Heavy-traffic limit of stationary distributions of a state-dependent queue

TL;DR

This paper investigates the heavy-traffic limit of stationary distributions in a highly general state-dependent queue, establishing conditions for tightness and explicit limits, with special results for multi-level queues.

Contribution

It extends heavy-traffic analysis to the most general state-dependent queues, providing new conditions for the limit distribution and demonstrating the results for multi-level queues.

Findings

Stationary distributions are tight if the drift's heavy-traffic limit exists and is negative.

Explicit heavy-traffic limits are derived under a density condition.

The density condition is always satisfied for multi-level queues.

Abstract

Inspired by the work of Atar and Miyazawa [1] (2026) as well as applications to energy-saving problems, we are interested in the heavy-traffic limit of the stationary queue length distribution, which is not addressed in [1]. In this paper, we consider this heavy-traffic limit for the single server queue which has the most general possible state-dependence. Namely, arrival and service speeds may take any values depending on the queue length. Here, the terminology, heavy-traffic limit, stands for a diffusion-scaled limit in heavy-traffic for processes, distributions and modeling primitives. This general model is referred to as a state-dependent queue. There are two motivations for this generalization. One is interest in the state-dependent queue itself because it allows finer control of service speed in application. Another is making it clear how the heavy-traffic limit is obtained under what conditions for the state-dependent queue. Thus, we start to study basic properties of this state-dependent queue, including its stability. We then take the sequence of the stationary distributions of its diffusion scaled queue-length processes. We have three main results for this sequence. We first show that it is tight if the heavy-traffic limit of their drifts exists and is negative as the queue length goes to infinity, where a drift is the arrival speed minus the service speed. We next assume the condition that the limit of every vaguely convergent subsequence has a density, which is referred to as a density condition, and show that the heavy-traffic limit of the stationary distributions is obtained in a closed form if and only if that negative drift condition holds. We then show that the density condition is always satisfied for the multi-level queue, so the problem is nicely solved for the multi-level queue.