Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-24T22:46:39.886Z Has data issue: false hasContentIssue false

Decay rates for quasi-birth-and-death processes with countably many phases and tridiagonal block generators

Published online by Cambridge University Press:  01 July 2016

Allan J. Motyer*
Affiliation:
University of Melbourne
Peter G. Taylor*
Affiliation:
University of Melbourne
*
Postal address: Department of Mathematics and Statistics, University of Melbourne, Parkville, VIC 3010, Australia.
Postal address: Department of Mathematics and Statistics, University of Melbourne, Parkville, VIC 3010, Australia.
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.

We consider the class of level-independent quasi-birth-and-death (QBD) processes that have countably many phases and generator matrices with tridiagonal blocks that are themselves tridiagonal and phase independent. We derive simple conditions for possible decay rates of the stationary distribution of the ‘level’ process. It may be possible to obtain decay rates satisfying these conditions by varying only the transition structure at level 0. Our results generalize those of Kroese, Scheinhardt, and Taylor, who studied in detail a particular example, the tandem Jackson network, from the class of QBD processes studied here. The conditions derived here are applied to three practical examples.

Type
General Applied Probability
Copyright
Copyright © Applied Probability Trust 2006 

References

Bean, N. G., Pollett, P. K. and Taylor, P. G. (2000). Quasistationary distributions for level dependent quasi-birth-and-death processes. Stoch. Models 16, 511541.Google Scholar
Chihara, T. S. (1978). An Introduction to Orthogonal Polynomials. Gordon and Breach, New York.Google Scholar
Fayolle, G., Iasnogorodski, R. and Malyshev, V. (1999). Random Walks in the Quarter-Plane. Springer, Berlin.Google Scholar
Flatto, L. and Hahn, S. (1984). Two parallel queues created by arrivals with two demands. SIAM J. Appl. Math. 44, 10411053.Google Scholar
Foley, R. D. and McDonald, D. R. (2001). Join the shortest queue: stability and exact asymptotics. Ann. Appl. Prob. 11, 569607.Google Scholar
Gail, H. R., Hantler, S. L. and Taylor, B. A. (1996). Spectral analysis of M/G/1 and G/M/1 type Markov chains. Adv. Appl. Prob. 28, 114165.Google Scholar
Haque, L., Zhao, Y. Q. and Liu, L. (2005). Sufficient conditions for a geometric tail in a QBD process with many countable levels and phases. Stoch. Models 21, 7799.Google Scholar
Kroese, D. P., Scheinhardt, W. R. W. and Taylor, P. G. (2004). Spectral properties of the tandem Jackson network, seen as a quasi-birth-and-death process. Ann. Appl. Prob. 14, 20572089.Google Scholar
Latouche, G. and Ramaswami, V. (1999). Introduction to Matrix Analytic Methods in Stochastic Modeling. ASA-SIAM, Philadelphia, PA.Google Scholar
Latouche, G. and Taylor, P. G. (2000). Level-phase independence in processes of GI/M/1 type. J. Appl. Prob. 37, 984998.Google Scholar
Latouche, G. and Taylor, P. G. (2002). Drift conditions for matrix-analytic models. Math. Operat. Res. 28, 346360.Google Scholar
Li, Q. L. and Zhao, Y. Q. (2003). β-invariant measures for transition matrices of GI/M/1 type. Stoch. Models 19, 201233.Google Scholar
Li, Q. L. and Zhao, Y. Q. (2005). Heavy-tailed asymptotics of stationary probability vectors of Markov chains of GI/G/1 type. Adv. Appl. Prob. 37, 482509.CrossRefGoogle Scholar
Li, Q. L. and Zhao, Y. Q. (2005). Light-tailed asymptotics of stationary probability vectors of Markov chains of GI/G/1 type. Adv. Appl. Prob. 37, 10751093.Google Scholar
Malyshev, V. A. (1973). Asymptotic behaviour of stationary probabilities for two-dimensional positive random walks. Siberian Math. J. 14, 156169.Google Scholar
Miyazawa, M. (2002). A Markov renewal approach to the asymptotic decay of the tail probabilities in risk and queuing processes. Prob. Eng. Inf. Sci. 16, 139150.Google Scholar
Neuts, M. F. (1981). Matrix-Geometric Solutions in Stochastic Models. Johns Hopkins University Press, Baltimore, MD.Google Scholar
Ramaswami, V. and Taylor, P. G. (1996). Some properties of the rate operators in level dependent quasi-birth-and-death processes with a countable number of phases. Stoch. Models 12, 143164.CrossRefGoogle Scholar
Sakuma, Y. and Miyazawa, M. (2005). On the effect of finite buffer truncation in a two-node Jackson network. J. Appl. Prob. 42, 199222.Google Scholar
Seneta, E. (1981). Non-Negative Matrices and Markov Chains. Springer, New York.Google Scholar
Takahashi, Y., Fujimoto, K. and Makimoto, N. (2001). Geometric decay of the steady-state probabilities in a quasi-birth-and-death process with a countable number of phases. Stoch. Models 17, 124.Google Scholar
Tweedie, R. L. (1982). Operator-geometric stationary distributions for Markov chains with applications to queueing models. Adv. Appl. Prob. 14, 368391.Google Scholar