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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 ›› 2022, Vol. 39 ›› Issue (1): 63-78.doi: 10.3969/j.issn.1005-3085.2022.01.005

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服务员休假的生产服务库存模型的稳态分析及最优生产策略

徐  浩,   岳德权    

  1. 燕山大学理学院，秦皇岛 066004
  • 出版日期:2022-02-15 发布日期:2022-04-15
  • 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)；河北省自然科学基金 (A2017203078)；河北省高等学校科技计划重点项目 (ZD2018042).

Steady-state Analysis and Optimal Production Strategy of Production Service Inventory Model with Server's Vacation

XU Hao,   YUE Dequan   

  1. School of Science, Yanshan University, Qinhuangdao 066004
  • Online:2022-02-15 Published:2022-04-15
  • Contact: D. Yue. E-mail address: ydq@ysu.edu.cn
  • Supported by:
    The National Natural Science Foundation of China (71971189); the Natural Science Foundation of Hebei Province (A2017203078); the Key Project of Science and Technology Program of Higher Education Institutions of Hebei Province (ZD2018042).

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摘要：

将服务员休假与生产服务库存相结合，研究带有服务员休假的 M/M/1 生产服务库存系统模型。需求 (顾客) 的到达过程服从 Poisson 过程，顾客的服务时间和每个产品的生产时间都服从指数分布。当系统库存为零时，服务员开始随机长度的休假。假设休假时间服从指数分布。首先，利用拟生灭过程理论给出了系统的稳态平衡条件。其次，对忽略服务时间的生产服务库存系统模型进行了稳态分析，得到了此系统的稳态概率的明显的解析表达式，进而证明了系统的稳态概率分布具有乘积解形式的结构。在此基础上，进一步得到了系统的一些稳态性能指标和费用函数的计算公式，数值求解了模型的最优 $(s,S)$ 库存策略，并研究系统的一些参数对性能指标、最优策略和最优费用的影响。

关键词: 生产服务库存系统, 服务员休假, 损失销售, 拟生灭过程, $({s,S})$ 策略

Abstract:

The production service inventory system with server's vacation by combining the server's vacation and the production-service inventory is examined in the paper. Demands (customers) arrive according to a Poisson process, and the service time of each customer and the production time of each product are all assumed to be exponentially distributed. When the inventory is depleted, the server goes to a random time of a vacation. It is assumed that the vacation time follows an exponential distribution. Firstly, the steady state condition of the system is obtained by using the quasi-birth-and-death process. Then, the steady-state analysis is established for the case of the system model with negligible service time and the explicit expressions of the steady state probabilities is obtained of this special case. Furthermore, it is proved that the steady-state probability distribution of the system has a structure of product form solution. On this basis, some performance indexes and cost function are calculated. Finally, the optimal $( {s, S})$ inventory strategy of the model is numerically obtained. The numerical analysis is performed to study the influence of some system parameters on performance indexes, the optimal inventory strategy and the optimal cost.

Key words: production-service inventory system, server's vacation, lost sales, QBD, $(s,S)$ strategy

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