延迟Min(N,D)-策略的M/G/1排队系统的队长分布与数值计算

doi:10.15960/j.cnki.issn.1007-6093.2016.02.002

运筹学学报

延迟Min(N,D)-策略的M/G/1排队系统的队长分布与数值计算

魏瑛源¹ 唐应辉^2,* 余玅妙³

1. 河西学院数学与统计学院, 甘肃张掖 734000; 2. 四川师范大学数学与软件科学学院, 成都 610066; 3. 四川理工学院理学院, 四川自贡 643000

收稿日期:2015-07-03 出版日期:2016-06-15 发布日期:2016-06-15
通讯作者: 唐应辉 tangyh@uestc.edu.cn
基金资助:
国家自然科学基金(Nos. 71171138, 71301111, 71571127)

Queue length distribution and numerical calculation of queueing system with delay Min(N,D)-policy

WEI Yingyuan¹ TANG Yinghui^2,*YU Miaomiao³

1. School of Mathematics and Statistics, Hexi University, Zhangye 734000, Gansu, China; 2. School of Mathematics and Software Science, Sichuan Normal University, Chengdu 610066, China; 3. School of Science, Sichuan University of Science and Engineering, Zigong 643000, Sichuan, China

Received:2015-07-03 Online:2016-06-15 Published:2016-06-15

摘要/Abstract

摘要：

考虑延迟Min(N, D)-策略的M/G/1排队系统. 运用更新过程理论、全概率分解技术和Laplace变换工具, 从任意初始状态出发, 研究了队长的瞬态和稳态性质, 获得了瞬态队长分布的Laplace变换的递推表达式和稳态队长分布的递推表达式, 同时求出了附加队长分布的显示表达式. 进一步讨论了当N \to \infty, 或D \to \infty, 或N=1且P{Y=0}=1, 或P{Y=0}=1时的特殊情形. 最后通过数值实例, 讨论了稳态队长分布对系统参数的敏感性, 并阐述了稳态队长分布的表达式在系统容量优化设计中的重要价值.

关键词: M/G/1排队, 延迟Min(N, D)-策略, 全概率分解技术, Laplace变换, 队长分布, 系统容量优化设计

Abstract:

This paper considers the M/G/1 queueing system under the delay Min(N,D)-policy. By using the renewal process theory, the total probability decomposition technique and the Laplace transform tool, we study the transient and equilibrium properties of the queue length from the beginning of the any initial state, and obtain both the recursion expressions of the Laplace transformation of the transient queue length distribution and the recursion expressions of the steady state queue length distribution. Meanwhile, we present the explicit expression of the additional queue-length distribution. Furthermore, we discuss some special cases, such as N \to \infty, or D \to \infty, or N=1 and P{Y=0}=1 or P{Y=0}=1, respectively. Finally, by numerical examples, we discuss the sensitivity of the steady state queue length distribution towards system parameters, and illustrate the important value of the expressions of the steady state queue length distribution in the system capacity optimum design.

Key words: M/G/1 queue, delay Min(N, D)-policy, total probability decomposition technique, Laplace transform, queue length distribution, system capacity optimum design

魏瑛源, 唐应辉, 余玅妙. 延迟Min(N,D)-策略的M/G/1排队系统的队长分布与数值计算[J]. 运筹学学报, doi: 10.15960/j.cnki.issn.1007-6093.2016.02.002.

WEI Yingyuan, TANG Yinghui, YU Miaomiao. Queue length distribution and numerical calculation of queueing system with delay Min(N,D)-policy[J]. Operations Research Transactions, doi: 10.15960/j.cnki.issn.1007-6093.2016.02.002.

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延迟Min(N,D)-策略的M/G/1排队系统的队长分布与数值计算

Queue length distribution and numerical calculation of queueing system with delay Min(N,D)-policy

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