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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 ›› 2023, Vol. 40 ›› Issue (4): 621-633.doi: 10.3969/j.issn.1005-3085.2023.04.008

• • 上一篇    下一篇

可选服务的多台工作休假排队系统驱动的流体模型稳态分析

李子坤1,2,  徐秀丽1   

  1. 1. 燕山大学理学院,秦皇岛 066004;
    2. 北华航天工业学院文理学院,廊坊 065000
  • 收稿日期:2021-02-05 接受日期:2022-09-17 出版日期:2023-08-15 发布日期:2023-10-15
  • 通讯作者: 徐秀丽 E-mail: xxl-ysu@163.com
  • 基金资助:
    河北省自然科学基金(A2019203313);河北省高等学校科学研究重点项目(ZD2019079);北华航天工业学院青年基金(KY202203).

Stationary Analysis of Fluid Model Driven by Multi-server Working Vacation Queue with Optional Service

LI Zikun1,2,  XU Xiuli1   

  1. 1. School of Science, Yanshan University, Qinhuangdao 066004;
    2. School of Liberal Arts and Sciences, North China Institute of Aerospace Engineering, Langfang 065000
  • Received:2021-02-05 Accepted:2022-09-17 Online:2023-08-15 Published:2023-10-15
  • Contact: X. Xu. E-mail: xxl-ysu@163.com
  • Supported by:
    The Natural Science Foundation of Hebei Province (A2019203313); the Key Project of Scientific Research in Higher Education of Hebei Province (ZD2019079); the Youth Fund of North China Institute of Aerospace Engineering (KY202203).

摘要:

在当前的大数据时代,流体排队模型已成为服务系统建模分析的有力工具。由于实时服务系统中服务台除了对顾客进行必要的基本服务外,可能还会有可选择的附加服务;另一方面,服务系统可能会因为执行其它任务等各种因素降低服务效率而低速运行。综合考虑上述情况,以具有可选服务和工作休假的多服务台排队系统作为外界随机环境构建了新的流体排队模型。运用拟生灭过程和矩阵几何解方法得到了驱动系统的稳态队长分布,进而给出了流体模型的净输入率结构及其稳态条件。利用概率分析方法和随机过程理论导出了流体模型稳态联合分布满足的矩阵微分方程,并根据其结构特点构建了几个重要的函数序列,进一步通过Laplace变换(LT)和Lapalce-Stieltjes变换(LST)方法得到了平衡状态下缓冲器中流体的平均库存量及空库概率的解析表达式。最后通过数值分析,展示了系统性能指标受参数变化的影响。结果表明在实际问题中可通过调整模型参数优化系统性能,提升服务效率。

关键词: 可选服务, 流体模型, 平均库存量, 空库概率

Abstract:

Fluid queue model has become a powerful tool for service system modeling and analysis. In the real-time service system, customers may have optional additional services in addition to the necessary basic services. On the other hand, the service system may operate at low speeds due to various factors such as performing other tasks, which reduce the service efficiency. Taking the above situation into consideration, a new fluid queue model is constructed based on a multi-server queuing system with optional services and working vacation as an external random environment. The steady-state queue length distribution of driving system is obtained by means of quasi birth-and-death process and matrix-geometric solution method, and the net input rate structure and steady state conditions of the fluid model are given. The matrix differential equation satisfied by the steady-state joint distribution of fluid model is derived by using probability analysis method and stochastic process theory. Several important function sequences are constructed according to the structural characteristics of the above matrix differential equation. With the Laplace transform (LT) and Laplace-Stieltjes transform (LST) methods, the steady-state distribution of fluid level in the buffer, mean fluid level and the probability of empty buffer are given. Finally, numerical analysis is presented to analyze the influence of system parameters on performance indices. The results show that the system performance can be optimized and the service efficiency can be improved by adjusting the model parameters in practical background.

Key words: optional service, fluid model, buffer content, the probability of empty buffer

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