The author studies the queueing process in a single‐server, bulk arrival and batch service queueing system with a compound Poisson input, bilevel service delay discipline, start‐up time, and a fixed accumulation level with control operating policy. It is assumed that when the queue length falls below a predefined level r(≥1), the system, with server capacity R, immediately stops service until the queue length reaches or exceeds the second predefined accumulation level N(≥r). Two cases, with N ≤ R and N ≥ R, are studied.The author finds explicitly the probability generating function of the stationary distribution of the queueing process and gives numerical examples.