A queueing system with random server capacity and multiple control

The authors studying queueing, input and output processes in a queueing system with bulk service and state dependent service delay. The input flow of customers, modulated by a semi-Markov process, is served by a single server that takes batches of a certain fixed size if available or waits until the queue accumulates enough customers for service. In the latter case, the batch taken for service is of random size dependent on the state of the system, while service duration depends both on the state of the system and on the batch size taken. The authors establish a necessary and sufficient condition for equilibrium of the system and obtain the following results: Explicit formulas for steady state distribution of the queueing process, intensity of the input and output processes, and mean values of idle and busy periods. They employ theory of semi-regenerative processes and illustrate the results by a number of examples. In one of them an optimization problem is discussed.