A multilevel control bulk queueing system with vacationing server

This article studies a Markov chain describing the evolution of the queue in a general single-server bulk queueing system with continuously operating (or, equivalently vacationing) server, semi-Markov modulated compound Poisson input, queue length dependent service time, and multilevel control server capacity. The authors establish a necessary and sufficient criterion for ergodicity of this Markov chain and find its stationary distribution. As an example a single-server queueing system is considered, and some explicit results are obtained in this case.