Path-ZVA: general, efficient and automated importance sampling for highly reliable Markovian systems

TL;DR

Path-ZVA is an importance sampling method that efficiently estimates rare event probabilities in Markov chains by leveraging shortest path analysis, reducing the state space needed for accurate simulation.

Contribution

The paper introduces Path-ZVA, a novel importance sampling technique that uses graph analysis to focus on relevant states, improving efficiency in estimating rare event probabilities in Markovian systems.

Findings

Path-ZVA outperforms existing importance sampling methods in several models.

The method is particularly effective when system reliability is due to component reliability.

Significant reduction in computational effort compared to traditional methods.

Abstract

We introduce Path-ZVA: an efficient simulation technique for estimating the probability of reaching a rare goal state before a regeneration state in a (discrete-time) Markov chain. Standard Monte Carlo simulation techniques do not work well for rare events, so we use importance sampling; i.e., we change the probability measure governing the Markov chain such that transitions `towards' the goal state become more likely. To do this we need an idea of distance to the goal state, so some level of knowledge of the Markov chain is required. In this paper, we use graph analysis to obtain this knowledge. In particular, we focus on knowledge of the shortest paths (in terms of `rare' transitions) to the goal state. We show that only a subset of the (possibly huge) state space needs to be considered. This is effective when the high dependability of the system is primarily due to high component reliability, but less so when it is due to high redundancies. For several models we compare our results to well-known importance sampling methods from the literature and demonstrate the large potential gains of our method.