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Product Form Solution of a Queuing-Inventory System with Lost Sales and Server Vacation

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Abstract

In this study, the authors consider an M/M/1 queuing system with attached inventory under an (s, S) control policy. The server takes multiple vacations whenever the inventory is depleted. It is assumed that the lead time and the vacation time follow exponential distributions. The authors formulate the model as a quasi-birth-and-dearth (QBD) process and derive the stability condition of the system. Then, the stationary distribution in product form for the joint process of the queue length, the inventory level, and the server’s status is obtained. Furthermore, the conditional distributions of the inventory level when the server is on and operational, and when it is off due to a vacation, are derived. Using the stationary distribution, the authors obtain some performance measures of the system. The authors investigate analytically the effect of the server’s vacation on the performance measures. Finally, several numerical examples are presented to investigate the effects of some parameters on the performance measures, the optimal policy, and the optimal cost.

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Authors and Affiliations

  1. School of Science, Yanshan University, Qinhuangdao, 066004, China

    Dequan Yue & Xiuli Xu

  2. School of Economics and Management, Yanshan University, Qinhuangdao, 066004, China

    Yuying Zhang

  3. The Kyoto College of Graduate Studies for Informatics, Kyoto City, 606-8225, Japan

    Wuyi Yue

Authors
  1. Dequan Yue
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  2. Yuying Zhang
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  3. Xiuli Xu
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  4. Wuyi Yue
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Correspondence to Dequan Yue.

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The authors declare no conflict of interest.

Additional information

This work was supported in part by the Natural Science Foundation of China under Grant No. 71971189, the Natural Science Foundation of Hebei Province under Grant No. A2019203313, the Key Project of Scientific Research in Higher Education of Hebei Province of China under Grant No. ZD2018042, and in part by MEXT, Japan.

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Yue, D., Zhang, Y., Xu, X. et al. Product Form Solution of a Queuing-Inventory System with Lost Sales and Server Vacation. J Syst Sci Complex 37, 729–758 (2024). https://doi.org/10.1007/s11424-024-1207-7

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  • DOI: https://doi.org/10.1007/s11424-024-1207-7

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