Summary: The ${M}^{x}/G/1$ queueing system is studied under the following two situations:

(1) At the end of a busy period, the server is turned off and inspects the length of the queue every time an arrival occurs. When the queue length reaches, or exceeds, a pre-specified value m for the first time, the server is turned on and serves the system until it is empty.

(2) At the end of a busy period, the server takes a sequence of vacations, each for a random amount of time. At the end of each vacation, he inspects the length of the queue. If the queue length is greater than, or equal to, a pre-specified value m at this time, he begins to serve the system until it is empty.

For both cases, the mean waiting time of an arbitrary customer for a given value of m is derived, and the procedure to find the stationary optimal policy under a linear cost structure is presented.