A Relation between Queue-Length Distributions during Server Vacations in Queues with Batch Arrivals, Batch Services, or Multiclass Arrivals: An Extension of Burke's Theorem

This paper presents a relation between queue-length distributions during vacations and those at vacation start and end times, in M/G/1 vacation queues with batch arrivals, batch services or multi-class arrivals. In order to obtain the relation, we develop a new approach that extends Burke's theorem to queues with batch arrivals, batch services or multi-class services. We also give an example to demonstrate how to apply the proposed relation to the analysis of such a queue.