On a fractional queueing model with catastrophes

TL;DR

This paper introduces a fractional $M/M/1$ queue model with catastrophes, deriving key probabilistic measures and discussing parameter estimation, extending classical queueing theory with fractional calculus.

Contribution

It formulates a fractional queueing model with catastrophes using fractional derivatives, providing analytical expressions for state probabilities, mean, and variance.

Findings

Derived explicit state probabilities for the fractional queue.

Calculated mean and variance of customers over time.

Discussed parameter estimation methods for the model.

Abstract

A $M/M/1$ queue with catastrophes is a modified $M/M/1$ queue model for which, according to the times of a Poisson process, catastrophes occur leaving the system empty. In this work, we study a fractional $M/M/1$ queue with catastrophes, which is formulated by considering fractional derivatives in the Kolmogorov's Forward Equations of the original Markov process. For the resulting fractional process, we obtain the state probabilities, the mean and the variance for the number of customers at any time. In addition, we discuss the estimation of parameters.