具有两类故障的M/M/1 + 1排队系统的均衡分析
Equilibrium Analysis for an M/M/1 + 1 Queueing System with Two Types of Failures
摘要: 本文研究了完全可视环境下具有两类故障的M/M/1 + 1排队系统，其中主服务台采用N策略与工作休假机制，并设有备用服务台。主服务台可能发生两类故障，备用服务台在主服务台故障时辅助服务，且在服务期间也可能发生不完全故障。通过建立二维马尔可夫链模型，推导出系统最优进队阈值与稳态概率分布的显式表达式，进一步计算系统的平均队长、顾客平均逗留时间及社会收益等性能指标。通过数值模拟探讨服务率，故障率等关键参数对系统与社会收益的影响。仿真实验验证了解析结果的正确性。研究结果为高可靠性、可调节服务系统的运营优化与资源配置提供了理论依据与决策支持。
Abstract: This paper studies an M/M/1 + 1 queueing system with two types of failures in a fully observable environment. The primary server operates under an N policy with a working vacation mechanism, and a standby server is employed. The primary server is susceptible to two types of failures. The standby server assists the primary server during its failures and may also experience partial failures. By establishing a two-dimensional Markov chain model, explicit expressions for the optimal joining threshold and the steady-state probability distribution of the system are derived. Furthermore, performance metrics such as the average queue length, average customer sojourn time, and social welfare are calculated. Numerical simulations are conducted to investigate the impact of key parameters, including service rates and failure rates, on system behavior and social welfare. Simulation experiments validate the correctness of the analytical results. The findings provide theoretical foundations and decision support for the operational optimization and resource allocation of highly reliable and adjustable service systems.
文章引用：李志洋. 具有两类故障的M/M/1 + 1排队系统的均衡分析[J]. 应用数学进展, 2026, 15(3): 367-386. https://doi.org/10.12677/aam.2026.153112

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