On the $M_t/M_t/K_t+M_t$ queue in heavy traffic

TL;DR

This paper investigates the long-term behavior of customer numbers in time-periodic many-server queues with abandonment, deriving limit theorems under heavy traffic conditions for various queue models.

Contribution

It provides new limit theorems for the periodic number-of-customers process in heavy traffic, extending to general and steady-state queue models.

Findings

Limit theorems for periodic customer numbers under fluid and diffusion scaling.

Asymptotic results for general time-dependent queues.

Steady-state limits for time-homogeneous queues.

Abstract

The focus of this paper is on the asymptotics of large-time numbers of customers in time-periodic Markovian many-server queues with customer abandonment in heavy traffic. Limit theorems are obtained for the periodic number-of-customers processes under the fluid and diffusion scalings. Other results concern limits for general time-dependent queues and for time-homogeneous queues in steady state.