Exact Sampling of the Infinite Horizon Maximum of a Random Walk Over a Non-linear Boundary

TL;DR

This paper introduces a novel exact sampling algorithm for the maximum of a mean-zero random walk over a non-linear boundary, with applications to simulating steady-state queues with infinite mean service times.

Contribution

It presents the first finite expected runtime algorithm for sampling the maximum of a random walk over a non-linear boundary and applies it to queue simulation with infinite mean service times.

Findings

Algorithm has finite expected running time.

First exact simulation method for queues with infinite mean service times.

Successfully samples the maximum of a random walk over a non-linear boundary.

Abstract

We present the first algorithm that samples $\max_{n\geq0}\{S_{n}-n^{\alpha}\},$ where $S_n$ is a mean zero random walk, and $n^{\alpha}$ with $\alpha\in(1/2,1)$ defines a nonliner boundary. We show that our algorithm has finite expected running time. We also apply the algorithm to construct the first exact simulation method for the steady-state departure process of a $GI/GI/\infty$ queue where the service time distribution has infinite mean.