Hostname: page-component-848d4c4894-4rdrl Total loading time: 0 Render date: 2024-06-30T13:56:41.467Z Has data issue: false hasContentIssue false
  • English
  • Français

Combinatorial techniques for M/G/1-type queues

Part of: Special processes Combinatorics

Published online by Cambridge University Press:  14 July 2016

G. Mercankosk ,
G. M. Nair  and
W. J. Soet
G. Mercankosk*
Affiliation:
University of Western Australia
G. M. Nair*
Affiliation:
Curtin University of Technology
W. J. Soet*
Affiliation:
Curtin University of Technology
*
∗ Postal address: TEN Research Group (EE), University of Western Australia, 35 Stirling Hwy, Crawley 6009, Australia.
∗∗ Postal address: Department of Mathematics and Statistics, Curtin University of Technology, GPO Box U 1987, Perth 6845, Australia.
∗∗ Postal address: Department of Mathematics and Statistics, Curtin University of Technology, GPO Box U 1987, Perth 6845, Australia.
Get access Rights & Permissions [Opens in a new window]

Abstract

The application of the generalised ballot theorem to queueing theory leads to elegant results for the simple M/G/1 queue. It is thought that such results are not possible for more general M/G/1-type queues. We, however, derive a batch ballot theorem which can be applied to derive the first passage distribution matrix, G, for the general M/G/1-type queue.

Keywords

M/G/1-type Markov queuesnonlinear matrix equationsballot theoremscombinatorial techniques

MSC classification

Primary: 60K25: Queueing theory
Secondary: 05A05: Permutations, words, matrices 05A19: Combinatorial identities, bijective combinatorics
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 3 , September 2001 , pp. 722 - 736
Copyright
Copyright © by the Applied Probability Trust 2001 

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

Akar, N., and Sohraby, K. (1997). An invariant subspace approach in M/G/1 and G/M/1 type Markov chains. Commum. Statist. Stoch. Models 13, 381–416.Google Scholar
Bailey, N. T. J. (1954). On queueing processes with bulk service. J. R. Statist. Soc. B 16, 80–87.Google Scholar
Bini, D., and Meini, B. (1996). On the solution of a nonlinear matrix equation arising in queueing problems. SIAM J. Matrix Anal. Appl. 17, 906–926.Google Scholar
Mendelson, H. (1982). A batch-ballot problem and applications. J. Appl. Prob. 19, 144–157.Google Scholar
Neuts, M. F. (1989). Structured Stochastic Matrices of (M/G/1) Type and their Applications. Marcel Dekker, New York.Google Scholar
Ramaswami, V. (1988). A stable recursion for the steady state vector in Markov chains of (M/G/1) type. Commum. Statist. Stoch. Models 4, 183–188.Google Scholar
Takács, L. (1962). A generalization of the ballot theorem and its application in the theory of queues. J. Amer. Statist. Assoc. 57, 327–337.Google Scholar
Takács, L. (1967). Combinatorial Methods in the Theory of Stochastic Processes. John Wiley, New York.Google Scholar