Stability and busy periods in a multiclass queue with state-dependent arrival rates

TL;DR

This paper studies a multiclass queue with state-dependent arrival rates based on the current job in service, providing stability conditions, busy period analysis, and tail asymptotics for heavy-tailed distributions.

Contribution

It introduces a novel queueing model with arrival rates depending on the in-service job, extending existing models and analyzing stability and busy period distributions.

Findings

Derived necessary and sufficient stability conditions.

Provided Laplace-Stieltjes transform for busy periods.

Established tail asymptotics for heavy-tailed service times.

Abstract

We introduce a multiclass single-server queueing system in which the arrival rates depend on the current job in service. The system is characterized by a matrix of arrival rates in lieu of a vector of arrival rates. Our proposed model departs from existing state-dependent queueing models in which the parameters depend primarily on the number of jobs in the system rather than on the job in service. We formulate the queueing model and its corresponding fluid model and proceed to obtain the necessary and sufficient conditions for stability via fluid models. Utilizing the natural connection with the multitype Galton-Watson processes, the Laplace-Stieltjes transform of busy periods in the system is given. We conclude with tail asymptotics for the busy period for heavy-tailed service time distributions for the regularly varying case.