PHASE: A Stochastic Formalism for Phase-Type Distributions

  • Gabriel Ciobanu
  • Armand Stefan Rotaru
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8829)


Models of non-Markovian systems expressed using stochastic formalisms often employ phase-type distributions in order to approximate the duration of transitions. We introduce a stochastic process calculus named PHASE which operates with phase-type distributions, and provide a step-by-step description of how PHASE processes can be translated into models supported by the probabilistic model checker PRISM. We then illustrate our approach by analysing the behaviour of a simple system involving both non-Markovian and Markovian transitions.


Model Checker Parallel Operator Discrete Event Simulation Parallel Composition Markovian Transition 
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.


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  1. 1.
    Asmussen, S., Nerman, O., Olsson, M.: Fitting Phase-type Distributions via the EM Algorithm. Scandinavian Journal of Statistics 23(4), 419–441 (1996)MATHGoogle Scholar
  2. 2.
    Baier, C., Katoen, J.-P., Hermanns, H.: Approximate Symbolic Model Checking of Continuous-Time Markov Chains (Extended Abstract). In: Baeten, J.C.M., Mauw, S. (eds.) CONCUR 1999. LNCS, vol. 1664, pp. 146–161. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  3. 3.
    Bernardo, M., Gorrieri, R.: Extended Markovian Process Algebra. In: Sassone, V., Montanari, U. (eds.) CONCUR 1996. LNCS, vol. 1119, pp. 315–330. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  4. 4.
    Bradley, J.T.: Semi-Markov PEPA: Modelling with generally distributed actions. International Journal of Simulation 6(3-4), 43–51 (2005)Google Scholar
  5. 5.
    Bravetti, M., D’Argenio, P.R.: Tutte le algebre insieme: Concepts, discussions and relations of stochastic process algebras with general distributions. In: Baier, C., Haverkort, B.R., Hermanns, H., Katoen, J.-P., Siegle, M. (eds.) Validation of Stochastic Systems. LNCS, vol. 2925, pp. 44–88. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  6. 6.
    Crovella, M.E.: Performance Evaluation with Heavy Tailed Distributions. In: Feitelson, D.G., Rudolph, L. (eds.) JSSPP 2001. LNCS, vol. 2221, pp. 1–10. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  7. 7.
    D’Argenio, P.: Algebras and Automata for Timed and Stochastic Systems. PhD thesis, University of Twente (1999)Google Scholar
  8. 8.
    D’Argenio, P., Katoen, J.-P., Brinksma, E.: A compositional approach to generalised semi-Markov processes. In: Guia, A., Spathopoulos, M., Smedinga, R. (eds.) Proceedings of WODES 1998, pp. 391–397. IEEE Press, New York (1998)Google Scholar
  9. 9.
    Doherty, G., Massink, M., Faconti, G.: Reasoning about interactive systems with stochastic models. In: Johnson, C. (ed.) DSV-IS 2001. LNCS, vol. 2220, pp. 144–163. Springer, Heidelberg (2001)Google Scholar
  10. 10.
    El-Rayes, A., Kwiatkowska, M., Norman, G.: Solving infinite stochastic process algebra models through matrix-geometric methods. In: Hillston, J., Silva, M. (eds.) Proceedings of PAPM 1999, Zaragoza, Spain, pp. 41–62. Prensas Universitarias de Zaragoza (1999)Google Scholar
  11. 11.
    Hahn, E.M., Hartmanns, A., Hermanns, H., Katoen, J.-P.: A compositional modelling and analysis framework for stochastic hybrid systems. In: Formal Methods in System Design (2012)Google Scholar
  12. 12.
    Harrison, P.G., Strulo, B.: Stochastic process algebra for discrete event simulation. In: Baccelli, F., Jean-Marie, A., Mitrani, I. (eds.) Quantitative Methods in Parallel Systems, pp. 18–37. Springer, Berlin (1995)CrossRefGoogle Scholar
  13. 13.
    Hermanns, H.: Interactive Markov Chains - The Quest for Quantified Quality. Springer, Berlin (2002)CrossRefGoogle Scholar
  14. 14.
    Hermanns, H., Katoen, J.-P.: Automated compositional Markov chain generation for a plain-old telephone system. Science of Computer Programming 36(1), 97–127 (2000)CrossRefMATHGoogle Scholar
  15. 15.
    Hillston, J.: A Compositional Approach to Performance Modelling. Cambridge University Press, Cambridge (1996)Google Scholar
  16. 16.
    Horváth, A., Telek, M.: PhFit: A General Phase-Type Fitting Tool. In: Computer Performance Evaluation: Modelling Techniques and Tools, pp. 82–91. Springer, Heidelberg (2002)Google Scholar
  17. 17.
    Katoen, J.-P., D’Argenio, P.R.: General distributions in process algebra. In: Brinksma, E., Hermanns, H., Katoen, J.-P. (eds.) EEF School 2000 and FMPA 2000. LNCS, vol. 2090, pp. 375–429. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  18. 18.
    Nelson, R.: Probability, Stochastic Processes, and Queueing Theory. Springer, New York (1995)CrossRefMATHGoogle Scholar
  19. 19.
    Neuts, M.F.: Matrix-geometric solutions in stochastic models: an algorithmic approach. Dover Publications (1981)Google Scholar
  20. 20.
    Norris, J.R.: Markov chains. Cambridge University Press, Cambridge (1998)MATHGoogle Scholar
  21. 21.
    Pulungan, M.R.: Reduction of Acyclic Phase-Type Representations. PhD thesis, Saarland University, Germany (2009)Google Scholar
  22. 22.
    Reinecke, P., Krauss, T., Wolter, K.: HyperStar: Phase-type Fitting Made Easy. In: Proceedings of QEST 2012, pp. 201–202. IEEE Computer Society, Washington, DC (2012)Google Scholar
  23. 23.
    Riaño, G., Pérez, J.F.: jPhase: an Object-Oriented Tool for Modeling Phase-Type Distributions. In: Meini, B., van Houdt, B. (eds.) Proceeding of SMCTools 2006, vol. 5, ACM, New York (2006)Google Scholar
  24. 24.
    Wolf, V.: Equivalences on phase type processes. PhD thesis, University of Mannheim, Germany (2008)Google Scholar
  25. 25.
    Younes, H.L.S.: Ymer: A statistical model checker. In: Etessami, K., Rajamani, S.K. (eds.) CAV 2005. LNCS, vol. 3576, pp. 429–433. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  26. 26.
    Zeng, K.: Logics and Models for Stochastic Analysis Beyond Markov Chains. PhD thesis, Technical University of Denmark, Denmark (2012)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Gabriel Ciobanu
    • 1
  • Armand Stefan Rotaru
    • 1
  1. 1.Institute of Computer ScienceRomanian AcademyIaşiRomania

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