Better Automated Importance Splitting for Transient Rare Events

  • Carlos E. Budde
  • Pedro R. D’Argenio
  • Arnd HartmannsEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10606)


Statistical model checking uses simulation to overcome the state space explosion problem in formal verification. Yet its runtime explodes when faced with rare events, unless a rare event simulation method like importance splitting is used. The effectiveness of importance splitting hinges on nontrivial model-specific inputs: an importance function with matching splitting thresholds. This prevents its use by non-experts for general classes of models. In this paper, we propose new method combinations with the goal of fully automating the selection of all parameters for importance splitting. We focus on transient (reachability) properties, which particularly challenged previous techniques, and present an exhaustive practical evaluation of the new approaches on case studies from the literature. We find that using Restart simulations with a compositionally constructed importance function and thresholds determined via a new expected success method most reliably succeeds and performs very well. Our implementation within the Modest Toolset supports various classes of formal stochastic models and is publicly available.



We are grateful to José Villén-Altamirano for very helpful discussions that led to our eventual design of the expected success method.

This work is supported by the 3TU.BSR project, ERC grant 695614 (POWVER), the NWO SEQUOIA project, and SeCyT-UNC projects 05/BP12 and 05/B497.


  1. 1.
    Amrein, M., Künsch, H.R.: A variant of importance splitting for rare event estimation: Fixed number of successes. ACM Trans. Model. Comput. Simul. 21(2), 13:1–13:20 (2011)Google Scholar
  2. 2.
    Bayes, A.J.: Statistical techniques for simulation models. Aust. Comput. J. 2(4), 180–184 (1970)Google Scholar
  3. 3.
    Budde, C.E.: Automation of Importance Splitting Techniques for Rare Event Simulation. Ph.D. thesis, Universidad Nacional de Córdoba, Córdoba, Argentina (2017)Google Scholar
  4. 4.
    Budde, C.E., D’Argenio, P.R., Monti, R.E.: Compositional construction of importance functions in fully automated importance splitting. In: VALUETOOLS (2016)Google Scholar
  5. 5.
    Cérou, F., Guyader, A.: Adaptive multilevel splitting for rare event analysis. Stoch. Anal. Appl. 25(2), 417–443 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cérou, F., Moral, P.D., Furon, T., Guyader, A.: Sequential Monte Carlo for rare event estimation. Stat. Comput. 22(3), 795–808 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    D’Argenio, P.R., Hartmanns, A., Legay, A., Sedwards, S.: Statistical approximation of optimal schedulers for probabilistic timed automata. In: Ábrahám, E., Huisman, M. (eds.) IFM 2016. LNCS, vol. 9681, pp. 99–114. Springer, Cham (2016). doi: 10.1007/978-3-319-33693-0_7 CrossRefGoogle Scholar
  8. 8.
    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). doi: 10.1007/978-3-319-44878-7_4 CrossRefGoogle Scholar
  9. 9.
    Garvels, M.J.J., Kroese, D.P.: A comparison of RESTART implementations. In: Winter Simulation Conference, WSC, pp. 601–608 (1998)Google Scholar
  10. 10.
    Garvels, M.J.J., van Ommeren, J.C.W., Kroese, D.P.: On the importance function in splitting simulation. Eur. Trans. Telecommun. 13(4), 363–371 (2002)CrossRefGoogle Scholar
  11. 11.
    Garvels, M.J.J.: The splitting method in rare event simulation. Ph.D. thesis, University of Twente, Enschede, The Netherlands (2000)Google Scholar
  12. 12.
    Glasserman, P., Heidelberger, P., Shahabuddin, P., Zajic, T.: A large deviations perspective on the efficiency of multilevel splitting. IEEE Trans. Autom. Control 43(12), 1666–1679 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Glasserman, P., Heidelberger, P., Shahabuddin, P., Zajic, T.: Multilevel splitting for estimating rare event probabilities. Oper. Res. 47(4), 585–600 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hahn, E.M., Hartmanns, A., Hermanns, H.: Reachability and reward checking for stochastic timed automata. In: ECEASST 70 (2014)Google Scholar
  15. 15.
    Hartmanns, A., Hermanns, H.: A Modest approach to checking probabilistic timed automata. In: QEST, pp. 187–196. IEEE Computer Society (2009)Google Scholar
  16. 16.
    Hartmanns, A., Hermanns, H.: The Modest Toolset: an integrated environment for quantitative modelling and verification. In: Ábrahám, E., Havelund, K. (eds.) TACAS 2014. LNCS, vol. 8413, pp. 593–598. Springer, Heidelberg (2014). doi: 10.1007/978-3-642-54862-8_51 CrossRefGoogle Scholar
  17. 17.
    Hérault, T., Lassaigne, R., Magniette, F., Peyronnet, S.: Approximate probabilistic model checking. In: Steffen, B., Levi, G. (eds.) VMCAI 2004. LNCS, vol. 2937, pp. 73–84. Springer, Heidelberg (2004). doi: 10.1007/978-3-540-24622-0_8 CrossRefGoogle Scholar
  18. 18.
    Jegourel, C., Larsen, K.G., Legay, A., Mikučionis, M., Poulsen, D.B., Sedwards, S.: Importance sampling for stochastic timed automata. In: Fränzle, M., Kapur, D., Zhan, N. (eds.) SETTA 2016. LNCS, vol. 9984, pp. 163–178. Springer, Cham (2016). doi: 10.1007/978-3-319-47677-3_11 CrossRefGoogle Scholar
  19. 19.
    Jegourel, C., Legay, A., Sedwards, S.: Importance splitting for statistical model checking rare properties. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 576–591. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-39799-8_38 CrossRefGoogle Scholar
  20. 20.
    Jegourel, C., Legay, A., Sedwards, S.: An effective heuristic for adaptive importance splitting in statistical model checking. In: Margaria, T., Steffen, B. (eds.) ISoLA 2014. LNCS, vol. 8803, pp. 143–159. Springer, Heidelberg (2014). doi: 10.1007/978-3-662-45231-8_11 Google Scholar
  21. 21.
    Kroese, D.P., Nicola, V.F.: Efficient estimation of overflow probabilities in queues with breakdowns. Perform. Eval. 36, 471–484 (1999)CrossRefzbMATHGoogle Scholar
  22. 22.
    Kwiatkowska, M., Norman, G., Parker, D.: PRISM 4.0: verification of probabilistic real-time systems. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 585–591. Springer, Heidelberg (2011). doi: 10.1007/978-3-642-22110-1_47 CrossRefGoogle Scholar
  23. 23.
    Kwiatkowska, M.Z., Norman, G., Segala, R., Sproston, J.: Automatic verification of real-time systems with discrete probability distributions. Theor. Comput. Sci. 282(1), 101–150 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    L’Ecuyer, P., Demers, V., Tuffin, B.: Rare events, splitting, and quasi-Monte Carlo. ACM Trans. Model. Comput. Simul. 17(2) (2007)Google Scholar
  25. 25.
    LeGland, F., Oudjane, N.: A sequential particle algorithm that keeps the particle system alive. In: EUSIPCO, pp. 1–4. IEEE (2005)Google Scholar
  26. 26.
    Paolieri, M., Horváth, A., Vicario, E.: Probabilistic model checking of regenerative concurrent systems. IEEE Trans. Softw. Eng. 42(2), 153–169 (2016)CrossRefGoogle Scholar
  27. 27.
    Reijsbergen, D., de Boer, P.-T., Scheinhardt, W., Haverkort, B.: Automated rare event simulation for stochastic Petri nets. In: Joshi, K., Siegle, M., Stoelinga, M., D’Argenio, P.R. (eds.) QEST 2013. LNCS, vol. 8054, pp. 372–388. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-40196-1_31 CrossRefGoogle Scholar
  28. 28.
    Rubino, G., Tuffin, B. (eds.): Rare Event Simulation Using Monte Carlo Methods. Wiley (2009)Google Scholar
  29. 29.
    Villén-Altamirano, J.: Rare event RESTART simulation of two-stage networks. Eur. J. Oper. Res. 179(1), 148–159 (2007)CrossRefzbMATHGoogle Scholar
  30. 30.
    Villén-Altamirano, M., Villén-Altamirano, J.: RESTART: a method for accelerating rare event simulations. In: Queueing, Performance and Control in ATM (ITC-13), pp. 71–76. Elsevier (1991)Google Scholar
  31. 31.
    Villén-Altamirano, M., Villén-Altamirano, J.: RESTART: a straightforward method for fast simulation of rare events. In: WSC, pp. 282–289. ACM (1994)Google Scholar
  32. 32.
    Villén-Altamirano, M., Villén-Altamirano, J.: Analysis of restart simulation: theoretical basis and sensitivity study. Eur. Trans. Telecommun. 13(4), 373–385 (2002)CrossRefzbMATHGoogle Scholar
  33. 33.
    Younes, H.L.S., Simmons, R.G.: Probabilistic verification of discrete event systems using acceptance sampling. In: Brinksma, E., Larsen, K.G. (eds.) CAV 2002. LNCS, vol. 2404, pp. 223–235. Springer, Heidelberg (2002). doi: 10.1007/3-540-45657-0_17 CrossRefGoogle Scholar
  34. 34.
    Zimmermann, A., Maciel, P.: Importance function derivation for RESTART simulations of Petri nets. In: RESIM 2012, pp. 8–15 (2012)Google Scholar
  35. 35.
    Zimmermann, A., Reijsbergen, D., Wichmann, A., Canabal Lavista, A.: Numerical results for the automated rare event simulation of stochastic Petri nets. In: RESIM, pp. 1–10 (2016)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Carlos E. Budde
    • 1
    • 2
  • Pedro R. D’Argenio
    • 2
    • 3
  • Arnd Hartmanns
    • 1
    Email author
  1. 1.University of TwenteEnschedeThe Netherlands
  2. 2.Universidad Nacional de CórdobaCórdobaArgentina
  3. 3.Saarland UniversitySaarbrückenGermany

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