Abstract: A queueing-inventory system is an integrating system that integrates a queueing process of customers into an inventory system. Upon arrival, customers require both the product and the service time for tasks such as inspection, preparation, packaging,and loading. Therefore, a queueing-inventory system is also called a service-inventory system. Compared to traditional inventory systems, queueing-inventory systems are universal and practical. This kind of systems has a wide application in many fields such as integrated supply chain management, airline and train ticketing systems, transportation, and healthcare services. The frequent occurrence of events such as pandemics, floods, and trade wars increases the uncertainty of customer demand, so as to lead to supply chain uncertainties and frequent stock-outs. This uncertainty challenges the efficiency of traditional queueing-inventory systems. Especially in the context of constructing new, high-quality, and efficient service systems, enhancing the system’s ability to adapt to demand uncertainty has become an urgent issue.
This study proposes a novel M/M/1/∞ queueing-inventory system model. It incorporates a mixed vacation policy with a r-random order size policy to enhance system adaptability and efficiency. The server takes vacation when the inventory is empty at the epoch of the completion of customer’s service. The mixed vacation policy means that if the inventory is still empty at the end of the server’s vacation, the server returns from the vacation with probability q, or takes another vacation with probability 1-q.The mixed vacation policy allows service interruptions during stock-outs, and the interrupted service can be resumed upon the completion of the replenishment of the inventory or under some other special conditions. For instance, if a server is temporarily closed due to breakdown, the service of the server will be resumed once the server is repaired. This flexible vacation policy is helpful not only for inventory management according to real-time inventory status, but also for dealing with the service interruption due to server’s breakdown or other situations. In highly dynamic and uncertain market environments, this strategy provides a flexible and effective means for companies to tackle challenges related to inventory shortages and service interruptions. The r-random order size policy dictates that an order of random sizes is placed when the inventory level falls to a threshold r. This provides a flexible mechanism to handle demand fluctuations. This is particularly crucial during periods like pandemics. When demand fluctuates sharply, this strategy significantly reduces the risk of stock-outs and enhances supply chain resilience. By combining these two strategies, companies can better manage uncertainties during unexpected events, ensuring service continuity and supply chain efficiency. This enables the companies to maintain competitiveness in a complex and volatile market environment.
In this paper, the system’s stability condition and the product form solution of the steady-state probability distribution are obtained by using the theory of the quasi-birth-and-death process. Performance measures are established to develop the expected cost function of the system. The expected cost function of the system between the random order size policyand r-random order size policy is compared by numerical examples. The impact of the threshold r, the probability q and other parameters on the expected cost function are also analyzed numerically. The integration of the mixed vacation and r-random order size policies enables the system to optimize resource usage while ensuring the quality of service, and also reduces the potential loss of the system due to service interruptions.

Key words: queueing-inventory system, server vacation, random order size policy, lost sales, cost function

摘要： 本文将随机补货策略和混合休假策略引入M/M/1/∞排队库存系统中,提出了一个新的排队库存系统模型。当该模型中现有库存水平达到阈值r时,系统即刻向外部供应商发出随机订货量的补货订单。当服务员服务完顾客后发现库存空时,服务员进行休假。当服务员休假结束时若系统的库存还是空的,服务员则以概率q结束休假,以概率1-q继续休假。本文通过拟生灭过程理论来求解系统的稳态平衡条件和稳态概率分布的乘积形式解。进而构建该系统的各项稳态性能指标和平均费用函数。通过数值算例对随机补货策略和r-随机补货策略下的平均费用的函数表达式进行分析,并讨论了阈值r、概率q等参数产生的影响。

关键词: 排队库存系统, 服务员休假, 随机补货策略, 损失销售, 费用函数