New fluctuation analysis of D-policy bulk queues with multiple vacations

This paper offers new studies of the queueing process in D-policy models with Poisson bulk input, general service time, and multiple vacations. The D-policy applies when the system, after being exhausted, suspends service until the total workload crosses D (≥ 0), thereby accumulating some number of customers. The key investigation is that of the queueing process during the “first service cycle”, which includes the first vacation period when all customers line up waiting for the server to return and a period when all those customers are processed. For this model, neither the servicing process is renewal, nor the queueing process is semiregenerative relative to the departure epochs. To overcome the problem, the paper explores a new fluctuation technique of multivariate marked counting processes. It includes the time dependent analysis of queueing and busy period processes, specially developed for this process, and it yields their stationary distributions in closed analytic forms.