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.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
E. Seneta, Finite approximations to infinite non-negative matrices, Proc. Cambridge Phil. Soc. 63(1967)983.
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.
Y. Ikebe, Inverse of Hessenberg matrices, Linear Algebra and Its Applications 24(1979)93.
C.C. Paige et al., Computation of the stationary distribution of a Markov chain, J. Statist. Comput. Stimul. 4(1975)173.
M.F. Neuts,Matrix Geometric Solutions in Stochastic Models — An Algorithmic Approach (Johns Hopkins University Press, Baltimore, 1981).
R.E. Funderlic, M. Neumann and R.J. Plemmons, LU factorizations of generalized diagonally dominant matrices, Num. Math. 40(1982)57.
J.H. Wilkinson,The Algebraic Eigenvalue Problem (Clarendon Press, Oxford, 1965).
G.W. Stewart,Introduction to Matrix Computation (Academic Press, New York-San Francisco-London, 1973).
P. Snyder and W.J. Stewart, Explicit and iterative numerical approaches to solving queueing models, Oper. Res., to appear.
This work was supported in part by NSF Grant MCS-83-00438.
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
Keywords and phrases
- Queueing models
- matrix methods
- block Hessenberg matrices
- stationary probabilities
- stability considerations