[论文解读] An Asymptotic Analysis of Queues with Delayed Information and Time Varying Arrival Rates

Jamol Pender, Richard H. Rand+1 more|arXiv (Cornell University)|Jan 19, 2017

Advanced Queuing Theory Analysis参考文献 20被引用 4

一句话总结

本文通过渐近方法分析了具有延迟信息和时变到达率的两个确定性流体排队模型，表明仅当到达率的频率为临界延迟频率的两倍时，时变到达率才会影响系统稳定性。主要贡献在于提出了一种改进的稳定性阈值，该阈值考虑了正弦到达波动引起的参数激励效应。

ABSTRACT

Understanding how delayed information impacts queueing systems is an important area of research. However, much of the current literature neglects one important feature of many queueing systems, namely non-stationary arrivals. Non-stationary arrivals model the fact that customers tend to access services during certain times of the day and not at a constant rate. In this paper, we analyze two two-dimensional deterministic fluid models that incorporate customer choice behavior based on delayed queue length information with time varying arrivals. In the first model, customers receive queue length information that is delayed by a constant Delta. In the second model, customers receive information about the queue length through a moving average of the queue length where the moving average window is Delta. We analyze the impact of the time varying arrival rate and show using asymptotic analysis that the time varying arrival rate does not impact the critical delay unless the frequency of the time varying arrival rate is twice that of the critical delay. When the frequency of the arrival rate is twice that of the critical delay, then the stability is enlarged by a wedge that is determined by the model parameters. As a result, this problem allows us to combine the theory of nonlinear dynamics, parametric excitation, delays, and time varying queues together to provide insight on the impact of information in queueing systems.

研究动机与目标

• 研究非平稳、时变到达率下，延迟队列长度信息对系统稳定性的影响。
• 通过引入时变到达率，扩展现有排队系统中延迟信息的模型，此类模型在医疗和交通等实际应用中普遍存在。
• 确定时变到达率在何种条件下会改变区分稳定与不稳定系统动态的临界延迟阈值。
• 应用渐近分析技术，推导出考虑振荡性到达率引起的参数激励效应的修正稳定性阈值。

提出的方法

• 使用二维确定性流体模型来表示在延迟信息下的队列长度动态行为。
• 基于延迟或移动平均队列长度信息，通过多项式对数函数建模客户选择行为。
• 应用匹配渐近展开法和双变量展开法，分析小振幅时变到达率下的系统行为。
• 将时变到达率视为小振幅ε的正弦扰动，围绕非时变平衡点展开系统。
• 推导出依赖于时变到达率频率和系统参数的修正临界延迟阈值。
• 通过延迟微分方程的数值积分，验证渐近分析对稳定性及霍普夫分岔的预测结果。

实验结果

研究问题

• RQ1 时变到达率如何影响具有延迟信息的排队系统中的临界延迟阈值？
• RQ2 在恒定到达率下稳定的系统，其时变到达率特性在何种条件下会引发不稳定？
• RQ3 时变到达率的频率是否与临界延迟频率相互作用，从而改变系统稳定性？
• RQ4 渐近分析能否准确预测具有延迟信息和时变到达率的流体模型中振荡行为（霍普夫分岔）的出现？

主要发现

• 仅当时变到达率的频率恰好为临界延迟频率的两倍时，时变到达率才会对系统稳定性产生影响。
• 当到达率频率为临界延迟频率的两倍时，稳定性阈值会因模型参数λ、μ和Δcr决定的楔形区域而扩大。
• 修正的临界延迟为Δ = Δcr ± ε√(α²Δcr² / (Δcrλ + 4Δcrμ + 4))，其中符号取决于λ、μ和Δcr，如公式(3.123)–(3.125)所定义。
• 数值模拟结果证实，渐近预测能准确捕捉在修正阈值处系统从稳定收敛转变为振荡、异步行为的过渡。
• 当到达率频率不匹配临界延迟频率的两倍时，稳定性动态行为与非时变情况近似等价。
• 该分析揭示了排队系统中非线性动力学、参数激励、时滞和时变输入之间的一种新型相互作用。