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Analysis of a discrete-time queue with server subject to vacations and breakdowns

This paper studies a Geo/G/1 queue in which the server operates multiple vacations and may break down while working. After all messages have been served in the system, the server leaves for a vacation. Upon returning from the vacation, the server activates immediately to serve if there are some messages waiting in the queue. Otherwise, the server takes another vacation. This queueing system has potential application in the inbound email contact center. By using the generating function technique, we derive some important system characteristics, such as system size distributions of server on vacation/busy/broken-down, the expected lengths of server vacation period and complete period, and message's waiting time in the queue. In particular, a cost model is developed to determine the optimal parameter values of different vacation distributions that minimize the cost, and some numerical examples are presented for illustrative purpose.