On many-server queues in heavy traffic

TL;DR

This paper proves a heavy-traffic limit theorem for the number of customers in large many-server queues, without assuming critical loading, and discusses convergence conditions and new results for infinite-server models.

Contribution

It introduces a heavy-traffic limit theorem for many-server queues with general conditions, extending previous results and including new findings for infinite-server scenarios.

Findings

Established convergence in distribution for large-server queues

Provided conditions for convergence in the topology of compact convergence

Presented new results related to infinite-server models

Abstract

We establish a heavy-traffic limit theorem on convergence in distribution for the number of customers in a many-server queue when the number of servers tends to infinity. No critical loading condition is assumed. Generally, the limit process does not have trajectories in the Skorohod space. We give conditions for the convergence to hold in the topology of compact convergence. Some new results for an infinite server are also provided.