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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 ›› 2023, Vol. 40 ›› Issue (4): 605-620.doi: 10.3969/j.issn.1005-3085.2023.04.007

• • 上一篇    下一篇

服务时间累积量控制的$Geo^{X}/G/1$排队的稳态队长

刘仁彬1,  唐应辉2   

  1. 1. 重庆理工大学理学院，重庆 400054;
    2. 四川师范大学数学与软件科学学院，成都 610066
  • 收稿日期:2021-01-22 接受日期:2022-06-02 出版日期:2023-08-15 发布日期:2023-10-05
  • 基金资助:
    国家自然科学基金(71571127)；重庆市自然科学基金面上项目(CSTB2022NSCQ-MSX1160).

The Steady-state Queue Size for the $Geo^{X}/G/1$ Queue Controlled by Service Time Backlog

LIU Renbin1,  TANG Yinghui2   

  1. 1. School of Science, Chongqing University of Technology, Chongqing 400054;
    2. School of Mathematics & Software Science, Sichuan Normal University, Chengdu 610066
  • Received:2021-01-22 Accepted:2022-06-02 Online:2023-08-15 Published:2023-10-05
  • Supported by:
    The National Natural Science Foundation of China (71571127); the General program of Chongqing Natural Science Foundation of China (CSTB2022NSCQ-MSX1160).

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摘要：

考虑服务台的启动由服务台闲期所有到达的服务时间累积之和(即顾客即将被服务的时间累积量)控制的离散时间批到达$Geo^{X}/G/1$排队服务系统。顾客成批到达系统，当顾客的被服务时间累积量超过某个非负整数$D$时，服务台立刻为顾客提供服务直到忙期结束(此策略被称为$D$策略)。该模型可为无线传感网络的从业者提供一些理论基础。首先，在准备工作中，讨论了忙期开始时刻的顾客数、服务时间累积量以及服务台的忙期和闲期分布；然后，通过闲、忙期到达顾客的分类和概率分析方法，研究了系统的离去时刻稳态队长和任意时刻$n^{+}$的稳态队长分布的概率母函数。作为特例，分析得到了离散时间$Geo^{X}/G/1$排队和$D$策略离散时间$Geo/G/1$排队的队长分布结果。最后，模拟分析了一类无线传感节点，并在数值上获得了节点的最低能耗。

关键词: 成批到达, 服务时间累积量, 队长分布, 无线传感节点, 最小能耗

Abstract:

A discrete time batch arrival $Geo^{X}/G/1$ queue is considered, in which the startup of server is controlled by the sum of service times of all arrivals (called service time backlog). The customers enter the system in batch arrival. When the service time backlog of all arriving customers exceeds a given non-negative integer $D$, the server starts its service and lasts until a busy period ends (this policy is called the $D$ policy). The model can offer some theoretical basis for the practitioners of wireless sensor network. Firstly, in the preparation work, the queue size and service time backlog at the start of a busy period, and the busy and idle periods are discussed. Then, by the classifications of the customers who arrive during the idle and busy periods, and the method of probabilistic analysis, the steady-state queue sizes at a departure time and an arbitrary time $n^{+}$ are studied. As two special cases, the steady-state queue sizes for the $Geo^{X}/G/1$ and $D$-policy $Geo/G/1$ queueing systems are derived. Finally, a kind of wireless sensor node is modelled, and the minimum power consumption is numerically gotten.

Key words: batch arrival, service time backlog, queue size distribution, wireless sensor node, minimum power consumption

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