A Note on Many-server Fluid Models with Time-varying Arrivals

TL;DR

This paper extends measure-valued fluid models for many-server queues to incorporate time-varying arrivals, revealing that different models are governed by the same convolution equation, thus unifying their analysis.

Contribution

It introduces a novel extension of measure-valued fluid models to handle time-varying arrivals and establishes their equivalence through a shared convolution equation.

Findings

The extended fluid model accurately captures time-varying arrival processes.

Both measure-valued models are characterized by the same convolution equation.

The analysis unifies different fluid models under a common framework.

Abstract

We extend the measure-valued fluid model, which tracks residuals of patience and service times, to allow for time-varying arrivals. The fluid model can be characterized by a one-dimensional convolution equation involving both the patience and service time distributions. We also make an interesting connection to the measure-valued fluid model tracking the elapsed waiting and service times. Our analysis shows that the two fluid models are actually characterized by the same one-dimensional convolution equation.