Hostname: page-component-848d4c4894-p2v8j Total loading time: 0.001 Render date: 2024-05-26T14:40:44.133Z Has data issue: false hasContentIssue false
  • English
  • Français

The Compensation Approach Applied to a 2 × 2 Switch

Published online by Cambridge University Press:  27 July 2009

O. J. Boxma  and
G. J. van Houtum
O. J. Boxma
Affiliation:
CWI, P.O. Box 4079, 1009 AB Amsterdam, The Netherlands, and Tilburg University, P.O. Box 90153, 5000 LE Tilburg, The Netherlands
G. J. van Houtum
Affiliation:
Eindhoven University of Technology, Department of Mathematics and Computing Science, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we analyze an asymmetric 2 × 2 buffered switch, fed by two independent Bernoulli input streams. We derive the joint equilibrium distribution of the numbers of messages waiting in the two output buffers. This joint distribution is presented explicitly, without the use of generating functions, in the form of a sum of two alternating series of product-form geometric distributions. The method used is the so-called compensation approach, developed by Adan, Wessels, and Zijm.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 7 , Issue 4 , October 1993 , pp. 471 - 493
Copyright
Copyright © Cambridge University Press 1993

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.Adan, I.J.B.F. (1991). A compensation approach for queueing problems. Ph.D. thesis, Eindhoven University of Technology, Eindhoven, The Netherlands.Google Scholar
2.Adan, I.J.B.F., van Houtum, G.J., Wessels, J., & Zijm, W.H.M. (1993). A compensation procedure for multiprogramming queues. OR Spektrum 15: 95–106.CrossRefGoogle Scholar
3.Adan, I.J.B.F., Wessels, J., & Zijm, W.H.M. (1990). Analysis of the symmetric shortest queue problem. Stochastic Models 6: 691–713.Google Scholar
4.Adan, I.J.B.F., Wessels, J., & Zijm, W.H.M. (1991). Analysis of the asymmetric shortest queue problem. Queueing Systems 8: 1–58.CrossRefGoogle Scholar
5.Adan, I.J.B.F., Wessels, J., & Zijm, W.H.M. (1993). Analysing multiprogramming queues by generating functions. SIAM Journal of Applied Mathematics 53: 1123–1131.CrossRefGoogle Scholar
6.Adan, I.J.B.F., Wessels, J., & Zijm, W.H.M. (1993). A compensation approach for two-dimensional Markov processes. Advances in Applied Probability (to appear).CrossRefGoogle Scholar
7.Cohen, J.W. (1993). On a class of two-dimensional nearest neighbouring random walks. CWI report BS-R9220, Festschrift for Lajos Takacs. Journal of Applied Probability (to appear).Google Scholar
8.Cohen, J.W. & Boxma, O.J. (1983). Boundary value problems in queueing system analysis. Amsterdam: North-Holland.Google Scholar
9.Flatto, L. & McKean, H.P. (1977). Two queues in parallel. Communications on Pure and Applied Mathematics 30: 255–263.Google Scholar
10.Hofri, M. (1978). A generating-function analysis of multiprogramming queues. International Journal of Computer and Informational Sciences 7: 121–155.CrossRefGoogle Scholar
11.Jaffe, S. (1989). Equilibrium results for a pair of coupled discrete-time queues. Ultracomputer Note, NYA Ultracomputer Research Lab, Courant Institute of Mathematical Sciences, New York.Google Scholar
12.Jaffe, S. (1992). The equilibrium distribution for a clocked buffered switch. Probability in the Engineering and Informational Sciences 6: 425–438.Google Scholar
13.Kingman, J.F.C. (1961). Two similar queues in parallel. Annals of Mathematical Statistics 32: 1314–1323.CrossRefGoogle Scholar
14.Malyshev, V.A. (1972). Classification of two-dimensional positive random walks and almost linear semimartingales. Doklady Akademii Nauk SSSR 202: 526–528 (English translation in Soviet Mathematics Doklady 13).Google Scholar