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带负顾客和N-策略的Geo^{lambda_1, lambda_2/Geo/1(MWV)排队系统分析及最优控制策略N*

潘取玉1  唐应辉1,2,*  兰绍军1   

  1. 1. 四川师范大学数学与软件科学学院, 成都  610068 2. 四川师范大学基础教学学院, 成都 610068 
  • 收稿日期:2017-01-10 出版日期:2017-09-15 发布日期:2017-09-15
  • 通讯作者: 唐应辉 tangyh@sicnu.edu.cn
  • 基金资助:

    国家自然科学基金 (No. 71571127), 国家自然科学青年基金 (No. 71301111)

Analysis and optimal control policy N* for Geo^{lambda_1, lambda_2}/Geo/1(MWV) queueing system with negative customers and N-policy

PAN Quyu1  TANG Yinghui1,2,*   LAN Shaojun1   

  1. 1. School of Mathematics and Software Science, Sichuan Normal University, Chengdu 610068, China 2. School of Fundamental Education, Sichuan Normal University, Chengdu  610068, China
  • Received:2017-01-10 Online:2017-09-15 Published:2017-09-15

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摘要：

分析了一个带有负顾客、N-策略控制的Geo/Geo/1多重工作休假排队系统, 其中正顾客在工作休假及正规忙期以不同的到达率进入系统. 利用拟生灭过程和矩阵几何解方法, 给出了该模型的稳态队长分布及平均队长, 以及系统分别处于假期和忙期的概率. 同时, 对该系统的忙期进行了分析, 并讨论了稳态队长分布在系统容量的优化设计中的应用. 最后, 在给定的费用结构下, 用数值计算例子确定了使系统长期单位时间内期望费用最小的最优控制策 N*.

关键词: 可变到达率, N-策略, 负顾客, 工作休假, 最优控制策略

Abstract:

This paper deals with a discrete-time Geo/Geo/1 working vacations queue with negative customers and N-policy control in which the positive customers arrive at the system in different input rates during the working vacation period and the normal busy period.  Employing the quasi birth-death process and the matrix-geometric solution method, we derive the steady-state queue length distribution and the expected queue length, as well as the steady-state probabilities that the system is in working vacation state and busy state. Meanwhile, the busy period and the application for the steady state queue length distribution in system capacity optimum design are discussed. Finally, through numerical calculation, it is determined the optimal control policy N* such that the long-run expected cost rate is minimum under a given cost structure.

Key words: variable arrival rate, N-policy, negative customer, working vacation, optimal control policy