• 论文 •

### 基于 Min$\left( N,D,V \right)$- 策略和单重休假且休假不中断的~$M/G/1$ 排队系统

1. 四川师范大学数学科学学院,成都  610068
• 出版日期:2021-02-25 发布日期:2021-04-19

LI Dan,TANG Yinghui. $M/G/1$ Queuing System Based on Min$(N,D,V)$-Policy and Single Server Vacation Without Interruption[J]. Journal of Systems Science and Mathematical Sciences, 2021, 41(2): 533-556.

### $M/G/1$ Queuing System Based on Min$(N,D,V)$-Policy and Single Server Vacation Without Interruption

LI Dan ,TANG Yinghui

1. School of Mathematical Sciences, Sichuan Normal University, Chengdu 610068
• Online:2021-02-25 Published:2021-04-19

This paper discusses the~$M/G/1$ queuing system for single server vacation without interruption under the control of Min$(N,D,V)$- policy, in which when the server is transferred on vacation, the server starts service immediately if the number of waiting customers is no less than a given positive integer threshold~$N$ or the total service times of waiting customers is no less than a given positive threshold ~$D$, whichever occurs first. Using the renewal process theory, the total probability decomposition technique and Laplace transform tool, we discuss the transient queue length distribution and the steady-state queue length distribution. Both the expressions of the Laplace transformation of the transient queue length distribution with respect to time~$t$ and the recursive expressions of the steady-state queue length distribution are obtained. Furthermore, the stochastic decomposition structure of the steady-state queue length and the explicit expression of the additional queue length distribution are presented. Meanwhile, some special cases, such as~$N\to \infty$, or~$D\to \infty$, or~$p\left\{ V=0 \right\}=1$, and etc., are discussed. The error in the Tang and Liu (2008) is also pointed. Finally, the explicit expression of the long-run expected cost rate is derived under a given cost structure. And by numerical calculation, we determine the optimal control policy $({{N}^{*}},{{D}^{*}})$ for minimizing the long-run expected cost rate.

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