连续时间马尔科夫链及其在排队论中的应用

Continuous Time Markov Chain and Its Applications in Queuing Theory

摘要: 马尔科夫链的无记忆性简化了复杂系统的分析，使之成为研究排队论的核心方法之一。本文首先介绍了连续时间马尔科夫链的基本知识。然后重点探讨了连续时间马尔科夫链在排队论中的实际应用，借助于生灭过程的平稳分布等概率方法，详细分析了在数学期望的意义下的最优库存管理问题，展现了该模型的重要实用价值，为数学在管理科学、机械工程中的融通发展提供了重要的理论支持，也对培养学生理论联系实际的能力具有重要的提升作用。

Abstract: The Markov chains has become one of the core models in studying queuing theory, due to the fact that its memoryless property helps to simplify the analysis of complex systems. In this article, some basic knowledge of continuous time Markov chains is recalled at first. Then, the practical application of continuous time Markov chains in queuing theory was discussed. By applying probability methods such as the stationary distribution of birth and death processes, the optimal inventory management problem is solved thoroughly in the sense of mathematical expectations. The obtained results show the significant practical value of the continuous time Markov chains model and provide important theoretical support for the integrated development between mathematics, management science and mechanical engineering. Moreover, it also plays an important role in enhancing students’ ability to integrate theory with practice.