Hostname: page-component-76fb5796d-qxdb6 Total loading time: 0 Render date: 2024-04-27T02:26:46.190Z Has data issue: false hasContentIssue false

On the quasi-stationary distribution for queueing networks with defective routing

Published online by Cambridge University Press:  17 February 2009

Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This note introduces quasi-local-balance for discrete-time Markov chains with absorbing states. From quasi-local-balance product-form quasi-stationary distributions are derived by analogy with product-form stationary distributions for Markov chains that satisfy local balance.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1997

References

[1]Boucherie, R. J. and van Dijk, N. M., “Product-forms for queueing networks with state dependent multiple job transitions”, Adv. Appl. Prob. 23 (1991) 152187.CrossRefGoogle Scholar
[2]Darroch, J. N. and Seneta, E., “On quasi-stationary distributions in absorbing discrete-time finite Markov chains”, J. Appl. Prob. 2 (1965) 88100.CrossRefGoogle Scholar
[3]Darroch, J. N. and Seneta, E., “On quasi-stationary distributions in absorbing continuous-time finite Markov chains”, J. Appl. Prob. 4 (1967) 192196.CrossRefGoogle Scholar
[4]Henderson, W. and Taylor, P. G., “Product-forms in networks of queues with batch arrival and batch services”, Queueing Systems 6 (1990) 7188.CrossRefGoogle Scholar
[5]Kesten, H., “A ratio limit theorem for (sub) Markov chains on {1, 2,…} with bounded jumps”, Adv. Appl. Prob. 27 (1995) 652691.CrossRefGoogle Scholar
[6]Seneta, E., Non-negative matrices and Markov chains (Springer-Verlag, New-York, 1981).CrossRefGoogle Scholar
[7]Seneta, E. and Vere-Jones, D., “On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states”, J. Appl. Prob. 3 (1966) 403434.CrossRefGoogle Scholar
[8]Vere-Jones, D., “Geometric ergodicity in denumerable Markov chains”, Quart. J. Math. Oxford 13 (1962) 728.CrossRefGoogle Scholar