On state dependent batch service queue with single and multiple vacation under Markovian arrival process

TL;DR

This paper analyzes an infinite buffer batch service queue with state-dependent service and vacation rates under a Markovian arrival process, providing analytical results and numerical validation of system performance.

Contribution

It introduces a comprehensive model combining state-dependent batch service and vacations with MAP arrivals, deriving joint distributions using advanced techniques.

Findings

Analytical expressions for joint distributions of system states.

Validation of model through numerical results.

Insights into system performance under various parameters.

Abstract

An infinite buffer batch service vacation queue has been studied where service rate of the batch is dependent on the size of the batch and vacation rate is dependent on the queue size at vacation initiation epoch. The arrivals follow the Markovian arrival process (MAP). For service rule, general bulk service (GBS) rule is considered. The service time and vacation time both are considered to be generally distributed. Several joint distributions of interest are obtained using the bivariate vector generating function method and the supplementary variable technique (SVT). Numerical results are presented to show the behavior of the system performance to validate the analytical results.