# Componentwise accurate fluid queue computations using doubling algorithms

- 261 Downloads
- 2 Citations

## Abstract

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 Classification

Primary 65F30 60K25## Notes

### Acknowledgments

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 [2]. Both authors are grateful to helpful comments of N. Bean.

## References

- 1.Alfa, A.S., Xue, J., Ye, Q.: Accurate computation of the smallest eigenvalue of a diagonally dominant \(M\)-matrix. Math. Comput.
**71**(237), 217–236 (2002)MATHMathSciNetCrossRefGoogle Scholar - 2.Arioli, M., Codenotti, B., Fassino, C.: The Padé method for computing the matrix exponential. Linear Algebra Appl.
**240**, 111–130 (1996)MATHMathSciNetCrossRefGoogle Scholar - 3.Bean, N.G., O’Reilly, M.M., Sargison, J.E.: A stochastic fluid flow model of the operation and maintenance of power generation systems. IEEE Trans. Power Syst.
**25**(3), 1361–1374 (2010)CrossRefGoogle Scholar - 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
- 5.Bini, D.A., Gemignani, L.: Solving quadratic matrix equations and factoring polynomials: new fixed point iterations based on Schur complements of Toeplitz matrices. Numer. Linear Algebra Appl.
**12**(2–3), 181–189 (2005)MATHMathSciNetCrossRefGoogle 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
- 8.Bini, D.A., Latouche, G., Meini, B.: Numerical Methods for Structured Markov Chains. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2005)CrossRefGoogle Scholar
- 9.Bini, D.A., Meini, B., Poloni, F.: Transforming algebraic Riccati equations into unilateral quadratic matrix equations. Numer. Math.
**116**(4), 553–578 (2010)MATHMathSciNetCrossRefGoogle Scholar - 10.Crabtree, D.E., Haynsworth, E.V.: An identity for the Schur complement of a matrix. Proc. Am. Math. Soc.
**22**, 364–366 (1969)MATHMathSciNetCrossRefGoogle 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
- 13.Govorun, M., Latouche, G., Remiche, M.-A.: Stability for fluid queues: characteristic inequalities. Stoch. Models
**29**(1), 64–88 (2013)MATHMathSciNetCrossRefGoogle Scholar - 14.Guo, X.-X., Lin, W.-W., Xu, S.-F.: A structure-preserving doubling algorithm for nonsymmetric algebraic Riccati equation. Numer. Math.
**103**, 393–412 (2006)MATHMathSciNetCrossRefGoogle Scholar - 15.Higham, N.J.: The Matrix Function Toolbox. http://www.ma.man.ac.uk/higham/mftoolbox
- 16.Higham, N.J.: Accuracy and Stability of Numerical Algorithms, 2nd edn. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2002)CrossRefGoogle Scholar
- 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
- 19.Latouche, G., Taylor, P.G.: A stochastic fluid model for an ad hoc mobile network. Queueing Syst.
**63**(1), 109–129 (2009)MATHMathSciNetCrossRefGoogle 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
- 21.Meyer, C.D.: Stochastic complementation, uncoupling Markov chains, and the theory of nearly reducible systems. SIAM Rev.
**31**(2), 240–272 (1989)MATHMathSciNetCrossRefGoogle Scholar - 22.Mitra, D.: Stochastic theory of a fluid model of producers and consumers coupled by a buffer. Adv. Appl. Probab.
**20**(3), 646–676 (1988)MATHCrossRefGoogle 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
- 25.Rogers, L.C.G.: Fluid models in queueing theory and Wiener-Hopf factorization of Markov chains. Ann. Appl. Probab.
**4**, 390–413 (1994)MATHMathSciNetCrossRefGoogle Scholar - 26.Shao, M., Gao, W., Xue, J.: Aggressively truncated Taylor series method for accurate computation of exponentials of essentially nonnegative matrices. SIAM J. Matrix Anal. Appl.
**35**(2), 317–338 (2014)MATHMathSciNetCrossRefGoogle Scholar - 27.van Foreest, N., Mandjes, M., Scheinhardt, W.: Analysis of a feedback fluid model for heterogeneous TCP sources. Stoch. Models
**19**(3), 299–324 (2003)MATHMathSciNetCrossRefGoogle Scholar - 28.Wang, W.-G., Wang, W.-C., Li, R.-C.: Alternating-directional doubling algorithm for \(M\)-matrix algebraic Riccati equations. SIAM J. Matrix Anal. Appl.
**33**(1), 170–194 (2012)MATHMathSciNetCrossRefGoogle Scholar - 29.Xue, J., Xu, S., Li, R.-C.: Accurate solutions of M-matrix algebraic Riccati equations. Numer. Math.
**120**, 671–700 (2012)MATHMathSciNetCrossRefGoogle Scholar - 30.Xue, J., Ye, Q.: Entrywise relative perturbation bounds for exponentials of essentially non-negative matrices. Numer. Math.
**110**(3), 393–403 (2008)MATHMathSciNetCrossRefGoogle Scholar - 31.Xue, J., Ye, Q.: Computing exponentials of essentially non-negative matrices entrywise to high relative accuracy. Math. Comput.
**82**(283), 1577–1596 (2013)MATHMathSciNetCrossRefGoogle Scholar