An M^X/G/1 queue with randomized working vacations and at most J vacations

Abstract

This paper treats a bulk arrival queue with randomized working vacation policy. Whenever the system becomes empty, the server takes a vacation. During the vacation period, customers are to be served at a lower rate. Once the vacation ends, the server will return to the normal working state and begin to serve the customers in the system if any. Otherwise, the server either remains idle with probability p or leaves for another vacation with probability 1−p. This pattern continues until the number of vacations taken reaches J. If the system is empty at the end of the Jth vacation, the server will wait idly for a new arrival. By using supplementary variable technique, we derive the system size distribution at arbitrary epoch, at departure epoch and at busy period initial epoch, as well as some important system characteristics. Numerical examples are provided to illustrate the influence of system parameters on several performance measures.

Keywords:

• queue
• batch arrival
• randomized working vacations
• supplementary variable
• system characteristics

2000 AMS Subject Classifications:

• 60K25
• 90B22

Acknowledgements

The authors would like to thank the editor and the anonymous referees for their valuable comments and suggestions, which improved the presentation of this paper. Many thanks are due to Prof. Jinting Wang who kindly gave some help to edit this paper. The authors also thank the support from National Natural Science Foundation of China (Nos. 11171179, 11171019, 11201123), Natural Science Foundation of Education Bureau of Anhui Province (No. KJ2013Z254) and the Tianyuan Fund for Mathematics (No. 11226200).