# Fractional Erlang Queues

## TL;DR

This paper introduces a fractional generalization of Erlang queues using inverse stable subordinators, deriving equations, analyzing properties, and providing simulation algorithms for this new queue model.

## Contribution

It presents the first fractional Erlang queue model, extending classical queues with fractional calculus and stable subordinators, and offers analytical and simulation tools.

## Key findings

- Derived fractional Kolmogorov forward equations
- Analyzed mean queue length and busy period distribution
- Developed algorithms for simulating fractional queue paths

## Abstract

We introduce a fractional generalization of the Erlang Queues $M/E_k/1$. Such process is obtained through a time-change via inverse stable subordinator of the classical queue process. We first exploit the (fractional) Kolmogorov forward equation for such process, then we use such equation to obtain an interpretation of this process in the queuing theory context. Then we also exploit the transient state probabilities and some features of this fractional queue model, such as the mean queue length, the distribution of the busy periods and some conditional distributions of the waiting times. Finally, we provide some algorithms to simulate their sample paths.