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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 ›› 2026, Vol. 42 ›› Issue (6): 1171-1188.doi: 10.3969/j.issn.1005-3085.2025.06.013cstr: 32411.14.cjem.CN61-1269/O1.2025.06.013

• • 上一篇    

具有启动时间和预留空闲时间的可修重试排队分析

田瑞玲,  吴欣宇   

  1. 燕山大学理学院，秦皇岛 066004
  • 收稿日期:2023-03-15 接受日期:2023-10-16 出版日期:2025-12-15 发布日期:2026-02-15
  • 基金资助:
    国家自然科学基金(71971189).

Analysis to a Repairable Retrial Queue with Setup Times and Reserved Idle Times

TIAN Ruiling,  WU Xinyu   

  1. School of Science, Yanshan University, Qinhuangdao 066004
  • Received:2023-03-15 Accepted:2023-10-16 Online:2025-12-15 Published:2026-02-15
  • Supported by:
    The National Natural Science Foundation of China (71971189).

摘要：

研究了具有启动时间和预留空闲时间的可修$M/M/1$重试排队模型。服务台在服务完成后保持一段空闲时间，此时到达的顾客可以立即接受服务。否则，服务台将被关闭以节省能源，直到有新的顾客到达才能激活服务台。服务台在忙期和启动期间可能以不同的速率发生故障。首先，运用母函数方法，得到系统的稳态分布和系统稳态性能指标。其次，基于收入–支出结构，得到顾客的均衡进队策略和使得社会收益最大化的社会最优进队策略。最后，采用遗传算法找到使系统成本最小的最佳参数组合。对于双目标优化模型，借助NSGA-II算法来寻找Pareto最优解集，旨在将预期成本和顾客的平均等待时间降至最低。提出最小成本与等待时间之间的回归方程并进行回归检验，为生产者提供决策。

关键词: 重试排队, 启动时间, 预留空闲时间, 服务台故障, 均衡策略

Abstract:

A repairable $M/M/1$ retrial queue with setup times and reserved idle time is studied. The server holds a period of idle time after finishing the service, and customers who arrive at this time can receive the service immediately. Otherwise, the server will be closed down to reduce operating costs until a new customer arrives to activate the system. The server may break down during the busy and setup period. Firstly, the steady-state probability and stationary performance measures of the system are obtained by using the probability generating function method. Secondly, based on the reward-cost structure, the customer's equilibrium strategy and the socially optimal strategy that maximizes social benefits are obtained. Finally, we use the genetic algorithm to find the best combination of parameters that minimize the cost of the system. In addition, we construct a bi-objective optimal model and makes it possible to achieve an appropriate balance between the system cost and the service quality. The regression equation between the minimum cost and the waiting time is proposed, which is helpful to determine the minimum cost that meets service quality.

Key words: retrial queue, setup time, reserved idle time, server failure, equilibrium strategy

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