Approximating zero-variance importance sampling in a reliability setting
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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.
KeywordsMonte Carlo Rare events Importance sampling
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- Asmussen, S., & Glynn, P. W. (2007). Stochastic simulation. New York: Springer. Google Scholar
- Booth, T. E. (1985). Exponential convergence for Monte Carlo particle transport? Transactions of the American Nuclear Society, 50, 267–268. Google Scholar
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- Rubinstein, R. Y., & Shapiro, A. (1993). Discrete event systems: sensitivity analysis and stochastic optimization by the score function method. New York: Wiley. Google Scholar