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《山东大学学报(理学版)》 ›› 2023, Vol. 58 ›› Issue (8): 104-110, 117.doi: 10.6040/j.issn.1671-9352.0.2021.786

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基于MPEWMA控制图的串联排队网络的监测与控制

高琦(),戴洪帅,武艳华*()   

  1. 山东财经大学统计与数学学院, 山东 济南 250014
  • 收稿日期:2021-11-28 出版日期:2023-08-20 发布日期:2023-07-28
  • 通讯作者: 武艳华 E-mail:1456283672@qq.com;wuyhup@126.com
  • 作者简介:高琦(1997—)，女，硕士研究生，研究方向为排队系统的统计推断. E-mail：1456283672@qq.com
  • 基金资助:
    山东省自然科学基金资助项目(ZR2019MA035);山东省自然科学基金资助项目(ZR2020MA036);山东省研究生教育优质课程资助项目(SDYKC21137)

MPEWMA chart for detecting tandem queuing network

Qi GAO(),Hongshuai DAI,Yanhua WU*()   

  1. School of Statistics and Mathmatics, Shandong University of Finance and Economics, Jinan 250014, Shandong, China
  • Received:2021-11-28 Online:2023-08-20 Published:2023-07-28
  • Contact: Yanhua WU E-mail:1456283672@qq.com;wuyhup@126.com

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摘要：

构造了可以检测具有中间输入的串联排队网络的多元泊松指数加权移动平均(multivariate Poisson exponentially weighted moving-average, MPEWMA)控制图, 该控制图能够对系统中的多个节点同时进行监控。通过MPEWMA控制图对外部到达速率λi的变点进行检测, 根据变点对过程分段, 在每段满足逐段平稳性的情况下, 采用极大似然法对分段后的参数λi进行估计。在成本最低的条件约束下, 由外部输入λi的估计值计算到达速率γi的值, 根据到达速率γi的变化及时优化服务过程, 合理调整服务速率, 减少顾客的等待时间。随机模拟结果表明, 当顾客的到达速率发生变化时, MPEWMA控制图能够及时检测出外部到达率λi的变化并发出警报, 对排队系统的平稳运行起着重要作用。

关键词: 串联排队网络, MPEWMA控制图, 变点检测, 系统优化

Abstract:

We constructed a multivariate Poisson exponential weighted moving average (MPEWMA) chart, which can detect a series queuing network with intermediate inputs. The MPEWMA chart can be used to monitor and control multiple nodes in the system simultaneously. We detect the change points of the external arrival rate λiby the MPEWMA chart, segment the process according to the change points, and estimate the segmented parameters using the MLE method when each segment meets the stationarity of each segment. Under the constraint of the lowest cost, the estimated value of the arrival rate γi is calculated from external inputs λi, and the service process is optimized in a timely manner based on changes in the arrival rate. The service rate is adjusted reasonably to reduce customer waiting time. The random simulation results indicate that when the customer's arrival rate changes, the MPEWMA chart can timely detect changes in external arrival rate λi and issue alerts, playing an important role in the smooth operation of the queuing system.

Key words: tandem queuing network, MPEWMA chart, change point detection, system optimization

中图分类号: 

  • O211.9

图1

具有中间输入的串联排队网络"

图2

两节点的具有中间输入的串联排队网络模型"

表1

不同位移下的ARL1和δ"

位移 [0,0] [0,1] [1,1] [0,2] [2,1] [2,2] [1,3] [2,3] [3,3] [3,4] [4,4]
δ 0 0.41 0.65 0.75 0.98 1.17 1.17 1.39 1.62 1.81 2.01
r=0.05 H=7.52 200.12 33.02 17.85 14.27 10.23 8.22 8.16 6.62 5.46 4.75 4.15
r=0.1 H=8.77 199.94 36.54 17.65 13.78 9.21 7.19 7.12 5.61 4.50 3.87 3.33
r=0.2 H=9.87 198.423 46.29 20.69 15.68 9.57 6.96 6.92 5.17 3.97 3.29 2.76

图3

矢控的平均运行步长和δ关系图"

图4

变点检测图"

表2

变点前后λ的估计值"

λ1i λi2
节点1 3.044 2.923
节点2 5.238 5.964

表3

系统优化表"

最优服务率μ* 平均等待时间WSE 最优成本z
γ21=5.24 9.28 0.139 105.24
γ22=11.20 19.88 0.065 211.66

图5

产品生产模型"

图6

变点检测图"

表4

系统优化表"

最优服务率μ* 平均等待时间EWs 最优成本z
γ21=15.39 27.96 0.044 875.42
γ21=17.02 30.92 0.039 964.25
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