On Derivative Estimation of the Mean Time to Failure in Simulations of Highly Reliable Markovian Systems

The mean time to failure (MTTF) of a Markovian system can be expressed as a ratio of two expectations. For highly reliable Markovian systems, the resulting ratio formula consists of one expectation that cannot be estimated with bounded relative error when using standard simulation, while the other, which we call a non-rare expectation, can be estimated with bounded relative error. We show that some derivatives of the nonrare expectation cannot be estimated with bounded relative error when using standard simulation, which in turn may lead to an estimator of the derivative of the MTTF that has unbounded relative error. However, if particular importance-sampling methods (e.g., balanced failure biasing) are used, then the estimator of the derivative of the non-rare expectation will have bounded relative error, which (under certain conditions) will yield an estimator of the derivative of the MTTF with bounded relative error. Subject classifications: Probability, stochastic model applicatio..

Data-Driven Methods and Applications for Optimization under Uncertainty and Rare-Event Simulation

For most of decisions or system designs in practice, there exist chances of severe hazards or system failures that can be catastrophic. The occurrence of such hazards is usually uncertain, and hence it is important to measure and analyze the associated risks. As a powerful tool for estimating risks, rare-event simulation techniques are used to improve the efficiency of the estimation when the risk occurs with an extremely small probability. Furthermore, one can utilize the risk measurements to achieve better decisions or designs. This can be achieved by modeling the task into a chance constrained optimization problem, which optimizes an objective with a controlled risk level. However, recent problems in practice have become more data-driven and hence brought new challenges to the existing literature in these two domains. In this dissertation, we will discuss challenges and remedies in data-driven problems for rare-event simulation and chance constrained problems. We propose a robust optimization based framework for approaching chance constrained optimization problems under a data-driven setting. We also analyze the impact of tail uncertainty in data-driven rare-event simulation tasks. On the other hand, due to recent breakthroughs in machine learning techniques, the development of intelligent physical systems, e.g. autonomous vehicles, have been actively investigated. Since these systems can cause catastrophes to public safety, the evaluation of their machine learning components and system performance is crucial. This dissertation will cover problems arising in the evaluation of such systems. We propose an importance sampling scheme for estimating rare events defined by machine learning predictors. Lastly, we discuss an application project in evaluating the safety of autonomous vehicle driving algorithms.PHDIndustrial & Operations EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/163270/1/zhyhuang_1.pd