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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2019, Vol. 36 ›› Issue (3): 344-358.doi: 10.3969/j.issn.1005-3085.2019.03.010

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带有启动时间、单重工作休假及休假终止的负顾客排队模型分析（英）

周宗好1,  周甄川1,  朱翼隽2,  石志岩2,  鲍志晖1   

  1. 1- 黄山学院数学与统计学院，黄山  245041 
    2- 江苏大学理学院，镇江  212013
  • 收稿日期:2016-06-06 接受日期:2019-02-22 出版日期:2019-06-15 发布日期:2019-08-15
  • 基金资助:
    安徽省教育厅重点基金(gxyq2017075;KJ2015A166;KJ2011Z364)；黄山大学基础研究基金(2014xkjq006).

Analysis of the Single Working Vacation and Vacation Interruption G-queue with Setup Times

ZHOU Zong-hao1,  ZHOU Zhen-chuan1,  ZHU Yi-jun2,  SHI Zhi-yan2,  BAO Zhi-hui1   

  1. 1- School of Mathematics and Statistics, Huangshan University, Huangshan 245041
    2- Faculty of Science, Jiangsu University, Zhenjiang 212013
  • Received:2016-06-06 Accepted:2019-02-22 Online:2019-06-15 Published:2019-08-15
  • Supported by:
    The Anhui Provincial Department of Education Key Fund (gxyq2017075; KJ2015A166; KJ2011Z364); the Fundamental Research Funds of Huangshan University (2014xkjq006).

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摘要： 本文建立带有启动时间的工作休假及休假中止的负顾客排队模型．在正规忙期里，如果服务完成一个顾客系统中没有顾客，系统将进入一个随机长度的工作休假．系统有两种情况从工作休假状态恢复到正常工作状态：一种情况是工作休假期间服务完成一个顾客后队列中有顾客；另一种情况是一个工作休假周期结束队列中有顾客，以前没有服务完成的服务时间无效．如果一个工作休假周期结束队列中没有顾客，系统则关闭，关闭后的首次接收服务前服务台必须有一个随机长度的启动时间．本文运用拟生灭过程和矩阵几何解的方法解析地求出了系统的稳态状态概率分布和平均排队长．通过分布函数法求出正规忙期里到达顾客的平均等待时间．另外，数值模拟了系统参数变化对以上排队指标的影响．

关键词: 单重工作休假, 休假终止, 拟生灭过程, 矩阵几何解

Abstract: This paper considers the single working vacation and vacation interruption G-queue with negative customers and setup times. When a service is completed during a regular busy period, if there is no customer in the system, the server begins a working vacation of random length. There are two ways that the system transfers to the regular service period, one is that there are customers in the queue after the completion of a service during the working vacation. Another is that there are customers in the queue after the completion of a vacation, but the service time before is invalid for the unfinished service. If there is no customer in the system after the completion of a service during the working vacation, the system turns off. The service of the first positive customer must take a setup time from the close-down server. Using the quasi birth death process and matrix-geometric-solution method, the steady-state distribution for the steady probabilities of the system and the queue length distribution are obtained. The mean waiting time of arriving customers in the regular busy period is derived by solving the distribution function. Additionally, we provide the numerical examples to illustrate the effect of the parameters on several performance characteristics mentioned above.

Key words: single working vacation, vacation interruption, QBD process, matrix-geometric-solution

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