Abstract
This paper generalizes a popular class of a single-server bulk-input-batch-service queues, with multiple vacations, ( N, r)-hysteretic control, and “gated service”. The latter applies to a two-stage service policy, which lets customers choose to join the service batch during the first stage, not in excess of server capacity ( R), and it locks up the gate during the second stage of service unconditionally. ( N, r)-hysteresis refers to the combination of r-quorum and N-policy, i.e., if the server stops processing customers when the queue drops below r ≤ R and leaves the system for a multiple vacation sequence; service is resumed upon one of the servers' returns, if the queue accumulates to at least N customers. The discrete and continuous time parameter queueing processes are analyzed by means of semiregenerative techniques, along with the recently established results on functionals of a marked Poisson process observed by a renewal process. The steady-state distribution is obtained in a closed analytical form. Other results and special cases include conservation equation and busy period distribution.
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