Analysis of Timed Properties Using the Jump-Diffusion Approximation

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10497)

Abstract

Density dependent Markov chains (DDMCs) describe the interaction of groups of identical objects. In case of large numbers of objects a DDMC can be approximated efficiently by means of either a set of ordinary differential equations (ODEs) or by a set of stochastic differential equations (SDEs). While with the ODE approximation the chain stochasticity is not maintained, the SDE approximation, also known as the diffusion approximation, can capture specific stochastic phenomena (e.g., bi-modality) and has also better convergence characteristics. In this paper we introduce a method for assessing temporal properties, specified in terms of a timed automaton, of a DDMC through a jump diffusion approximation. The added value is in terms of runtime: the costly simulation of a very large DDMC model can be replaced through much faster simulation of the corresponding jump diffusion model. We show the efficacy of the framework through the analysis of a biological oscillator.

Keywords

Diffusion approximation Stochastic differential equations with jumps Statistical model checking 

References

  1. 1.
    Alur, R., Dill, D.L.: A theory of timed automata. Theor. Comput. Sci. 126(2), 183–235 (1994)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Amparore, E.G., Beccuti, M., Donatelli, S.: (Stochastic) model checking in GreatSPN. In: Ciardo, G., Kindler, E. (eds.) PETRI NETS 2014. LNCS, vol. 8489, pp. 354–363. Springer, Cham (2014). doi: 10.1007/978-3-319-07734-5_19 CrossRefGoogle Scholar
  3. 3.
    Angius, A., Balbo, G., Beccuti, M., Bibbona, E., Horvath, A., Sirovich, R.: Approximate analysis of biological systems by hybrid switching jump diffusion. Theor. Comput. Sci. 587, 49–72 (2015)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Ballarini, P.: Analysing oscillatory trends of discrete-state stochastic processes through HASL statistical model checking. STTT 17(4), 505–526 (2015)CrossRefGoogle Scholar
  5. 5.
    Ballarini, P., Barbot, B., Duflot, M., Haddad, S., Pekergin, N.: HASL: a new approach for performance evaluation and model checking from concepts to experimentation. Perform. Eval. 90, 53–77 (2015)CrossRefGoogle Scholar
  6. 6.
    Ballarini, P., Djafri, H., Duflot, M., Haddad, S., Pekergin, N.: COSMOS: a statistical model checker for the hybrid automata stochastic logic. In: Proceedings of the QEST 2011, pp. 143–144. IEEE Computer Society (2011)Google Scholar
  7. 7.
    Beccuti, M., Bibbona, E., Horvath, A., Sirovich, R., Angius, A., Balbo, G.: Analysis of petri net models through stochastic differential equations. In: Ciardo, G., Kindler, E. (eds.) PETRI NETS 2014. LNCS, vol. 8489, pp. 273–293. Springer, Cham (2014). doi: 10.1007/978-3-319-07734-5_15 CrossRefGoogle Scholar
  8. 8.
    Bibbona, E., Sirovich, R.: Strong approximation of density dependent Markov chains on bounded domains by jump diffusion processes. Technical report, Università di Torino (2017)Google Scholar
  9. 9.
    Bortolussi, L., Hillston, J.: Model checking single agent behaviours by fluid approximation. Inf. Comput. 242, 183–226 (2015)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Bortolussi, L., Lanciani, R.: Model checking markov population models by central limit approximation. In: Joshi, K., Siegle, M., Stoelinga, M., D’Argenio, P.R. (eds.) QEST 2013. LNCS, vol. 8054, pp. 123–138. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-40196-1_9 CrossRefGoogle Scholar
  11. 11.
    Bortolussi, L., Lanciani, R.: Fluid model checking of timed properties. In: Sankaranarayanan, S., Vicario, E. (eds.) FORMATS 2015. LNCS, vol. 9268, pp. 172–188. Springer, Cham (2015). doi: 10.1007/978-3-319-22975-1_12 CrossRefGoogle Scholar
  12. 12.
    Chen, T., Han, T., Katoen, J.-P., Mereacre, A.: Model checking of continuous-time Markov chains against timed automata specifications. Log. Meth. Comput. Sci. 7(1), 1–34 (2011)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Donatelli, S., Haddad, S., Sproston, J.: Model checking timed and stochastic properties with CSL\({^{\rm TA}}\). IEEE T. Software Eng. 35(2), 224–240 (2009)CrossRefGoogle Scholar
  14. 14.
    Gressens, P., Steenwinckel, J.V., Schang, A., Sigaut, S., Degos, V., Lebon, S., Schwendimann, L., Le Charpentier, T., Hagberg, H., Soussi, N., Fleiss, B.: Microglial Wnt signaling inhibition promotes microglia activation and oligodendrocyte maturation blockade. J. Neurochem. 134, 122 (2015)Google Scholar
  15. 15.
    Jensen, P.B., Pedersen, L., Krishna, S., Jensen, M.H.: A Wnt oscillator model for somitogenesis. Biophys. J. 98(6), 943–950 (2010)CrossRefGoogle Scholar
  16. 16.
    Kolesnichenko, A., de Boer, P., Remke, A., Haverkort B.R.: A logic for model-checking mean-field models. In: Proceedings of the DSN 2013, pp. 1–12. IEEE Computer Society (2013)Google Scholar
  17. 17.
    Kurtz, T.G.: Solutions of ordinary differential equations as limits of pure jump Markov processes. J. Appl. Probab. 1(7), 49–58 (1970)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Kurtz, T.G.: Limit theorems and diffusion approximations for density dependent Markov chains. In: Wets, R.J.B. (ed.) Stochastic Systems: Modeling, Identification and Optimization, I, pp. 67–78. Springer, Heidelberg (1976)CrossRefGoogle Scholar
  19. 19.
    Kurtz, T.G.: Strong approximation theorems for density dependent Markov chains. Stoc. Proc. Appl. 6(3), 223–240 (1978)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Mikeev, L., Neuhäußer, M.R., Spieler, D., Wolf, V.: On-the-fly verification and optimization of DTA-properties for large Markov chains. Form. Method. Syst. Des. 43(2), 313–337 (2013)CrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Paolo Ballarini
    • 1
  • Marco Beccuti
    • 2
  • Enrico Bibbona
    • 3
  • Andras Horvath
    • 2
  • Roberta Sirovich
    • 4
  • Jeremy Sproston
    • 2
  1. 1.Laboratoire MICS, CentraleSupèlecUniversité Paris SaclayParisFrance
  2. 2.Dipartimento di InformaticaUniversità di TorinoTurinItaly
  3. 3.Dipartimento di Scienze Matematiche“G. L. Lagrange”, Politecnico di TorinoTurinItaly
  4. 4.Dipartimento di MatematicaUniversità di TorinoTurinItaly

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