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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 (4): 778-792.doi: 10.3969/j.issn.1005-3085.2025.04.014cstr: 32411.14.cjem.CN61-1269/O1.2025.04.014

• • 上一篇    

基于顾客批量需求和损失销售的工作休假排队库存系统

魏伊宁,   岳德权,   迟艳伟   

  1. 燕山大学理学院，秦皇岛 066004
  • 收稿日期:2022-11-08 接受日期:2023-03-29 出版日期:2025-08-15 发布日期:2025-10-15
  • 通讯作者: 岳德权 E-mail: ydq@ysu.edu.cn
  • 基金资助:
    国家自然科学基金 (71971189)；河北省高等学校科技计划重点项目 (ZD2018042).

Queueing-inventory System with Working Vacation Based on Customer's Batch Demand and Lost Sales

WEI Yining,   YUE Dequan,   CHI Yanwei   

  1. School of Science, Yanshan University, Qinhuangdao 066004
  • Received:2022-11-08 Accepted:2023-03-29 Online:2025-08-15 Published:2025-10-15
  • Contact: D. Yue. E-mail address: ydq@ysu.edu.cn
  • Supported by:
    The National Natural Science Foundation of China (71971189); the Key Project of Science and Technology Program of Higher Education Institutions of Hebei Province (ZD2018042).

摘要：

研究了具有顾客批量需求、损失销售和服务员多重工作休假的 $M/M$/1 排队库存系统，其中库存为零时服务员进行多重工作休假。顾客到达服从 Poisson 过程，服务员服务时间、补货时间和工作休假时间均服从指数分布，顾客需求批量服从一般分布。利用拟生灭过程和矩阵几何解法，求出了系统稳态条件和稳态概率分布，进而得到了系统的稳态性能指标的计算公式。基于性能指标建立了系统的长期平均费用函数，通过数值算例分析了系统的最优库存策略和最优费用。

关键词: 排队库存系统, 损失销售, 批量需求, 多重工作休假, $({s,S})$策略

Abstract:

In this paper, we study a continuous review $M/M/1$ queueing-inventory system with customer batch demand, lost sales and multiple working vacations, where the server takes multiple working vacations when the inventory is zero. Customers arrive according to a Poisson process. The service time, the lead time and the working vacation time are assumed to follow exponential distributions, and the batch size of demands is assumed to follow a general distribution. The steady-state condition and the steady-state probability distribution of the system are derived by using quasi-birth-and-death process and matrix geometry solution method, and then the computational formulae for the steady-state performance indicators of the system are obtained. The long-term average cost function of the system is established based on the performance measures, and the optimal inventory strategy and the optimal cost of the system are analyzed by numerical examples.

Key words: queueing-inventory system, lost sales, batch demand, multiple working vacations, $(s,S)$ policy

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