On Cyclic Reduction Applied to a Class of Toeplitz-Like Matrices Arising in Queueing Problems

  • Dario Bini
  • Beatrice Meini
Conference paper

Abstract

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.

Keywords

Convolution Tate Deconvolution Bini 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    J. L. Barlow, On the smallest positive singular value of a singular M-matrix with applications to ergodic Markov Chains, SIAM J. Alg. Discr. Meth., 7 (1986), pp. 414–424.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    J. L. Barlow, Error bounds for the computation of null vectors with application to Markov chains, SIAM J. Matrix Anal. Appl., 14 (1993), pp. 798–812.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    D. Bini, V. Pan, Matrix and Polynomial Computations, Vol. 1: Fundamental Algorithms, Birkhäuser, Boston 1994.Google Scholar
  4. [4]
    E. Çinlar, Introduction To Stochastic Processes, Prentice-hall, Englewood Cliffs, N.J., 1975.MATHGoogle Scholar
  5. [5]
    C.A. O’Cinneide, Entrywise perturbation theory and error analysis for Markov chains, Nuttier. Math., 65, (1993), pp. 109–120.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    R. E. Funderlic, J. B. Mankin, Solution of homegeneous systems of linear equations arising in compartmental models, SIAM J. Sci. Stat. Comput., 2 (1981), pp. 375–383.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    R. E. Funderlic, R. J. Plemmons, LU decomposition of M-matrices by elimination without pivoting, Linear Algebra Appl., 41 (1981), pp. 99–110.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    R. E. Funderlic, R. J. Plemmons, A combined direct-iterative method for certain M-matrix linear systems, SIAM J. Alg. Discr. Meth., 5 (1984), pp. 33–42.MathSciNetCrossRefGoogle Scholar
  9. [9]
    W. K. Grassman, M. I. Taksar, D. P. Heyman, Regenerative analysis and steady state distribution for Markov chains, Oper. Res., 33 (1985), pp. 1107–1116.MathSciNetCrossRefGoogle Scholar
  10. [10]
    W. J. Harrod, Rank modification methods for certain singular systems of linear equations, Ph. D. thesis, Univ. of Tennessee, Knoxville. TN, Department of Mathematics, 1984.Google Scholar
  11. [11]
    W. J. Harrod, R. J. Plemmons, Comparison of some direct methods for computing stationary distributions of Markov chains, J. Sci. Stat. Comput., 5 (1984), pp. 453–469.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    I.S. Iohvidov, Hankel and Toeplitz Matrices and Forms, Algebraic Theory, Birkhäuser, Boston 1982.MATHGoogle Scholar
  13. [13]
    T. Kailath, A. Viera, M. Morf, Inverses of Toeplitz operators, innovations and orthogonal polynomials, SIAM Rev., 20, (1978), pp. 106–119.MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    L. Kaufman, Matrix methods for queueing problems, J. Sci. Stat. Comput, 4 (1983), pp. 525–552.MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    D.E. Knuth, The Art of Computer Programming: Seminumerical Algorithms, 2, Addison Wesley, Reading, Mass., 1981.MATHGoogle Scholar
  16. [16]
    R. Kouri, D. F. McAllister, W. J. Stewart, A numerical comparison of block iterative methods with aggregation for computing stationary probability vectors for nearly completely decomposable Markov Chains, SIAM J. Alg. Discr. Meth., 5 (1984), pp. 164–186.CrossRefGoogle Scholar
  17. [17]
    G. Latouche, V. Ramaswami, A logarithmic reduction algorithm for quasi-birthdeath processes, J. Appl. Prob., 30, (1993), pp. 650–674.MathSciNetMATHCrossRefGoogle Scholar
  18. [18]
    E. Linzer, On the stability of transform-based circular deconvolution, SIAM J. Numer. Anal., 29, (1992), pp. 1482–1492.MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    M.F. Neuts, Structured Stochastic Matrices of M/G/1 Type and Their Applications,Dekker Inc., New York, 1989.MATHGoogle Scholar
  20. [20]
    D. J. Rose, Convergent regular splittings for singular M-matrices, SIAM J. Alg. Discr. Meth., 5 (1984), pp. 133–144.MATHCrossRefGoogle Scholar
  21. [21]
    G.W. Stewart, On the solution of block Hessenberg systems, CS-TR-2973, Department of Computer Science, University of Maryland, 1992, To appear in Numerical Linear Algebra and Applications.Google Scholar
  22. [22]
    R. S. Varga, D.-Y. Cai, On the LU factorization of M-matrices, Numer. Math., 38 (1981), pp. 179–192.MathSciNetMATHCrossRefGoogle Scholar
  23. [23]
    J. Ye, San-qi Li, Analisys of multi-media traffic queues with finite buffer and overload control—Part I: algorithm, Proc. IEEE Infocurn 91, Bal Harbour, 1464–1474.Google Scholar

Copyright information

© Springer Science+Business Media New York 1995

Authors and Affiliations

  • Dario Bini
    • 1
  • Beatrice Meini
    • 1
  1. 1.Dipartimento di MatematicaUniversità di PisaPisaItaly

Personalised recommendations