Single-server queues with controlled bulk service, random accumulation level, and modulated input

Abstract

A class of bulk queueing models with one server of random capacity is introduced and studied. It is assumed that upon completion of a previous service, the server takes a group of random size from units that are available. Or, the server waits until the queue attains a desired level. In addition, the input is supposed to be modulated by the queueing process and both service times of groups of units and batch sizes are under control. After establishing an ergodicity criterion for both the Queueing process with continuous time parameter and the embedded process, the author obtains explicit formulas for the stationary distributions of both processes by using semi–regenerative techniques. Various characteristics of related processes (idle and busy periods and intensity of the input) are obtained and examples and special cases are discussed