$1/f^{3/2}$ noise in heavy traffic M/M/1 queue

TL;DR

This paper analyzes the power spectral density of the M/M/1 queue in heavy traffic, revealing a $1/f^{3/2}$ noise pattern through spectral analysis and numerical validation, linking queue dynamics to complex noise phenomena.

Contribution

It provides explicit spectral relations for the M/M/1 queue, demonstrating the emergence of $1/f^{3/2}$ noise from a simple queuing model, supported by numerical simulations.

Findings

The spectral density exhibits a $1/f^{3/2}$ noise pattern.

Numerical simulations confirm the theoretical spectral analysis.

Similar behavior observed in a continuous time random walk on a ring.

Abstract

We study the power spectral density of continuous time Markov chains and explicit its relationship with the eigenstructure of the infinitesimal generator. This result helps us understand the dynamics of the number of customers for a M/M/1 queuing process in the heavy traffic regime.Closed-form relations for the power law scalings associated to the eigenspectrum of the M/M/1 queue generator are obtained, providing a detailed description of the power spectral density structure, which is shown to exhibit a $1/f^{3/2}$ noise.We confirm this result by numerical simulation.We also show that a continuous time random walk on a ring exhibits very similar behavior.It is remarkable than a complex behavior such as $1/f$ noise can emerge from the M/M/1 queue, which is the ''simplest'' queuing model.