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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2025, Vol. 42 ›› Issue (3): 577-594.doi: 10.3969/j.issn.1005-3085.2025.03.011doi: 32411.14.1005-3085.2025.03.011

• • 上一篇    

具有插队行为和单重休假的$M/M/1/m+1$排队系统的等待时间分布函数研究

吴文青1,  徐海文1,  余玅妙2,  郑克龙1   

  1. 1. 中国民用航空飞行学院理学院，广汉  618307
    2. 四川师范大学数学科学学院，成都 610068
  • 收稿日期:2022-11-08 接受日期:2023-05-31 出版日期:2025-06-15 发布日期:2025-06-15
  • 通讯作者: 徐海文 E-mail: hwxu@cafuc.edu.cn
  • 基金资助:
    国家自然科学基金 (72001181)；四川省科技计划项目 (2022YFG0324)；四川省心理学会立项项目 (SCSXLXH202402012).

Analysis of the Waiting Time Distributions of Customers in an $M/M/1/m+1$ Queue with Customer Interjections and Single Vacation

WU Wenqing1,  XU Haiwen1,  YU Miaomiao2,  ZHENG Kelong1   

  1. 1. School of Science, Civil Aviation Flight University of China, Guanghan 618307
    2. School of Mathematical Sciences, Sichuan Normal University, Chengdu 610068
  • Received:2022-11-08 Accepted:2023-05-31 Online:2025-06-15 Published:2025-06-15
  • Contact: H. Xu. E-mail address: hwxu@cafuc.edu.cn
  • Supported by:
    The National Natural Science Foundation of China (72001181); the Sichuan Province Science and Technology Plan Project (2022YFG0324); the Sichuan Psychological Association Project (SCSXLXH202402012).

摘要：

研究了具有插队行为和服务员单重休假的有限容量$M/M/1/m+1$排队系统中顾客的等待时间分布函数。将进入系统的顾客按照插队与否分为常规顾客和插队顾客，常规顾客进入系统后在等待队尾排队等待服务，插队顾客进入系统后总是尽可能的靠近队首插队接受服务。系统中有一个服务员，且采取单重休假策略。利用负指数分布、位相型分布的性质、吸收时间的马尔可夫链推导了处于等待队列位置$n$的顾客、常规顾客、插队顾客的等待时间分布函数的矩阵表达式，并在此基础上数值模拟了等待时间分布函数随时间的变化情况。

关键词: $M/M/1/m+1$排队系统, 插队行为, 单重休假, 等待时间分布函数

Abstract:

This paper studies the waiting time distributions of customers in an $M/M/1/m+1$ queueing system with customer interjections and server's single vacation. The customers who enter the system are divided into regular customers and interjection customers according to whether they interject the queue or not. After entering the system, the regular customers queue up at the end of the waiting line and wait for service, while the interjection customers queue up as close as possible to the head of the waiting line to receive service. There is one server who takes single vacation in the system. By using the properties of negative exponential distribution, the phase type distribution and the Markov chain with absorbing state, the matrix expressions of waiting time distribution of customers in waiting queue position $n$, regular customers and interjection customers are derived. Further, the waiting time distributions with time $t$ are plotted.

Key words: $M/M/1/m+1$ queue system, customer interjection, single vacation, waiting time distribution

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