摘 要：随着现代服务系统复杂性的不断提升，排队论作为研究随机服务系统的有效工具，在理论与实践领域均具有重要意义。本文以随机过程模型为核心，深入探讨其在排队论中的应用机制，旨在通过构建更精确的数学模型来优化系统性能分析与设计。研究基于泊松过程、马尔可夫链等经典随机过程理论，结合实际应用场景提出了一种改进的多服务台排队模型，该模型能够动态反映顾客到达和服务时间的随机特性。通过对模型的数值模拟与实证分析，结果表明所提方法在预测系统稳态性能和瞬态行为方面具有较高精度，并能有效降低系统资源浪费。本文的主要创新点在于将非齐次泊松过程引入传统排队模型，从而更好地适应现实场景中顾客流量的时变特性。这一改进为复杂服务系统的优化设计提供了新的理论支持，同时拓展了随机过程模型在排队论领域的应用边界。
关键词：排队论；随机过程模型；非齐次泊松过程；多服务台排队模型；系统性能优化
Analysis of the Application of Stochastic Process Models in Queueing Theory
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Abstract：With the increasing complexity of modern service systems, queuing theory, as an effective tool for studying stochastic service systems, plays a significant role in both theoretical and practical domains. This paper focuses on stochastic process models at its core, exploring their application mechanisms in queuing theory to optimize system performance analysis and design through the construction of more accurate mathematical models. Based on classical stochastic process theories such as Poisson processes and Markov chains, combined with real-world application scenarios, an improved multi-server queuing model is proposed, which dynamically reflects the stochastic characteristics of customer arrivals and service times. Through numerical simulations and empirical analyses of the model, the results indicate that the proposed method achieves high precision in predicting the steady-state performance and transient behavior of the system while effectively reducing resource wastage. The primary innovation of this study lies in incorporating non-homogeneous Poisson processes into traditional queuing models, thereby better accommodating the time-varying nature of customer traffic in real-world scenarios. This enhancement provides new theoretical support for the optimal design of complex service systems and extends the application boundaries of stochastic process models in the field of queuing theory.
Keywords: Queueing Theory；Stochastic Process Model；Non-Homogeneous Poisson Process；Multi-Server Queueing Model；System Performance Optimization
目 录
引言 1
一、随机过程模型基础分析 1
(一)随机过程的基本概念 1
(二)排队论中的随机特性 2
(三)模型选择与适用性 2
二、排队系统建模方法研究 2
(一)排队系统的分类与特征 3
(二)基于泊松过程的建模分析 3
(三)马尔可夫链在排队中的应用 3
三、性能指标与优化分析 4
(一)系统性能的关键指标 4
(二)平均等待时间的计算方法 5
(三)优化策略的设计与实现 5
四、实际应用案例探讨 6
(一)通信网络中的排队问题 6
(二)生产线调度的应用分析 6
(三)医疗服务系统的优化实践 7
结论 7
参考文献 8
致谢 8
