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Control policies for the $$M^ x/G/1$$ queueing system. (English) Zbl 0671.60099
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.

MSC:
 60K25 Queueing theory (aspects of probability theory) 90B22 Queues and service in operations research
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