On a first passage problem in general queueing systems with multiple vacations

The author studies a generalized single-server queueing system with bulk arrivals and batch service, where the server takes vacations each time the queue level falls below r (¸•1) in accordance with the multiple vacation discipline. The input to the system is assumed to be a compound Poisson process modulated by the system and the service is assumed to be state dependent. One of the essential part in the analysis of the system is the employment of new techniques related to te first excess level processes. A preliminary analysis of such processes and recent results of the author on modulated processes enabled the author to obtain all major characteristics for the queueing process explicitly. Various examples and applications are discussed.