Abstract:

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.

Key words: queueing-inventory systems, steady-state analysis, product-form solution, matrix-geometric solution, game theory