Distribution-valued heavy-traffic limits for the $G/\mathit{GI}/\infty$ queue

TL;DR

This paper develops a novel approach using tempered distribution-valued processes to analyze the heavy-traffic behavior of the $G/\text{GI}/\infty$ queue, revealing that the limits are Ornstein-Uhlenbeck processes.

Contribution

It introduces a new distribution-valued framework for heavy-traffic analysis of infinite-server queues, deriving fluid and diffusion limits as Ornstein-Uhlenbeck processes.

Findings

Diffusion limits are tempered distribution-valued Ornstein-Uhlenbeck processes.

The approach captures age and residual service time distributions in heavy traffic.

The methodology extends functional limit theorems to distribution-valued processes.

Abstract

We study the $G/\mathit{GI}/\infty$ queue in heavy-traffic using tempered distribution-valued processes which track the age and residual service time of each customer in the system. In both cases, we use the continuous mapping theorem together with functional central limit theorem results in order to obtain fluid and diffusion limits for these processes in the space of tempered distribution-valued processes. We find that our diffusion limits are tempered distribution-valued Ornstein-Uhlenbeck processes.