftoolbox
- 17.Higham, N.J.: The scaling and squaring method for the matrix exponential revisited. SIAM J. Matrix Anal. Appl. 26(4), 1179–1193 (2005, electronic)Google Scholar
- 18.Higham, N.J.: Functions of Matrices. Theory and Computation. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2008)Google Scholar
- 20.Mandjes, M., Mitra, D., Scheinhardt, W.: Simple models of network access, with applications to the design of joint rate and admission control. In: INFOCOM 2002. Proceedings of Twenty-First Annual Joint Conference of the IEEE Computer and Communications Societies. IEEE, vol. 1, pp. 3–12 (2002)Google Scholar
- 23.Poloni, F., Reis, T.: The SDA method for numerical solution of Lur’e equations. Technical Report 1101.1231, arXiv.org (2011)Google Scholar
- 24.Ramaswami, V.: Matrix analytic methods for stochastic fluid flows. In: Smith, D., Hey, P. (eds.) Proceedings of the 16th International Teletraffic Congress. Teletraffic Engineering in a Competitive World, pp. 1019–1030. Elsevier Science B.V, Edinburgh (1999)Google Scholar