具有两类顾客和同步多重工作休假的可修排队

A Repairable Queueing Model with Two Types of Customers and Synchronous Multiple Working Vacations

摘要: 本文研究了一个具有两类顾客、同步多重工作休假以及可修故障机制的可修排队系统。为更贴合现实服务环境，模型假设两类顾客均服从泊松到达过程，服务时间在正常与休假模式下均服从指数分布，系统采用同步休假策略，并考虑服务台在正常忙期发生的指数型故障与维修过程。基于系统状态转移率矩阵构建出三维连续时间马尔可夫链，并证明其生成元矩阵具有拟生灭(QBD)结构。利用矩阵几何解法推导系统的稳态分布，从而得到系统平均队长、平均逗留时间、休假概率、忙期长度及可用度等性能指标的解析表达式。通过数值实验分析模型参数对系统性能的敏感性，并进一步构建个人效益函数与系统整体效益函数，探讨最优参数设置。结果表明，提高优先级顾客的服务率并适当增加服务台数量能够有效提升系统整体效益。

Abstract: Queueing models with two classes of customers of different priority levels are widely observed in real-life scenarios, for instance, priority users in ride-hailing services, emergency patients in hospitals and vehicles with prior reservations for charging stations. Consequently, many scholars at home and abroad have incorporated dual customer classes as a key strategy in the construction and analysis of queueing models. In this paper, based on possible situations that may occur in real-world queueing systems, non-preemptive priority policy, multiple working vacation policy and repairable server policy are incorporated into the queueing model to construct a system that accurately reflects practical conditions. The main contents of the paper are as follows: model description of the state transition rate matrix of the system, analysis of the three-dimensional Markov chain, and the derivation of the system’s steady-state distribution using the matrix-geometric method and iterative techniques. Based on the steady-state results, performance measures are derived and numerically analyzed. Finally, individual and social benefit functions are established for further analysis, and suggestions for improving overall efficiency are also proposed.