Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-22T17:27:46.587Z Has data issue: false hasContentIssue false
  • English
  • Français

The solution of certain two-dimensional Markov models

Published online by Cambridge University Press:  01 July 2016

G. Fayolle ,
P. J. B. King  and
I. Mitrani
G. Fayolle*
Affiliation:
INRIA
P. J. B. King*
Affiliation:
University of Newcastle upon Tyne
I. Mitrani*
Affiliation:
University of Newcastle upon Tyne
*
∗Postal address: INRIA, Domaine de Voluceau-Rocquencourt, B.P. 105–78150 Le Chesnay, France.
∗∗Postal address: Computing Laboratory, University of Newcastle upon Tyne, Newcastle upon Tyne NE1 74U, U.K.
∗∗Postal address: Computing Laboratory, University of Newcastle upon Tyne, Newcastle upon Tyne NE1 74U, U.K.
Get access Rights & Permissions [Opens in a new window]

Abstract

A class of two-dimensional birth-and-death processes, with applications in many modelling problems, is defined and analysed in the steady state. These are processes whose instantaneous transition rates are state-dependent in a restricted way. Generating functions for the steady-state distribution are obtained by solving a functional equation in two variables. That solution method lends itself readily to numerical implementation. Some aspects of the numerical solution are discussed, using a particular model as an example.

Keywords

ANALYTIC FUNCTIONBIRTH-AND-DEATH PROCESSGENERATING FUNCTIONRIEMANN-HILBERT PROBLEMSTATE DEPENDENCY
Type
Research Article
Information
Advances in Applied Probability , Volume 14 , Issue 2 , June 1982 , pp. 295 - 308
Copyright
Copyright © Applied Probability Trust 1982 

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.)

References

[1] Courtois, P. J. (1977) Decomposability: Queueing and Computer Systems Applications. Academic Press, New York.Google Scholar
[2] Fayolle, G. and Iasnogorodski, R. (1979) Two coupled processors: the reduction to a Riemann-Hilbert problem. Z. Wahrscheinlichkeitsth. 47, 325–351.CrossRefGoogle Scholar
[3] Fayolle, G. (1979) Méthodes analytiques pour les files d'attente couplées. , Université Paris VI.Google Scholar
[4] Hine, J. H., Mitrani, I. and Tsur, S. (1979) The control of response times in multi-class systems by memory allocation. Commun. Assoc. Comput. Mach. 22, 416–424.Google Scholar
[5] Iasnogorodski, R. (1979) Problèmes-frontières dans les files d'attente. , Université Paris VI.Google Scholar
[6] Mitrani, I. and King, P. J. B. (1981) Multiprocessor system with preemptive priorities.CrossRefGoogle Scholar
[7] Smirnov, V. I. (1964) A Course of Higher Mathematics, Vol. 3. Part 2. Pergamon Press, Oxford.Google Scholar