Approximate analysis of networks of PH¦PH|1¦K queues: Theory & tool support

  • Boudewijn R. Haverkort
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 977)


We address the approximate analysis of open networks of PH¦PH¦1 and PH¦PH¦1¦K queues. We start from the analysis of open queueing networks (QNs) as proposed by Whitt, where large QNs are decomposed into individual GI¦G¦1 queues, characterized by the first and second moment of the service and interarrival time distribution. We extend this approach in two ways.

First of all, we use PH¦PH¦1 queues, instead of GI¦G¦1 queues, so that the individual queues can be solved exactly, using matrix-geometric techniques. Secondly, we allow for the inclusion of finite-buffer queues. In doing so, the proposed decomposition becomes an iterative process.

We present the mathematical background of the approach as well as a tool implementation (Qnaut). It turns out that our approach not only yields accurate results (within a few percents from simulation results) but also is very fast in obtaining them (in comparison with simulation).


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    L. Gün, A.M. Makowski, “Matrix-Geometric Solutions for Finite-Capacity Queues with Phase-Type Distributions”, in: Performance'87, Editors: P.J. Courtois, G. Latouche, pp.269–282, 1988.Google Scholar
  2. 2.
    B.R. Haverkort, A.P.A. van Moorsel, D.-J. Speelman, “Xmgm: A Performance Analysis Tool Based on Matrix Geometric Methods”, Proceedings of the 2nd International Workshop on Modelling, Analysis and Simulation of Computer and Telecommunication Systems, IEEE Computer Society Press, pp.152–157, 1994.Google Scholar
  3. 3.
    B.R. Haverkort, “Approximate Analysis of Networks of PH¦PH¦1¦K Queues: Test Results”, Particicpants Proceedings of the 3rd International Workshop on Queueing Networks with Finite Capacity, Bradford, UK, July 6–7, 1995.Google Scholar
  4. 4.
    G. Heijenk, M. El Zarki, I.G. Niemegeers, “Modelling Segmentation and Reassembly Processes in Communication Networks”, Proceedings ITC-14, North-Holland, pp.513–524, 1994.Google Scholar
  5. 5.
    K. Kant, Introduction to Computer System Performance Evaluation, McGraw-Hill, Inc., 1992.Google Scholar
  6. 6.
    W. Krämer, M. Langenbach-Belz, “Approximate Formulae for the Delay in the Queueing System GI¦G¦1”, Proceedings ITC-8, pp.235–1/8, 1976.Google Scholar
  7. 7.
    U. Krieger, B. Müller-Clostermann, M. Sczittnick, “Modelling and Analysis of Communication Systems Based on Computational Methods for Markov Chains”, IEEE JSAC 8(9), pp.1630–1648, 1990.Google Scholar
  8. 8.
    K.T. Marshall, “Some Inequalities in Queueing”, Operations Research 16(3), pp.651–665, 1968.Google Scholar
  9. 9.
    M.F. Neuts, Matrix Geometric Solutions in Stochastic Models: An Algorithmic Approach, Johns Hopkins University Press, Baltimore, 1981.Google Scholar
  10. 10.
    H.G. Perros, Queueing Networks with Blocking, Oxford University Press, 1994.Google Scholar
  11. 11.
    H.C. Tijms, Stochastic Modelling and Analysis: A Computational Approach, John Wiley & Sons, 1986.Google Scholar
  12. 12.
    K.S. Trivedi, Probability & Statistics with Reliability, Queueing and Computer Science Applications, Prentice-Hall, 1982.Google Scholar
  13. 13.
    A.J. Weerstra, “Using Matrix-Geometric Methods to Enhance the QNA Method for Solving Large Queueing Networks; Vol. I & II”, M.Sc. thesis, University of Twente, 1994.Google Scholar
  14. 14.
    W. Whitt, “The Queueing Network Analyzer”, The Bell System Technical Journal 62(9), pp.2779–2815, 1983.Google Scholar
  15. 15.
    W. Whitt, “Performance of the Queueing Network Analyzer”, The Bell System Technical Journal 62(9), pp.2817–2843, 1983.Google Scholar
  16. 16.
    M. El Zarki, N. Shroff, “Performance Analysis of Packet-Loss Recovery Schemes in Interconnected LAN-WAN-LAN Networks”, Proceedings of the Third IFIP WG 6.4 Conference on High-Speed Networking, Editors: A. Danthine. O. Spaniol, North-Holland, pp.337–351, 1991.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Boudewijn R. Haverkort
    • 1
  1. 1.Department of Computer ScienceUniversity of TwenteAE Enschedethe Netherlands

Personalised recommendations