The M/M/1 queue with working vacations and vacation interruptions

In this paper, we study the M/M/1 queue with working vacations and vacation interruptions. The working vacation is introduced recently, during which the server can still provide service on the original ongoing work at a lower rate. Meanwhile, we introduce a new policy: the server can come back from the vacation to the normal working level once some indices of the system, such as the number of customers, achieve a certain value in the vacation period. The server may come back from the vacation without completing the vacation. Such policy is called vacation interruption. We connect the above mentioned two policies and assume that if there are customers in the system after a service completion during the vacation period, the server will come back to the normal working level. In terms of the quasi birth and death process and matrix-geometric solution method, we obtain the distributions and the stochastic decomposition structures for the number of customers and the waiting time and provide some indices of systems. This work was supported in part by National Natural Science Foundation of China under Grant No. 10671170. Jihong Li is studying for her Ph.D degree in the College of Economics and Management, Yanshan University in China. Her research interests are queueing models with vacations, production management and performance analysis of Communication Networks. She has published several papers in international journals, such as Applied Mathematics and Computation (AMC) and the other journals in China. Naishuo Tian is a professor in the College of Sciences, Yanshan University in China. His research interests are stochastic process, queueing systems, and performance evaluation models in computer networks and supply chain systems. He has published many papers in several international journals including Queueing Systems, Stochastic Models, Performance Evaluation, Operations Research Letters, EJOR, INFOR and several other journals in China.