具有多级适应性休假和修正的 Min$(N,D)$-策略的 Geo/G/1 离散时间排队分析

数学物理学报 ›› 2025, Vol. 45 ›› Issue (5): 1671-1697.

具有多级适应性休假和修正的 Min$(N,D)$-策略的 Geo/G/1 离散时间排队分析

魏瑛源^1,^*(),余玅妙²()

¹河西学院数学学院甘肃张掖 734000
²四川师范大学数学科学学院成都 610068

收稿日期:2024-11-01 修回日期:2025-05-14 出版日期:2025-10-26 发布日期:2025-10-14
通讯作者: * 魏瑛源, E-mail:wyy_70@126.com
作者简介:余玅妙,E-mail: mmyu75@163.com
基金资助:
教育部人文社会科学研究规划基金(24YJA630121);河西学院校长基金创新团队项目(CXTD2022013)

Analysis of Geo/G/1 Queue with Multiple Adaptive Vacations and Modified Min$(N, D)$-Policy

Yingyuan Wei^1,^*(),Miaomiao Yu²()

¹School of Mathematics, Hexi University, Gansu Zhangye $734000$
²School of Mathematical Sciences, Sichuan Normal University, Chengdu 610068

Received:2024-11-01 Revised:2025-05-14 Online:2025-10-26 Published:2025-10-14
Supported by:
MOE (Ministry of Education in China) Project of Humanities and Social Sciences(24YJA630121);Faculty Research Grants Awarded by Principal's Funds(CXTD2022013)

摘要/Abstract

摘要：

考虑服务员具有多级适应性休假和系统采用修正的 Min$(N, D)$-控制策略的离散时间 Geo/G/1 排队系统, 运用更新过程理论、全概率分解技术和 $z$- 变换工具, 从任意初始状态开始, 研究队长的瞬时性态和平稳性态, 得到了任意时刻 $n^+$ 处队长瞬态分布的 $z$-变换表达式, 然后在队长瞬态分布的基础上, 使用洛必达法则, 通过直接运算得到了队长稳态分布的递推表达式, 同时给出了稳态队长的随机分解结构和附加队长分布的显示表达式, 并得到了不同时刻 $n^-$、$n$、$n^+$ 处和外部观测点处队长稳态分布之间的重要关系. 进一步, 以便于作数值计算的队长稳态分布的递推表达式为基础, 借助于数值算例, 讨论了系统容量的优化设计. 最后, 建立了费用结构模型, 应用更新报酬过程理论获得了系统长期运行下单位时间内所产生的期望费用函数, 并通过数值实例, 确定了使期望费用最小的最优控制策略.

关键词: Geo/G/1 离散时间排队, 修正的 Min$(N, D)$-策略, 多级适应性休假, 队长分布, 系统容量优化设计, 最优控制策略

Abstract:

This paper considers a discrete-time Geo/G/1 queueing system in which the server takes multiple adaptive vacations and the system adopts modified Min$(N, D)$-control policy. By using the renewal process theory、total probability decomposition technique and $z$-transform tool, we study the transient and equilibrium properties of the queue length from the beginning of the arbitrary initial state, and obtain the expressions of the $z$-transformation of the transient queue length distribution at arbitrary time epoch $n^+$. Then, the recursive expressions of the steady-state queue length distribution are obtained by using L'Hospital's rule. Meanwhile, both the probability generating function of the stochastic decomposition structure of the steady-state queue length and the explicit expressions of the additional queue length distribution are presented. Additionally, the important relations between the steady-state queue length distributions at different time epochs $n^-$、$n$、$n^+$ and outside observer's are also reported. Furthermore, numerical examples are implemented to discuss the system capacity design based on the recursive formulas of the steady-state queue length distribution for calculating conveniently. Finally, employing the renewal reward theorem, the function of the long-run expected cost per unit time is derived under a given cost structure, and numerical calculation are provided to determine the optimal control policy for minimizing the long-run expected cost rate.

Key words: discrete-time queue, modified Min$(N, D)$-policy, multiple adaptive vacations, queue length distribution, system capacity optimum design, optimal control policy

中图分类号:

O226

魏瑛源, 余玅妙. 具有多级适应性休假和修正的 Min$(N,D)$-策略的 Geo/G/1 离散时间排队分析[J]. 数学物理学报, 2025, 45(5): 1671-1697.

Yingyuan Wei, Miaomiao Yu. Analysis of Geo/G/1 Queue with Multiple Adaptive Vacations and Modified Min$(N, D)$-Policy[J]. Acta mathematica scientia,Series A, 2025, 45(5): 1671-1697.

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