Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Queueing models, block hessenberg matrices and the method of neuts

  • 59 Accesses

Abstract

We introduce a numerical method to compute the stationary probability vector of queueing models whose infinitesimal generator is of block Hessenberg form. It is shown that the stationary probability vector is equal to the first column of the inverse of the coefficient matrix. Furthermore, it is shown that the first column of the inverse of an upper (or lower) Hessenberg matrix may be obtained in a relatively small number of operations. Together, these results allow us to define a powerful algorithm for solving certain queueing models. The efficiency of this algorithm is discussed and a comparison with the method of Neuts is undertaken. A relationship with the method of Gaussian elimination is established and used to develop some stability results.

This is a preview of subscription content, log in to check access.

References

  1. [1]

    E. Seneta, Finite approximations to infinite non-negative matrices, Proc. Cambridge Phil. Soc. 63(1967)983.

  2. [2]

    E. Seneta, Computing the stationary distribution for infinite Markov chains, in:Large-Scale Matrix Problems, ed. A. Bjorck, R.J. Plemmons and H. Schneider (1981) p. 259.

  3. [3]

    Y. Ikebe, Inverse of Hessenberg matrices, Linear Algebra and Its Applications 24(1979)93.

  4. [4]

    C.C. Paige et al., Computation of the stationary distribution of a Markov chain, J. Statist. Comput. Stimul. 4(1975)173.

  5. [5]

    M.F. Neuts,Matrix Geometric Solutions in Stochastic Models — An Algorithmic Approach (Johns Hopkins University Press, Baltimore, 1981).

  6. [6]

    R.E. Funderlic, M. Neumann and R.J. Plemmons, LU factorizations of generalized diagonally dominant matrices, Num. Math. 40(1982)57.

  7. [7]

    J.H. Wilkinson,The Algebraic Eigenvalue Problem (Clarendon Press, Oxford, 1965).

  8. [8]

    G.W. Stewart,Introduction to Matrix Computation (Academic Press, New York-San Francisco-London, 1973).

  9. [9]

    P. Snyder and W.J. Stewart, Explicit and iterative numerical approaches to solving queueing models, Oper. Res., to appear.

Download references

Author information

Additional information

This work was supported in part by NSF Grant MCS-83-00438.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Cao, W., Stewart, W.J. Queueing models, block hessenberg matrices and the method of neuts. Ann Oper Res 8, 265–284 (1987). https://doi.org/10.1007/BF02187097

Download citation

Keywords and phrases

  • Queueing models
  • matrix methods
  • block Hessenberg matrices
  • stationary probabilities
  • stability considerations