Rare Event Simulation with Fully Automated Importance Splitting

  • Carlos E. Budde
  • Pedro R. D’Argenio
  • Holger Hermanns
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9272)


Probabilistic model checking is a powerful tool for analysing probabilistic systems but it can only be efficiently applied to Markov models. Monte Carlo simulation provides an alternative for the generality of stochastic processes, but becomes infeasible if the value to estimate depends on the occurrence of rare events. To combat this problem, intelligent simulation strategies exist to lower the estimation variance and hence reduce the simulation time. Importance splitting is one such technique, but requires a guiding function typically defined in an ad hoc fashion by an expert in the field. We present an automatic derivation of the importance function from the model description. A prototypical tool was developed and tested on several Markov models, compared to analytically and numerically calculated results and to results of typical ad hoc importance functions, showing the feasibility and efficiency of this approach. The technique is easily adapted to general models like GSMPs.


Rare Event Goal State Importance Sampling Importance Function Tandem Queue 
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.
    Cérou, F., Guyader, A.: Adaptive multilevel splitting for rare event analysis. Stochastic Analysis and Applications 25(2), 417–443 (2007)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Garvels, M.J.J.: The splitting method in rare event simulation. PhD thesis, University of Twente (2000)Google Scholar
  3. 3.
    Garvels, M.J.J., Van Ommeren, J.-K.C.W., Kroese, D.P.: On the importance function in splitting simulation. Eur. Trans. Telecommun. 13(4), 363–371 (2002)CrossRefGoogle Scholar
  4. 4.
    Glasserman, P., Heidelberger, P., Shahabuddin, P., Zajic, T.: Multilevel splitting for estimating rare event probabilities. Operations Research 47(4), 585–600 (1999)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    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) CrossRefGoogle Scholar
  6. 6.
    Kahn, H., Harris, T.E.: Estimation of particle transmission by random sampling. National Bureau of Standards Applied Mathematics Series 12, 27–30 (1951)Google Scholar
  7. 7.
    Kroese, D.P., Nicola, V.F.: Efficient estimation of overflow probabilities in queues with breakdowns. Performance Evaluation 36, 471–484 (1999)CrossRefMATHGoogle Scholar
  8. 8.
    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) CrossRefGoogle Scholar
  9. 9.
    Law, A.M., Kelton, W.D., Kelton, W.D.: Simulation modeling and analysis, vol. 2. McGraw-Hill, New York (1991)Google Scholar
  10. 10.
    L’Ecuyer, P., Demers, V., Tuffin, B.: Rare events, splitting, and quasi-Monte Carlo. ACM Trans. Model. Comput. Simul. 17(2) (April 2007)Google Scholar
  11. 11.
    L’Ecuyer, P., Le Gland, F., Lezaud, P., Tuffin, B.: Splitting techniques. In: Rare Event Simulation using Monte Carlo Methods, pp. 39–61. J. Wiley & Sons (2009)Google Scholar
  12. 12.
    L’Ecuyer, P., Mandjes, M., Tuffin, B.: Importance sampling in rare event simulation. In: Rare Event Simulation using Monte Carlo Methods, pp. 17–38. J. Wiley & Sons (2009)Google Scholar
  13. 13.
    L’Ecuyer, P., Tuffin, B.: Approximating zero-variance importance sampling in a reliability setting. Annals of Operations Research 189(1), 277–297 (2011)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    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) CrossRefGoogle Scholar
  15. 15.
    Villén-Altamirano, J.: RESTART method for the case where rare events can occur in retrials from any threshold. Int. J. Electron. Commun. (AEÜ) 52, 183–189 (1998)Google Scholar
  16. 16.
    Villén-Altamirano, J.: Rare event RESTART simulation of two-stage networks. European Journal of Operational Research 179(1), 148–159 (2007)CrossRefMATHGoogle Scholar
  17. 17.
    Villén-Altamirano, M., Martínez-Marrón, A., Gamo, J., Fernández-Cuesta, F.: Enhancement of the accelerated simulation method restart by considering multiple thresholds. In: Proc. 14th Int. Teletraffic Congress, pp. 797–810 (1994)Google Scholar
  18. 18.
    Villén-Altamirano, M., Villén-Altamirano, J.: RESTART: A method for accelerating rare event simulations. Analysis 3, 3 (1991)MATHGoogle Scholar
  19. 19.
    Villén-Altamirano, M., Villén-Altamirano, J.: The Rare Event Simulation Method RESTART: Efficiency Analysis and Guidelines for Its Application. In: Kouvatsos, D.D. (ed.) Next Generation Internet: Performance Evaluation and Applications. LNCS, vol. 5233, pp. 509–547. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  20. 20.
    Wilson, E.B.: Probable inference, the law of succession, and statistical inference. Journal of the American Statistical Association 22(158), 209–212 (1927)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Carlos E. Budde
    • 1
  • Pedro R. D’Argenio
    • 1
  • Holger Hermanns
    • 2
  1. 1.FaMAF, Universidad Nacional de Córdoba – CONICETCórdobaArgentina
  2. 2.Fakultät für Mathematik und InformatikUniversität des SaarlandesSaarbrückenGermany

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