The Distribution of Queue-length for the M/G/1 Queueing System under Bi-level Threshold Control

This paper studies a like- queue production system under bi-level control policy, where server operates N-policy with an early startup. The server is turned off whenever the system becomes empty. When the accumulated number of customer reaches m (< N), the system commences a startup with random time. After the startup, if there is N or more customers in the queue, the server begins to server the customers immediately. Otherwise the sever remains dormant until the number of customers reaches N. We model this system M/G/1 queue under (m, N) policy. By a concise decomposition method, the transient and steady-state queue-length distribution is obtained. Several common queueing systems are special cases of the bi-level threshold in this study. Finally, a numerical example is give to illustrate the importance of queue-length distribution in design and control.