Random Fluid Limit of an Overloaded Polling Model

TL;DR

This paper analyzes the fluid limit behavior of an overloaded cyclic polling model, revealing its oscillatory and random nature, and introduces a method to determine the finiteness of moments in an M/G/1 queue.

Contribution

It provides the first fluid asymptotics for an overloaded polling system and links it to multitype branching processes, highlighting the model's oscillatory and stochastic properties.

Findings

Fluid limit is oscillatory near zero.

The fluid limit is inherently random.

A new method for finiteness of moments in M/G/1 queues.

Abstract

In the present paper, we study the evolution of an overloaded cyclic polling model that starts empty. Exploiting a connection with multitype branching processes, we derive fluid asymptotics for the joint queue length process. Under passage to the fluid dynamics, the server switches between the queues infinitely many times in any finite time interval causing frequent oscillatory behavior of the fluid limit in the neighborhood of zero. Moreover, the fluid limit is random. Additionally, we suggest a method that establishes finiteness of moments of the busy period in an M/G/1 queue.