Transient behavior of fractional queues and related processes

TL;DR

This paper introduces a fractional generalization of the M/M/1 queue, incorporating non-Markovian features, with algorithms and statistical methods to estimate parameters, demonstrated on real financial data.

Contribution

It presents a novel fractional queue model with simulation algorithms and parameter estimation techniques, extending classical queue theory to better fit real-world systems.

Findings

The fractional queue model captures non-Markovian dynamics.

Algorithms enable practical simulation of the model.

Parameter estimators are derived and validated on S&P data.

Abstract

We propose a generalization of the classical M/M/1 queue process. The resulting model is derived by applying fractional derivative operators to a system of difference-differential equations. This generalization includes both non-Markovian and Markovian properties, which naturally provide greater flexibility in modeling real queue systems than its classical counterpart. Algorithms to simulate M/M/1 queue process and the related linear birth-death process are provided. Closed-form expressions of the point and interval estimators of the parameters of these fractional stochastic models are also presented. These methods are necessary to make these models usable in practice. The proposed fractional M/M/1 queue model and the statistical methods are illustrated using S&P data.