State-Dependent Rates and Semi-Product-Form via the Reversed Process

  • Nigel Thomas
  • Peter Harrison
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6342)


We consider the problem of finding a decomposed solution to a queueing model where the action rates may depend on the global state space. To do this we consider regular cycles in the underlying state space and show that a semi-product-form solution exists when the functions describing the action rates have specific forms. The approach is shown in detail for two queues and shown to extend to larger systems. Although not all the results for semi-product-form solutions are entirely new, the method by which they are derived is both novel, intuitive and leads to generalisations.


Reversed Process Minimal Cycle External Arrival Steady State Probability Distribution Reversed Network 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Balbo, G., Bruell, S., Sereno, M.: Embedded processes in generalized stochastic Petri nets. In: Proc. 9th Intl. Workshop on Petri Nets and Performance Models, pp. 71–80 (2001)Google Scholar
  2. 2.
    Balsamo, S., Harrison, P.G., Marin, A.: Systematic Construction of Product-Form Stochastic Petri-Nets (submitted for publication)Google Scholar
  3. 3.
    Balsamo, S., Marin, A.: Product-form solutions for models with joint-state dependent transition rates. In: Al-Begain, K., Fiems, D., Knottenbelt, W.J. (eds.) Analytical and Stochastic Modeling Techniques and Applications. LNCS, vol. 6148, pp. 87–101. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  4. 4.
    Baskett, F., Chandy, K., Muntz, R., Palacios, F.: Open, Closed, and Mixed Networks of Queues with Different Classes of Customers. Journal of the ACM 22(2), 248–260 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bonald, T., Proutiere, A.: Insensitivity in processor-sharing networks. Performance Evaluation 49, 193–209 (2002)CrossRefzbMATHGoogle Scholar
  6. 6.
    Boucherie, R.J.: A Characterisation of Independence for Competing Markov Chains with Applications to Stochastic Petri Nets. IEEE Trans. on Software Eng. 20(7), 536–544 (1994)CrossRefGoogle Scholar
  7. 7.
    Boucherie, R.J., van Dijk, N.M.: Product-forms for queueing networks with state-dependent multiple job transitions. Advances in Applied Probability 23, 152–187 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chao, X., Miyazawa, M., Pinedo, M.: Queueing Networks: Customers, Signals and product-form Solutions. Wiley, Chichester (1999)zbMATHGoogle Scholar
  9. 9.
    Fourneau, J.-M., Plateau, B., Stewart, W.J.: An algebraic condition for product form in stochastic automata networks without synchronizations. Performance Evaluation 85, 854–868 (2008)CrossRefGoogle Scholar
  10. 10.
    Harrison, P.G.: Turning back time in Markovian process algebra. In: Theoretical Computer Science (January 2003)Google Scholar
  11. 11.
    Harrison, P.G., Lee, T.T.: Separable equilibrium state probabilities via time reversal inmarkovian process algebra. Theoretical Computer Science (2005)Google Scholar
  12. 12.
    Harrison, P.G.: Compositional reversed Markov processes, with applications to G-networks. Performance Evaluation (2004)Google Scholar
  13. 13.
    Harrison, P.G.: Product-forms and functional rates. Performance Evaluation 66, 660–663 (2009)CrossRefGoogle Scholar
  14. 14.
    Henderson, W., Taylor, P.G.: Embedded Processes in Stochastic Petri Nets. IEEE Trans. on Software Eng. 17(2), 108–116 (1991)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Henderson, W., Taylor, P.G.: State-dependent Coupling of Quasireversible Nodes. Queueing Systems: Theory and Applications 37(1/3), 163–197 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kelly, F.P.: Reversibility and stochastic networks. Wiley, Chichester (1979)zbMATHGoogle Scholar
  17. 17.
    Serfozo, R.: Markovian network processes: congestion-dependent routing and processing. Queueing Systems: Theory and Applications 5(1-3), 5–36 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Harrison, P.G., Thomas, N.: Product-form solution in PEPA via the reversed process. In: Next Generation Internet: Performance Evaluation and Applications. LNCS, vol. 5233. Springer, Heidelberg (2010)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Nigel Thomas
    • 1
  • Peter Harrison
    • 2
  1. 1.School of Computing ScienceNewcastle UniversityUK
  2. 2.Department of ComputingImperial CollegeLondonUK

Personalised recommendations