Analysis of a Splitting Estimator for Rare Event Probabilities in Jackson Networks

TL;DR

This paper analyzes a splitting algorithm for estimating rare overflow probabilities in Jackson networks, showing it requires polynomial evaluations related to bottleneck stations, and compares favorably to direct computation methods.

Contribution

It provides the first rigorous analysis demonstrating that the splitting algorithm's computational complexity depends polynomially on the network's bottleneck stations, improving efficiency over direct linear system solutions.

Findings

Splitting algorithm requires O(n^{2β+1}) evaluations for accuracy.

Analysis compares splitting favorably to solving linear systems with O(n^d) variables.

Bottleneck stations significantly influence the computational complexity.

Abstract

We consider a standard splitting algorithm for the rare-event simulation of overflow probabilities in any subset of stations in a Jackson network at level n, starting at a fixed initial position. It was shown in DeanDup09 that a subsolution to the Isaacs equation guarantees that a subexponential number of function evaluations (in n) suffice to estimate such overflow probabilities within a given relative accuracy. Our analysis here shows that in fact O(n^{2{\beta}+1}) function evaluations suffice to achieve a given relative precision, where {\beta} is the number of bottleneck stations in the network. This is the first rigorous analysis that allows to favorably compare splitting against directly computing the overflow probability of interest, which can be evaluated by solving a linear system of equations with O(n^{d}) variables.