Annals of Operations Research

, Volume 189, Issue 1, pp 277–297 | Cite as

Approximating zero-variance importance sampling in a reliability setting

  • Pierre L’Ecuyer
  • Bruno Tuffin


We consider a class of Markov chain models that includes the highly reliable Markovian systems (HRMS) often used to represent the evolution of multicomponent systems in reliability settings. We are interested in the design of efficient importance sampling (IS) schemes to estimate the reliability of such systems by simulation. For these models, there is in fact a zero-variance IS scheme that can be written exactly in terms of a value function that gives the expected cost-to-go (the exact reliability, in our case) from any state of the chain. This IS scheme is impractical to implement exactly, but it can be approximated by approximating this value function. We examine how this can be effectively used to estimate the reliability of a highly-reliable multicomponent system with Markovian behavior. In our implementation, we start with a simple crude approximation of the value function, we use it in a first-order IS scheme to obtain a better approximation at a few selected states, then we interpolate in between and use this interpolation in our final (second-order) IS scheme. In numerical illustrations, our approach outperforms the popular IS heuristics previously proposed for this class of problems. We also perform an asymptotic analysis in which the HRMS model is parameterized in a standard way by a rarity parameter ε, so that the relative error (or relative variance) of the crude Monte Carlo estimator is unbounded when ε→0. We show that with our approximation, the IS estimator has bounded relative error (BRE) under very mild conditions, and vanishing relative error (VRE), which means that the relative error converges to 0 when ε→0, under slightly stronger conditions.


Monte Carlo Rare events Importance sampling 


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  1. Ahamed, I., Borkar, V. S., & Juneja, S. (2006). Adaptive importance sampling for Markov chains using stochastic approximation. Operations Research, 54(3), 489–504. CrossRefGoogle Scholar
  2. Alexopoulos, C., & Shultes, B. C. (2001). Estimating reliability measures for highly-dependable Markov systems, using balanced likelihood ratios. IEEE Transactions on Reliability, 50(3), 265–280. CrossRefGoogle Scholar
  3. Asmussen, S., & Glynn, P. W. (2007). Stochastic simulation. New York: Springer. Google Scholar
  4. Booth, T. E. (1985). Exponential convergence for Monte Carlo particle transport? Transactions of the American Nuclear Society, 50, 267–268. Google Scholar
  5. Booth, T. E. (1987). Generalized zero-variance solutions and intelligent random numbers. In Proceedings of the 1987 winter simulation conference (pp. 445–451). New York: IEEE Press. Google Scholar
  6. Cancela, H., Rubino, G., & Tuffin, B. (2002). MTTF estimation by Monte Carlo methods using Markov models. Monte Carlo Methods and Applications, 8(4), 312–341. CrossRefGoogle Scholar
  7. Carrasco, J. A. (1992). Failure distance-based simulation of repairable fault tolerant systems. In Proceedings of the 5th international conference on modeling techniques and tools for computer performance evaluation (pp. 351–365). Amsterdam: Elsevier. Google Scholar
  8. Gertsbakh, I. B. (1984). Asymptotic methods in reliability theory: A review. Advances in Applied Probability, 16, 147–175. CrossRefGoogle Scholar
  9. Glynn, P. W., & Iglehart, D. L. (1989). Importance sampling for stochastic simulations. Management Science, 35, 1367–1392. CrossRefGoogle Scholar
  10. Goyal, A., Shahabuddin, P., Heidelberger, P., Nicola, V. F., & Glynn, P. W. (1992). A unified framework for simulating Markovian models of highly reliable systems. IEEE Transactions on Computers, C-41, 36–51. CrossRefGoogle Scholar
  11. Heidelberger, P. (1995). Fast simulation of rare events in queueing and reliability models. ACM Transactions on Modelling and Computer Simulation, 5(1), 43–85. CrossRefGoogle Scholar
  12. Juneja, S., & Shahabuddin, P. (2001a). Fast simulation of Markov chains with small transition probabilities. Management Science, 47(4), 547–562. CrossRefGoogle Scholar
  13. Juneja, S., & Shahabuddin, P. (2001b). Splitting-based importance sampling algorithms for fast simulation of Markov reliability models with general repair policies. IEEE Transactions on Reliability, 50(3), 235–245. CrossRefGoogle Scholar
  14. Juneja, S., & Shahabuddin, P. (2006). Rare event simulation techniques: An introduction and recent advances. In S. G. Henderson & B. L. Nelson (Eds.) Simulation, handbooks in operations research and management science (pp. 291–350). Amsterdam: Elsevier, Chap. 11. Google Scholar
  15. Kollman, C., Baggerly, K., Cox, D., & Picard, R. (1999). Adaptive importance sampling on discrete Markov chains. Annals of Applied Probability, 9(2), 391–412. CrossRefGoogle Scholar
  16. L’Ecuyer, P., & Buist, E. (2005). Simulation in Java with SSJ. In M. E. Kuhl & N. M. Steiger, F. B. Armstrong, & J. A. Joines (Eds.), Proceedings of the 2005 winter simulation conference (pp. 611–620). New York: IEEE Press. CrossRefGoogle Scholar
  17. L’Ecuyer, P., & Tuffin, B. (2006). Splitting with weight windows to control the likelihood ratio in importance sampling. In Proceedings of Value Tools 2006: International conference on performance evaluation methodologies and tools (p. 7). New York: ACM. Google Scholar
  18. L’Ecuyer, P., & Tuffin, B. (2007). Effective approximation of zero-variance simulation in a reliability setting. In Proceedings of the 2007 European simulation and modeling conference (pp. 48–54). Ghent: EUROSIS. Google Scholar
  19. L’Ecuyer, P., & Tuffin, B. (2008). Approximate zero-variance simulation. In Proceedings of the 2008 winter simulation conference (pp. 170–181). New York: IEEE Press. CrossRefGoogle Scholar
  20. L’Ecuyer, P., Blanchet, J. H., Tuffin, B., & Glynn, P. W. (2008). Asymptotic robustness of estimators in rare-event simulation. ACM Transactions on Modelling and Computer Simulation (to appear). Google Scholar
  21. Nakayama, M. K. (1996). General conditions for bounded relative error in simulations of highly reliable Markovian systems. Advances in Applied Probability, 28, 687–727. CrossRefGoogle Scholar
  22. Nakayama, M. K., & Shahabuddin, P. (2004). Quick simulation methods for estimating the unreliability of regenerative models of large highly reliable systems. Probability in the Engineering and Information Sciences, 18, 339–368. Google Scholar
  23. Nicola, V. F., Shahabuddin, P., & Nakayama, M. K. (2001). Techniques for fast simulation models of highly dependable systems. IEEE Transactions on Reliability, 50(3), 246–264. CrossRefGoogle Scholar
  24. Rubinstein, R. Y., & Shapiro, A. (1993). Discrete event systems: sensitivity analysis and stochastic optimization by the score function method. New York: Wiley. Google Scholar
  25. Shahabuddin, P. (1994a). Fast transient simulation of Markovian models of highly dependable systems. Performance Evaluation, 20, 267–286. CrossRefGoogle Scholar
  26. Shahabuddin, P. (1994b). Importance sampling for the simulation of highly reliable Markovian systems. Management Science, 40(3), 333–352. CrossRefGoogle Scholar
  27. Shahabuddin, P., Nicola, V. F., Heidelberger, P., Goyal, A., & Glynn, P. W. (1988). Variance reduction in mean time to failure simulations. In Proceedings of the 1988 winter simulation conference (pp. 491–499). New York: IEEE Press. CrossRefGoogle Scholar
  28. Spanier, J. (1971). A new multistage procedure for systematic variance reduction in Monte Carlo. SIAM Journal on Numerical Analysis, 8, 548–554. CrossRefGoogle Scholar
  29. Tuffin, B. (2004). On numerical problems in simulations of highly reliable Markovian systems. In Proceedings of the 1st international conference on quantitative evaluation of systems (QEST) (pp. 156–164). Los Alamitos: IEEE Comput. Soc. Press. CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Département d’Informatique et de Recherche OpérationnelleUniversité de MontréalMontréalCanada
  2. 2.INRIA Rennes—Bretagne AtlantiqueCampus Universitaire de BeaulieuRennes CedexFrance

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