Convergence to Equilibrium States for Fluid Models of Many-server Queues with Abandonment

TL;DR

This paper proves that solutions to fluid models of many-server queues with abandonment converge to equilibrium states under mild conditions, using measure-valued process techniques to track remaining times.

Contribution

It establishes convergence conditions for fluid models of many-server queues with abandonment, extending prior understanding of their long-term behavior.

Findings

Convergence of fluid models to equilibrium states is proven.

A mild condition for convergence is identified.

The measure-valued process framework is employed for analysis.

Abstract

Fluid models have become an important tool for the study of many-server queues with general service and patience time distributions. The equilibrium state of a fluid model has been revealed by Whitt (2006) and shown to yield reasonable approximations to the steady state of the original stochastic systems. However, it remains an open question whether the solution to a fluid model converges to the equilibrium state and under what condition. We show in this paper that the convergence holds under a mild condition. Our method builds on the framework of measure-valued processes developed in Zhang (2013), which keeps track of the remaining patience and service times.