本文综述了排队库存系统(queueing-inventory system, QIS)的理论研究与应用进展, 涵盖其数学建模、稳态分析方法及在多领域的实际应用。排队库存系统基于排队论与库存管理, 研究始于1992年Sigman和Simichi-Levi以及Melikov和Molchanov的工作, 2006年Schwarz等明确定义了其框架。本文回顾了三种主要分析方法: 乘积形式解、矩阵几何解和近似乘积形式解。乘积形式解通过分解队列长度与库存水平的联合分布, 适用于M/M/$\cdot$模型等场景; 矩阵几何解基于准生灭过程, 利用率矩阵(R)求解稳态分布, 从解析解扩展至数值算法; 近似乘积形式解则通过状态空间分解处理复杂系统。此外, 本文探讨了博弈论在QIS中的应用, 如Stackelberg博弈分析顾客策略行为与最优库存控制。在应用层面, 研究覆盖食品制造(3D打印)、医疗服务(疫情废物管理)、血液供应链及运输系统, 创新模型如流体库存、批量马尔可夫到达过程等显著提升了系统效率与资源优化。综上, QIS研究在理论深度与应用广度上均取得重要进展, 为库存管理与服务优化提供了坚实支持。

Queueing-inventory systems (QIS) have emerged as a critical interdisciplinary framework that integrates queueing theory with inventory management to address dynamic service-inventory interactions in complex operational environments. This review aims to consolidate and evaluate the theoretical advancements and practical implementations of QIS, with particular emphasis on steady-state analysis techniques and their deployment across diverse application domains. Originating from the foundational contributions of Sigman and Simchi-Levi, and Melikov and Molchanov in 1992, and formally conceptualized by Schwarz et al. in 2006, QIS has evolved into a mature analytical framework. Three primary analytical approaches are examined in depth: product-form solutions, matrix-geometric methods, and approximate product-form solutions. Product-form solutions facilitate analytical tractability by decoupling the joint distribution of queue lengths and inventory levels, particularly effective in M/M/1 and related models. Matrixgeometric methods, based on quasi-birth-and-death (QBD) processes, leverage the rate matrix R to compute steady-state probabilities, with developments progressing from closed-form derivations to iterative numerical algorithms. Approximate product-form solutions are employed to handle more complex systems through state-space decomposition and bounding techniques, providing a balance between accuracy and computational efficiency. The review further explores the incorporation of game-theoretic models, particularly Stackelberg games, into QIS frameworks to capture strategic customer behavior and hierarchical decision-making in inventory control. Practical implementations of QIS span a wide range of sectors, including food manufacturing (e.g., 3D food printing), healthcare (e.g., medical waste disposal during the pandemic), blood supply chains, and urban transportation systems. Recent modeling innovations, such as fluid inventory models and batch Markovian arrival processes, have significantly improved system responsiveness and resource optimization.