Hostname: page-component-7dd5485656-wjfd9 Total loading time: 0 Render date: 2025-10-27T23:27:24.413Z Has data issue: false hasContentIssue false
  • English
  • Français

Solution of finite QBD processes

Part of: Computer system organization Markov processes

Published online by Cambridge University Press:  14 July 2016

Vittoria de Nitto Personè  and
Vincenzo Grassi
Vittoria de Nitto Personè*
Affiliation:
Università di Roma
Vincenzo Grassi*
Affiliation:
Università di Roma
*
∗ Postal address: Dipartimento di Ingegneria Elettronica, Universita' Degli Studi di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Roma, Italy.
∗ Postal address: Dipartimento di Ingegneria Elettronica, Universita' Degli Studi di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Roma, Italy.
Get access Rights & Permissions [Opens in a new window]

Abstract

We provide an explicit matrix analytic solution for finite quasi birth and death (QBD) processes, directly expressed in terms of process parameters. We show that this solution has the same asymptotic complexity of previously proposed non-explicit solutions and is more general than some of them. Moreover, it can be easily extended to the case of generalized QBD processes.

Keywords

MARKOV PROCESSFINITE QBD PROCESSMATRIX-ANALYTIC SOLUTION

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 68M20: Performance evaluation; queueing; scheduling

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 4 , December 1996 , pp. 1003 - 1010
Copyright
Copyright © Applied Probability Trust 1996 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

Work partially supported by MURST funds for national project ‘Performance and Reliability V&V for Software Engineering', and for local project ‘Performance Engineering of Software Systems'.

References

De Nitto Personè, V. and Iazeolla, G. (1994) Performance analysis of the parallel cyclic two-stage queueing model. Perf. Eval. 19, 167–193.Google Scholar
Grassmann, W. K. and Heyman, D. P. (1990) Equilibrium distribution of block-structured Markov chains with repeating rows. J. Appl. Prob. 27, 557–576.Google Scholar
Gun, L. and Makowski, A. M. (1987) Matrix-geometric solution for finite queues with phase-type distributions. In Performance '87. ed. Courtois, P.-J. and Latouche, G. Elsevier, Amsterdam. pp. 269–282.Google Scholar
Hajek, B. (1982) Birth-and-death processes on the integers with phases and general boundaries. J. Appl. Prob. 19, 488–499.Google Scholar
Heffes, H. and Lucantoni, D. M. (1986) A Markov modulated characterization of packetized voice and data traffic and related statistical multiplexer performance. IEEE J. Sel. Areas Commun. 6, 803–809.Google Scholar
Le Boudec, J. Y. (1989) A generalization of matrix geometric solutions for Markov models. IBM Research Report RZ 1903 (#67353).Google Scholar
Li, S. Q. (1989) Overload control in a finite message storage buffer. IEEE Trans. Commun. 37, 1330–1338.Google Scholar
Nelson, R. (1991) Matrix-geometric solutions in Markov models – a mathematical tutorial. IBM Research Report RC 1677 (#74293).Google Scholar
Neuts, M. F. (1981) Matrix-Geometric Solutions in Stochastic Models – An Algorithmic Approach. The Johns Hopkins University Press, Baltimore, MD.Google Scholar