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Mathematical Problems in Engineering
Volume 2017, Article ID 3246934, 14 pages
Research Article

On the Queue with Vacations and Multiple Service Phases

School of Science, Nanjing University of Science and Technology, Nanjing 210094, China

Correspondence should be addressed to Jianjun Li; moc.361@ilj_j

Received 6 March 2017; Revised 27 June 2017; Accepted 9 July 2017; Published 8 August 2017

Academic Editor: J.-C. Cortés

Copyright © 2017 Jianjun Li and Liwei Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper considers a queue with vacations and multiple service phases. Whenever the system becomes empty, the server takes a vacation, causing the system to move to vacation phase . If the server returns from a vacation to find no customer waiting, another vacation begins. Otherwise, the system jumps from phase to some service phase with probability Using the matrix geometric solution method and semi-Markov process, we obtain the distributions of the stationary system size at both arrival and arbitrary epochs. The distribution of the stationary waiting time of an arbitrary customer is also derived. In addition, we present some performance measures such as mean waiting time of an arbitrary customer, mean length of the type- cycle, and mean number of customers in the system at the end of phase . Finally, some numerical examples are presented.