The transient analysis of the queue-length distribution in the batch arrival system with N-policy, multiple vacations and setup times

A batch arrival queueing system of the M(X)/G/1 type with unlimited queue is considered. After each busy period the server begins a multiple vacation period, consisting of independent single vacations, when the service process is blocked. The server begins successive single vacations as far as at the end of one of them the number of customers waiting in the queue equals at least N. The service of the first customer after the vacation period is preceded by a setup time. The analysis of the queue-size distribution on the first vacation cycle is directed to the analysis of the same characteristic in the corresponding "usual" system with unremovable server on its first busy period. The renewal-theory approach is used to obtain results in the general case. As main result the explicit representation for the LT of queue-size distribution is derived for the original system.