Skip to main content

Mean Field for Performance Models with Generally-Distributed Timed Transitions

  • Conference paper

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8657))

Abstract

In this paper we extend the mean-field limit of a class of stochastic models with exponential and deterministic delays to include exponential and generally-distributed delays. Our main focus is the rigorous proof of the mean-field limit.

This is a preview of subscription content, log in via an institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Anselmi, J., Verloop, M.: Energy-aware capacity scaling in virtualized environments with performance guarantees. Perf. Eval. 68(11), 1207–1221 (2011)

    Article  Google Scholar 

  2. Azuma, K.: Weighted sums of certain dependent random variables. Tohoku Mathematical Journal 19(3), 357–367 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  3. Barbuti, R., Caravagna, G., Maggiolo-Schettini, A., Milazzo, P.: Delay stochastic simulation of biological systems: A purely delayed approach. In: Priami, C., Back, R.-J., Petre, I., de Vink, E. (eds.) Transactions on Computational Systems Biology XIII. LNCS, vol. 6575, pp. 61–84. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  4. Benaïm, M., Le Boudec, J.-Y.: A class of mean field interaction models for computer and communication systems. Performance Evaluation 65(11-12), 823–838 (2008)

    Article  Google Scholar 

  5. Bortolussi, L., Hillston, J.: Fluid approximation of ctmc with deterministic delays. In: Int. Conf. on Quantitative Evaluation of Systems, pp. 53–62 (2012)

    Google Scholar 

  6. Caravagna, G., Hillston, J.: Bio-PEPAd: A non-Markovian extension of Bio-PEPA. Theoretical Computer Science 419, 26–49 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  7. Choi, H., Kulkarni, V., Trivedi, K., Marsan, M.A.: Transient analysis of deterministic and stochastic petri nets. In: Ajmone Marsan, M. (ed.) ICATPN 1993. LNCS, vol. 691, pp. 166–185. Springer, Heidelberg (1993)

    Chapter  Google Scholar 

  8. Chung, F., Lu, L.: Concentration inequalities and martingale inequalities: A survey. Internet Mathematics 3(1), 79–127 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  9. Cox, D.R.: The analysis of non-Markovian stochastic processes by the inclusion of supplementary variables. Mathematical Proceedings of the Cambridge Philosophical Society 51(03), 433–441 (1955)

    Article  MATH  Google Scholar 

  10. Durrett, R.: Probability: Theory and Examples. Cambridge series on statistical and probabilistic mathematic. Cambridge University Press (2010)

    Google Scholar 

  11. Ellis, R.: Entropy, Large Deviations, and Statistical Mechanics. Classics in Mathematics. Springer (2005)

    Google Scholar 

  12. Ethier, S.N., Kurtz, T.G.: Markov Processes: Characterization and Convergence. Wiley (2005)

    Google Scholar 

  13. Gast, N., Bruno, G.: A mean field model of work stealing in large-scale systems. SIGMETRICS Perform. Eval. Rev. 38(1), 13–24 (2010)

    Article  Google Scholar 

  14. Glynn, P.W.: A GSMP formalism for discrete event systems. Proceedings of the IEEE 77(1), 14–23 (1989)

    Article  Google Scholar 

  15. Harrison, P.G., Strulo, B.: SPADES - a process algebra for discrete event simulation. Journal of Logic and Computation 10(1), 3–42 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  16. Hayden, R.A.: Mean field for performance models with deterministically-timed transitions. In: 2012 Ninth International Conference on Quantitative Evaluation of Systems (QEST), pp. 63–73 (September 2012)

    Google Scholar 

  17. Hayden, R.A., Bradley, J.T.: A fluid analysis framework for a markovian process algebra. Theoretical Computer Science 411, 2260–2297 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  18. Kurtz, T.G.: Strong approximation theorems for density dependent Markov chains. Stochastic Processes and their Applications 6(3), 223–240 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  19. Maset, S.: Numerical solution of retarded functional differential equations as abstract cauchy problems. J. Comput. Appl. Math. 161(2), 259–282 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  20. Matthes, K.: Zur theorie der bedienungsprozesse. In: 3rd Prague Conf. on Inf. Theory, Statistical Decision Functions and Random Processes, pp. 512–528 (1962)

    Google Scholar 

  21. Schlicht, R., Winkler, G.: A delay stochastic process with applications in molecular biology. Journal of Mathematical Biology 57(5), 613–648 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  22. Whitt, W.: Internet supplement to Stochastic-Process Limits (2002)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this paper

Cite this paper

Hayden, R.A., Horváth, I., Telek, M. (2014). Mean Field for Performance Models with Generally-Distributed Timed Transitions. In: Norman, G., Sanders, W. (eds) Quantitative Evaluation of Systems. QEST 2014. Lecture Notes in Computer Science, vol 8657. Springer, Cham. https://doi.org/10.1007/978-3-319-10696-0_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-10696-0_8

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-10695-3

  • Online ISBN: 978-3-319-10696-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics