Jeganathan K, Koffer V Anzen, Lakshmi K Prasanna, Anbazhagan N, Joshi Gyanendra Prasad, Cho Woong
Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai, Tamil Nadu, 600005, India.
Department of Mathematics with Computer Applications, Ethiraj College for Women, Chennai, Tamil Nadu, 600008, India.
Sci Rep. 2025 May 6;15(1):15860. doi: 10.1038/s41598-025-99748-5.
This paper studies the multi-server queueing-inventory system, which consists of n junior servers, a senior server, and two distinct waiting halls dedicated for customers and junior servers. The junior servers sometimes face difficulties while providing the service to the customers. To tackle the issues, junior servers need guidance from the senior server. When the senior server is already assisting a junior server, the other junior servers who are approaching must wait in a waiting area of size n until they consult with the senior server. We analyze the performance of such a system by modeling it into a quasi-birth-and-death process, and its steady-state probability vector has been computed. The convexity in the expected total cost for systems employing both homogeneous and state-dependent service rates has been established. Our findings demonstrate that systems employing state-dependent service rate exhibit superior operational efficiency compared to homogeneous service rates. Additionally, we conducted a comparative analysis of the expected waiting times on both homogeneous and state-dependent service rates. The expected number of junior servers in both server and consultation modes is analyzed. Furthermore, we examined the influence of changing probabilities of junior servers seeking the consultation with the senior server and compared our model with the traditional queueing-inventory model with multiple servers. In the comparison, our model performs more efficiently which shows that implementing a senior server in the multi-server system would improve the efficiency of the system in many ways to boost up the business profit.
本文研究了多服务器排队库存系统，该系统由n个初级服务器、一个高级服务器以及两个分别为客户和初级服务器设置的不同等待大厅组成。初级服务器在为客户提供服务时有时会遇到困难。为了解决这些问题，初级服务器需要高级服务器的指导。当高级服务器已经在协助一个初级服务器时，其他前来咨询的初级服务器必须在大小为n的等待区域等待，直到他们与高级服务器进行咨询。我们通过将其建模为准生死过程来分析这样一个系统的性能，并计算了其稳态概率向量。已经建立了采用均匀服务率和状态依赖服务率的系统在预期总成本方面的凸性。我们的研究结果表明，与均匀服务率相比，采用状态依赖服务率的系统具有更高的运营效率。此外，我们对均匀服务率和状态依赖服务率下的预期等待时间进行了比较分析。分析了处于服务模式和咨询模式下的初级服务器的预期数量。此外，我们研究了初级服务器寻求高级服务器咨询的概率变化的影响，并将我们的模型与传统的多服务器排队库存模型进行了比较。在比较中，我们的模型表现得更高效，这表明在多服务器系统中设置一个高级服务器将在许多方面提高系统效率，从而增加商业利润。
