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基于特征根方法的M/G_N/1个性化服务排队顾客逗留时间分布函数的数值计算

邹雪华1 余玅妙1,*  唐应辉2   周杰3   

  1. 1. 四川理工学院数学与统计学院, 四川自贡 643000; 2. 四川师范大学数学与软件科学学院, 成都 610068; 3. 四川师范大学商学院, 成都 610101
  • 收稿日期:2017-05-19 出版日期:2018-03-15 发布日期:2018-03-15
  • 通讯作者: 余玅妙 E-mail: mmyu75@163.com
  • 基金资助:

    国家自然科学基金(Nos. 71301111, 71571127, 71601135), 四川理工学院人才引进项目(No.2017RCL55), 四川理工学院研究生创新基金(No. y2016024)

 Calculation of the customer's sojourn time distribution function in M/G_N/1 queue with customized services using roots method

ZOU Xuehua1  YU Miaomiao1,* TANG Yinghui2  ZHOU Jie3   

  1. 1. School of Mathematics and Statistics, Sichuan University of Science and Engineering, Zigong 643000, Sichuan, China; 2. School of Mathematics & Software Science, Sichuan Normal University, Chengdu 610068, China; 3. School of Business, Sichuan Normal University, Chengdu 610101, China
  • Received:2017-05-19 Online:2018-03-15 Published:2018-03-15

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摘要：

以多语种便民服务热线为实际应用背景, 研究个性化服务M/G_N/1排队系统中顾客逗留时间分布函数的数值计算方法. 首先, 利用嵌入Markov链技术和Pollaczek-Khintchine变换公式给出顾客逗留时间的Laplace-Stieltjes(LS)变换. 其次, 根据个性化服务时间分布函数的具体类型, 给出上述LS变换的有理函数表达形式. 通过求解有理函数分母之具有负实部的零点, 即所谓的特征根, 最终使用部分分式分解方法和复分析中的留数理论给出顾客逗留时间的概率分布函数.

关键词: 排队系统, 个性化服务, 逗留时间分布函数, 特征根, Pade逼近

Abstract:

Taking the multilingual convenience service hotline for a practical example, we study the numerical method for calculating the customer's sojourn time distribution function in M/G_N/1 queue with customized service. Firstly, we give the Laplace-Stieltjes (LS) transform of the customer's sojourn time by using the embedded Markov chain technique and Pollaczek-Khintchine formula. Secondly, according to the specific type of customized service time distribution function, we give the rational form of the LS transform that mentioned above. By solving the zeros with negative real parts of the denominator of the rational function, namely, the so-called characteristic roots, we finally give the customer's sojourn time probability distribution function by using the method of partial fraction and residue theory.

Key words: queueing system, customized service, sojourn time distribution function, characteristic roots, Pade approximation