The Mean Value of the Maximum

  • Henrik Bohnenkamp
  • Boudewijn Haverkort
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2399)


This paper treats a practical problem that arises in the area of stochastic process algebras. The problem is the efficient computation of the mean value of the maximum of phase-type distributed random variables. The maximum of phase-type distributed random variables is again phase-type distributed, however, its representation grows exponentially in the number of considered random variables. Although an efficient representation in terms of Kronecker sums is straightforward, the computation of the mean value requires still exponential time, if carried out by traditional means. In this paper, we describe an approximation method to compute the mean value in only polynomial time in the number of considered random variables and the size of the respective representations. We discuss complexity, numerical stability and convergence of the approach.


Generator Matrix Label Transition System Dominant Eigenvalue Positive Random Variable Sojourn Time Distribution 
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  1. 1.
    Alexander Bell and Boudewijn R. Haverkort. Serial and parallel out-of-core solution of linear systems arising from GSPNs. In Adrian Tentner, editor, High Performance Computing (HPC 2001) — Grand Challenges in Computer Simulation, pages 242–247, San Diego, CA, USA, April 2001. The Society for Modeling and Simulation International.Google Scholar
  2. 2.
    Henrik Bohnenkamp. Compositional Solution of Stochastic Process Algebra Models. PhD thesis, Department of Computer Science, Rheinisch-Westfälische Technische Hochschule Aachen, Germany, 2001. Preliminary Version.Google Scholar
  3. 3.
    Henrik C. Bohnenkamp and Boudewijn R. Haverkort. Semi-numerical solution of stochastic process algebra models. In Joost-Pieter Katoen, editor, Proceedings of the 5th International AMAST Workshop, ARTS’99, volume 1601 of Lecture Notes in Computer Science, pages 228–243. Springer-Verlag, May 1999.Google Scholar
  4. 4.
    Hendrik Brinksma. On the design of extended LOTOS. PhD thesis, University of Twente, The Netherlands, 1988.Google Scholar
  5. 5.
    M. Davio. Kronecker products and shuffle algebra. IEEE Transactions on Computers, C-30(2):116–125, February 1981.Google Scholar
  6. 6.
    F.R. Gantmacher. Matrizentheorie. Springer Verlag (translated from Russian; originally published in 1966), 1986.Google Scholar
  7. 7.
    Peter G. Harrison and Naresh M. Patel. Performance Modelling of Communication Networks and Computer Architectures. Addison-Wesley, 1993.Google Scholar
  8. 8.
    Holger Hermanns. Interactive Markov Chains. PhD thesis, Universität Erlangen-Nürnberg, Germany, 1998.Google Scholar
  9. 9.
    Ronald A. Howard. Dynamic Probabilistic Systems, volume 2: Semimarkov and Decision Processes. John Wiley & Sons, 1971.Google Scholar
  10. 10.
    Ronald A. Howard. Dynamic Probabilistic Systems, volume 1: Markov Models. John Wiley & Sons, 1971.Google Scholar
  11. 11.
    A. Jensen. Markoff chains as an aid in the study of Markoff processes. Skand. Aktuarietidskrift, 36:87–91, 1953.Google Scholar
  12. 12.
    Marcel F. Neuts. Matrix-Geometric Solutions in Stochastic Models-An Algorithmic Approach. Series in the Mathematical Sciences. Johns Hopkins University Press, Baltimore, 1981.Google Scholar
  13. 13.
    Robin A. Sahner. A Hybrid, Combinatorial-Markov Method of solving Performance and Reliability Models. PhD thesis, Duke University, Computer Science Department, 1986.Google Scholar
  14. 14.
    Robin A. Sahner, Kishor S. Trivedi, and Antonio Puliafito. Performance and Reliability Analysis of Computer Systems. An Example-Based Approach Using the SHARPE Software Package. Kluwer Academic Publishers, 1996.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Henrik Bohnenkamp
    • 1
  • Boudewijn Haverkort
    • 2
  1. 1.University of TwenteThe Netherlands
  2. 2.Technical University of AachenGermany

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