# Rare-Event Simulation for Distribution Networks

## TL;DR

This paper develops importance sampling and Monte Carlo algorithms to efficiently estimate the probability of rare high-cost events in equilibrium allocations of distribution networks with Gaussian demands.

## Contribution

It introduces novel algorithms with proven asymptotic efficiency for rare-event probability estimation in network equilibrium models.

## Key findings

- Algorithms demonstrate strong numerical performance.
- Asymptotic efficiency of the proposed methods is established.
- Effective estimation of rare-event probabilities in network models.

## Abstract

We model equilibrium allocations in a distribution network as the solution of a linear program (LP) which minimizes the cost of unserved demands across nodes in the network. The constraints in the LP dictate that once a given node's supply is exhausted, its unserved demand is distributed among neighboring nodes. All nodes do the same and the resulting solution is the equilibrium allocation. Assuming that the demands are random (following a jointly Gaussian law), our goal is to study the probability that the optimal cost (i.e. the cost of unserved demands in equilibrium) exceeds a large threshold, which is a rare event. Our contribution is the development of importance sampling and conditional Monte Carlo algorithms for estimating this probability. We establish the asymptotic efficiency of our algorithms and also present numerical results which illustrate strong performance of our procedures.