Journal of Systems Science and Complexity

, Volume 30, Issue 6, pp 1382–1402 | Cite as

The structure of departure process and optimal control strategy N* for Geo/G/1 discrete-time queue with multiple server vacations and Min(N, V)-Policy

  • Shaojun Lan
  • Yinghui TangEmail author


This paper considers the departure process and the optimal control strategy for a discretetime Geo/G/1 queueing model in which the system operates under the control of multiple server vacations and Min(N, V)-policy. Using the law of total probability decomposition, the renewal theory and the probability generating function technique, the transient and the steady-state probabilities that the server is busy at any epoch n+ are derived. The authors also obtain the explicit expression of the probability generating function for the expected number of departures occurring in the time interval (0+, n+] from any initial state. Meanwhile, the relationship among departure process, server’s state process and service renewal process in server busy period is found, which shows the special structure of departure process. Especially, some corresponding results of departure process for special discrete-time queues are directly gained by our results. Furthermore, the approximate expansion for calculating the expected number of departures is presented. In addition, some other important performance measures, including the expected length of server busy period, server’s actual vacation period and busy cycle period etc., are analyzed. Finally, some numerical results are provided to determine the optimum value N* for minimizing the system cost under a given cost structure.


Cost optimization departure process discrete-time queue Min(N, V)-policy multiple server vacations 


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Copyright information

© Institute of Systems Science, Academy of Mathematics and Systems Science, CAS and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.School of Mathematics and Software ScienceSichuan Normal UniversityChengduChina
  2. 2.School of Mathematics and StatisticsSichuan University of Science and EngineeringZigongChina
  3. 3.School of Fundamental EducationSichuan Normal UniversityChengduChina

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