First passage times to congested states of many-server systems in the Halfin-Whitt regime

TL;DR

This paper analyzes the time it takes for a large, heavily loaded multi-server queueing system to reach congested states, using a diffusion approximation in the Halfin-Whitt heavy-traffic regime.

Contribution

It introduces a diffusion model combining Brownian motion and Ornstein-Uhlenbeck processes to approximate first passage times in large queueing systems.

Findings

Derived explicit formulas for first passage times.

Validated the diffusion approximation against queueing system simulations.

Provided insights into congestion onset in many-server queues.

Abstract

We consider the heavy-traffic approximation to the $GI/M/s$ queueing system in the Halfin-Whitt regime, where both the number of servers $s$ and the arrival rate $\lambda$ grow large (taking the service rate as unity), with $\lambda=s-\beta\sqrt{s}$ and $\beta$ some constant. In this asymptotic regime, the queue length process can be approximated by a diffusion process that behaves like a Brownian motion with drift above zero and like an Ornstein-Uhlenbeck process below zero. We analyze the first passage times of this hybrid diffusion process to levels in the state space that represent congested states in the original queueing system.