Markovian bulk-arrival and bulk-service queues with general state-dependent control

We study a modified Markovian bulk-arrival and bulk-service queue incorporating general state-dependent control. The stopped bulk-arrival and bulk-service queue is first investigated, and the relationship between this stopped queue and the full queueing model is examined and exploited. Using this relationship, the equilibrium behaviour for the full queueing process is studied and the probability generating function of the equilibrium distribution is obtained. Queue length behaviour is also examined, and the Laplace transform of the queue length distribution is presented. The important questions regarding hitting times and busy period distributions are answered in detail, and the Laplace transforms of these distributions are presented. Further properties regarding the busy period distributions including expectation and conditional expectation of busy periods are also explored.