Heavy traffic limit of stationary distribution of the multi-level single server queue

TL;DR

This paper derives a closed-form heavy traffic limit for the stationary distribution of a multi-level single server queue, confirming the interchange of process and distribution limits in heavy traffic.

Contribution

It provides the first explicit stationary distribution limit for the multi-level queue, extending previous process limit results and confirming the distributional interchange in heavy traffic.

Findings

Stationary distribution converges to a known reflected diffusion distribution.

Method from Miyazawa (2025) applies successfully to this multi-level queue.

Results match the stationary distribution of the limiting diffusion process.

Abstract

Atar and Miyazawa recently introduced a single server queue with queue length dependent arrival and service processes, and name it a multi-level queue. They prove that the heavy traffic limit of its queue length process weakly converges to a reflected diffusion with discontinuously state-dependent drift and deviations. We derive the heavy traffic limit of the stationary queue length distribution of this multi-level queue in a closed form, which agrees with the stationary distribution of the reflected diffusion obtained by Miyazawa (2024, Journal of the Indian Society for Probability and Statistics). Thus, those results show the limit interchange of process and stationary distribution in heavy traffic. The multi-level queue is a simpler version of the 2-level GI/G/1 queue of Miyazawa (2025, Advances in Applied Probability, to appear) and its extension for multi-levels. For this 2-level queue in heavy traffic, the process limit is unknown, and the distributional limit is obtained for limited cases under extra conditions. Nevertheless, it is shown that the method developed in Miyazawa (2025) perfectly works for the present multi-level queue.