Approximating zero-variance importance sampling in a reliability setting

Abstract

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.

Probabilistic Bounded Relative Error Property for Learning Rare Event Simulation Techniques

“…Then BRE will be obtained. But, following exactly the proof of Theorem 2 in [9], thanks to our assumptions, we can show that p 0 (ε) → 1 as ε → 0, hence the result. This basically comes from for any path x are such that…”

“…We are interested in efficient importance sampling estimators of these performance measures. More on this type of model and specific IS schemes can be found in [9,10,12,15]. …”

Some Recent Results in Rare Event Estimation

“…The first type of reliability models we are interested in are the so-called highly reliable Markovian systems (HRMS) often used to represent the evolution of multicomponent systems in reliability settings [30] and whose state space increases exponentially with the number of component classes. Assume that we have a number c of types of components, and n i components of type i.…”

“…In this context, c(y, z) = 0 (no cost associated to a transition), but µ(y) = 1 if y is a failed state while µ(y) = 0 if y is the state with all components operational. For all states y that are not failed or fully operational, we proposed in [30] to replace µ(y) byμ(y), the probability of the most likely path from y to a failed state (which can be computed in general in a polynomial time). To get the intuition behind our approximation, remark that as → 0,μ(y) = Θ(µ(y)), i.e., the probability of the path is of the same order in terms of as µ(y).…”

“…It can be proved that the produced estimator satisfies bounded relative error (BRE) under mild conditions, and can even have vanishing relative error (VRE) under slightly stronger conditions [30]. BRE (also called strong efficiency) means that the standard deviation of the estimator divided by its mean is kept bounded as → 0, while for VRE it converges to 0 when → 0 [25].…”