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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 ›› 2024, Vol. 41 ›› Issue (6): 1155-1169.doi: 10.3969/j.issn.1005-3085.2024.06.012

• • 上一篇    下一篇

具有随机检修$\langle p,Y \rangle$-策略M/G/1系统队长的瞬态与稳态分析

李占宇,   唐应辉   

  1. 四川师范大学数学科学学院，成都 610068
  • 收稿日期:2022-04-20 接受日期:2022-09-29 出版日期:2024-12-15 发布日期:2024-12-15
  • 通讯作者: 唐应辉 E-mail: tangyh@sicnu.edu.cn
  • 基金资助:
    国家自然科学基金 (71571127)；四川师范大学学科建设专项基金 (XKZX2021-04).

Transient and Steady State Analysis of M/G/1 Queueing System with Randomized Overhaul $\langle p,Y \rangle $-policy

LI Zhanyu,  TANG Yinghui   

  1. School of Mathematical Sciences, Sichuan Normal University, Chengdu 610068
  • Received:2022-04-20 Accepted:2022-09-29 Online:2024-12-15 Published:2024-12-15
  • Contact: Y. Tang. E-mail address: tangyh@sicnu.edu.cn
  • Supported by:
    The National Natural Science Foundation of China (71571127); the Specialized Project for Subject of Sichuan Normal University (XKZX2021-04).

摘要：

研究一个具有随机检修$\langle p,Y\rangle$-策略的M/G/1排队系统，当系统变空时，以概率$p(0\le p\le 1)$对系统进行检修，且检修时间是具有任意分布的随机变量。首先，分析了队长的嵌入马尔可夫链，得到了其稳态分布的概率母函数。其次，讨论了在任意时刻$t$队长的瞬态分布，得到了队长的瞬态分布关于时间$t$的拉普拉斯变换表达式。在队长瞬态分析的基础上，应用洛必达法则，通过直接计算获得了在任意时刻队长的稳态分布的递推式，给出了稳态队长的随机分解结构。最后，建立了系统的费用模型，并通过数值实例得到了使系统费用最少的最优检修策略。

关键词: M/G/1排队, 随机检修$\langle p,Y\rangle$-策略, 全概率分解, 队长分布, 最优检修策略

Abstract:

This paper considers the M/G/1 queueing system with randomized overhaul $\langle p,Y \rangle $-policy, in which when the system becomes empty, the system is overhauled with probability $p ( 0\le p\le 1 )$ and the length of overhauling time is a random variable with general distribution. Firstly, we analyze the embedded Markov chain of queue length, and obtain the probability generating function of its steady-state distribution. Secondly, the transient distribution of the queue size at any time $t$ is discussed, and the expressions of the Laplace transform of the transient queue length distribution with respect to time $t$ are presented. Meanwhile, based on the transient analysis of the queue length, the recursive formulas of the steady-state distribution of the queue length are obtained by employing L'Hospital rule. Furthermore, the stochastic decomposition structure of the steady-state queue size is presented. Finally, numerical examples are provided to determine the optimal overhaul policy for economizing the system cost under a given cost structure.

Key words: M/G/1 queue, randomized overhaul $\langle p,Y \rangle $-policy, total probability decomposition, queue length distribution, optimal overhaul policy

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