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

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 behaviour of the system performance to validate the analytical results.