Fractional Bessel Process with Constant Drift: Spectral Analysis and Queueing Applications

TL;DR

This paper introduces a fractional Bessel process with constant drift, providing spectral analysis and demonstrating its relevance in modeling subdiffusive phenomena and queueing systems.

Contribution

It extends classical Bessel processes to fractional settings with explicit spectral representation, enabling new analytical insights and applications.

Findings

Derived explicit spectral representation of the process

Established stationary distribution and correlation structure

Applied results to queueing theory problems

Abstract

We introduce a fractional Bessel process with constant negative drift, defined as a time-changed Bessel process via the inverse of a stable subordinator, independent of the base process. This construction yields a model capable of capturing subdiffusive behavior and long-range dependence, relevant in various complex systems. We derive an explicit spectral representation of its transition density, extending the non-fractional setting. Using this representation, we establish several analytical properties of the process, including its stationary distribution and correlation structure. These results provide new insights into the behavior of fractional diffusions and offer analytical tools for applications in queueing theory, mathematical finance, and related domains. In particular, we demonstrate their applicability through a concrete problem in queueing theory.