Input/Output Stochastic Automata with Urgency: Confluence and Weak Determinism

  • Pedro R. D’ArgenioEmail author
  • Raúl E. Monti
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11187)


In a previous work, we introduced an input/output variant of stochastic automata (IOSA) that, once the model is closed (i.e., all synchronizations are resolved), the resulting automaton is fully stochastic, that is, it does not contain non-deterministic choices. However, such variant is not sufficiently versatile for compositional modelling. In this article, we extend IOSA with urgent actions. This extension greatly increases the modularization of the models, allowing to take better advantage on compositionality than its predecessor. However, this extension introduces non-determinism even in closed models. We first show that confluent models are weakly deterministic in the sense that, regardless the resolution of the non-determinism, the stochastic behaviour is the same. In addition, we provide sufficient conditions to ensure that a network of interacting IOSAs is confluent without the need to analyse the larger composed IOSA.


  1. 1.
    Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, Cambridge (1998). Scholar
  2. 2.
    Behrmann, G., David, A., Larsen, K.G.: A tutorial on Uppaal. In: Bernardo, M., Corradini, F. (eds.) SFM-RT 2004. LNCS, vol. 3185, pp. 200–236. Springer, Heidelber (2004). Scholar
  3. 3.
    Bengtsson, J., et al.: Verification of an audio protocol with bus collision using Uppaal. In: Alur, R., Henzinger, T.A. (eds.) CAV 1996. LNCS, vol. 1102, pp. 244–256. Springer, Heidelberg (1996). Scholar
  4. 4.
    Bohnenkamp, H.C., D’Argenio, P.R., Hermanns, H., Katoen, J.: MODEST: a compositional modeling formalism for hard and softly timed systems. IEEE Trans. Softw. Eng. 32(10), 812–830 (2006). 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). Scholar
  6. 6.
    Budde, C.E.: Automation of importance splitting techniques for rare event simulation. Ph.D. thesis, Universidad Nacional de Córdoba (2017)Google Scholar
  7. 7.
    Budde, C.E., D’Argenio, P.R., Monti, R.E.: Compositional construction of importance functions in fully automated importance splitting. In: Puliafito, A., Trivedi, K.S., Tuffin, B., Scarpa, M., Machida, F., Alonso, J. (eds.) Proceedings of 10th EAI International Conference on Performance Evaluation Methodologies and Tools, VALUETOOLS 2016, October 2016, Taormina. ICST (2017).
  8. 8.
    Budde, C.E., Dehnert, C., Hahn, E.M., Hartmanns, A., Junges, S., Turrini, A.: JANI: quantitative model and tool interaction. In: Legay, A., Margaria, T. (eds.) TACAS 2017. LNCS, vol. 10206, pp. 151–168. Springer, Heidelberg (2017). Scholar
  9. 9.
    Crouzen, P.: Modularity and determinism in compositional markov models. Ph.D. thesis, Universität des Saarlandes, Saarbrücken (2014)Google Scholar
  10. 10.
    D’Argenio, P.R.: Algebras and automata for timed and stochastic systems. Ph.D. thesis, Universiteit Twente (1999)Google Scholar
  11. 11.
    D’Argenio, P.R., Katoen, J.P.: A theory of stochastic systems, part I: Stochastic automata. Inf. Comput. 203(1), 1–38 (2005). Scholar
  12. 12.
    D’Argenio, P.R., Katoen, J., Brinksma, E.: An algebraic approach to the specification of stochastic systems (extended abstract). In: Gries, D., de Roever, W.P. (eds.) PROCOMET 1998. IFIP Conference Proceedings, vol. 125, pp. 126–147. Chapman & Hall, Boca Raton (1998). Scholar
  13. 13.
    D’Argenio, P.R., Lee, M.D., Monti, R.E.: Input/Output stochastic automata. In: Fränzle, M., Markey, N. (eds.) FORMATS 2016. LNCS, vol. 9884, pp. 53–68. Springer, Cham (2016). Scholar
  14. 14.
    D’Argenio, P.R., Sánchez Terraf, P., Wolovick, N.: Bisimulations for non-deterministic labelled Markov processes. Math. Struct. Comput. Sci. 22(1), 43–68 (2012). Scholar
  15. 15.
    Desharnais, J., Edalat, A., Panangaden, P.: Bisimulation for labelled Markov processes. Inf. Comput. 179(2), 163–193 (2002). Scholar
  16. 16.
    Giry, M.: A categorical approach to probability theory. In: Banaschewski, B. (ed.) Categorical Aspects of Topology and Analysis. LNM, vol. 915, pp. 68–85. Springer, Heidelberg (1982). Scholar
  17. 17.
    van Glabbeek, R.J., Smolka, S.A., Steffen, B.: Reactive, generative and stratified models of probabilistic processes. Inf. Comput. 121(1), 59–80 (1995). Scholar
  18. 18.
    Hahn, E.M., Hartmanns, A., Hermanns, H., Katoen, J.: A compositional modelling and analysis framework for stochastic hybrid systems. Form. Methods Syst. Des. 43(2), 191–232 (2013). Scholar
  19. 19.
    Hartmanns, A.: On the analysis of stochastic timed systems. Ph.D. thesis, Saarlandes University, Saarbrücken (2015).
  20. 20.
    Hermanns, H.: Interactive Markov Chains: And the Quest for Quantified Quality. LNCS, vol. 2428. Springer, Heidelberg (2002). Scholar
  21. 21.
    Larsen, K.G., Skou, A.: Bisimulation through probabilistic testing. Inf. Comput. 94(1), 1–28 (1991). Scholar
  22. 22.
    Law, A.M., Kelton, W.D.: Simulation Modeling and Analysis, 3rd edn. McGraw-Hill Higher Education, New York City (1999)zbMATHGoogle Scholar
  23. 23.
    Milner, R.: Communication and Concurrency. Prentice-Hall, Englewood Cliffs (1989)zbMATHGoogle Scholar
  24. 24.
    Ruijters, E., Stoelinga, M.: Fault tree analysis: a survey of the state-of-the-art in modeling, analysis and tools. Comput. Sci. Rev. 15, 29–62 (2015). Scholar
  25. 25.
    Wolovick, N.: Continuous probability and nondeterminism in labeled transition systems. Ph.D. thesis, Universidad Nacional de Córdoba, Argentina (2012)Google Scholar
  26. 26.
    Wu, S., Smolka, S.A., Stark, E.W.: Composition and behaviors of probabilistic I/O automata. Theor. Comput. Sci. 176(1–2), 1–38 (1997). Scholar
  27. 27.
    Wang, Y.: Real-time behaviour of asynchronous agents. In: Baeten, J.C.M., Klop, J.W. (eds.) CONCUR 1990. LNCS, vol. 458, pp. 502–520. Springer, Heidelberg (1990). Scholar

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Authors and Affiliations

  1. 1.Universidad Nacional de Córdoba, FAMAFCórdobaArgentina
  2. 2.CONICETCórdobaArgentina
  3. 3.Saarland University, Department of Computer ScienceSaarbrückenGermany

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