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.
