We consider an M/G/1 queue where the server may take repeated vacations. Whenever a busy period terminates (i.e. when the queue empties) the server takes a vacation of random duration. At the end of each vacation the server may either take a new vacation or resume service provided that the system is nonempty. The decision to turn the server on/off may depend on all the history of the process (which includes the number of customers and all past decisions). The optimization problem typically arises when imposing a cost structure that involves a holding cost per unit time and per customer and a cost for turning the server on (a shut-down cost may also be included in the latter cost). One may wish to restrict to threshold policies where the server is turned on at the end of each vacation if and only if the queue-length is greater than or equal to a fixed threshold. A few recent papers address the problem of optimally choosing the threshold. The objective of this paper is to establish the optimality of threshold policies over all policies for two long-run average cost criteria.