# Linear Stochastic Fluid Networks: Rare-Event Simulation and Markov Modulation

## TL;DR

This paper develops efficient importance sampling algorithms for rare-event probabilities in Markov-modulated linear stochastic fluid networks, providing theoretical guarantees and analyzing their performance in different regimes.

## Contribution

It introduces importance sampling methods with provable efficiency for Markov-modulated fluid networks and extends techniques from light-tailed rare-event simulation.

## Key findings

- Algorithm is asymptotically efficient in slow modulation regime
- Number of runs grows polynomially for unmodulated networks
- Provides recursion for moments of storage levels

## Abstract

We consider a linear stochastic fluid network under Markov modulation, with a focus on the probability that the joint storage level attains a value in a rare set at a given point in time. The main objective is to develop efficient importance sampling algorithms with provable performance guarantees. For linear stochastic fluid networks without modulation, we prove that the number of runs needed (so as to obtain an estimate with a given precision) increases polynomially (whereas the probability under consideration decays essentially exponentially); for networks operating in the slow modulation regime, our algorithm is asymptotically efficient. Our techniques are in the tradition of the rare-event simulation procedures that were developed for the sample-mean of i.i.d. one-dimensional light-tailed random variables, and intensively use the idea of exponential twisting. In passing, we also point out how to set up a recursion to evaluate the (transient and stationary) moments of the joint storage level in Markov-modulated linear stochastic fluid networks.