[论文解读] A Robust Queueing Network Analyzer Based on Indices of Dispersion

Whitt, Ward, You, Wei|arXiv (Cornell University)|Jan 1, 2019

Advanced Queuing Theory Analysis参考文献 44被引用 6

一句话总结

本论文提出了一种基于分散指数（IDC）的鲁棒排队网络分析器（RQNA），用于近似具有非更新到达、一般服务时间及反馈的单类开放排队网络中的稳态性能。通过利用IDC函数对流量变异性进行建模，并求解内部流量的线性方程，RQNA在高负载、反馈密集的系统中相比先前方法显著提升了准确性——尤其在消除反馈后表现尤为出色，实现了具有竞争力的结果。

ABSTRACT

In post-industrial economies, modern service systems are dramatically changing the daily lives of many people. Such systems are often complicated by uncertainty: service providers usually cannot predict when a customer will arrive and how long the service will be. Fortunately, useful guidance can often be provided by exploiting stochastic models such as queueing networks. In iterating the design of service systems, decision makers usually favor analytical analysis of the models over simulation methods, due to the prohibitive computation time required to obtain optimal solutions for service operation problems involving multidimensional stochastic networks. However, queueing networks that can be solved analytically require strong assumptions that are rarely satisfied, whereas realistic models that exhibit complicated dependence structure are prohibitively hard to analyze exactly. In this thesis, we continue the effort to develop useful analytical performance approximations for the single-class open queueing network with Markovian routing, unlimited waiting space and the first-come first-served service discipline. We focus on open queueing networks where the external arrival processes are not Poisson and the service times are not exponential. We develop a new non-parametric robust queueing algorithm for the performance approximation in single-server queues. With robust optimization techniques, the underlying stochastic processes are replaced by samples from suitably defined uncertainty sets and the worst-case scenario is analyzed. We show that this worst-case characterization of the performance measure is asymptotically exact for approximating the mean steady-state workload in G/G/1 models in both the light-traffic and heavy-traffic limits, under mild regularity conditions. In our non-parametric Robust Queueing formulation, we focus on the customer flows, defined as the continuous-time processes counting customers in or out of the network, or flowing from one queue to another. Each flow is partially characterized by a continuous function that measures the change of stochastic variability over time. This function is called the index of dispersion for counts. The Robust Queueing algorithm converts the index of dispersion for counts into approximations of the performance measures. We show the advantage of using index of dispersion for counts in queueing approximation by a renewal process characterization theorem and the ordering of the mean steady-state workload in GI/M/1 models. To develop generalized algorithm for open queueing networks, we first establish the heavy-traffic limit theorem for the stationary departure flows from a GI/GI/1 model. We show that the index of dispersion for counts function of the stationary departure flow can be approximately characterized as the convex combination of the arrival index of dispersion for counts and service index of dispersion for counts with a time-dependent weight function, revealing the non-trivial impact of the traffic intensity on the departure processes. This heavy-traffic limit theorem is further generalized into a joint heavy-traffic limit for the stationary customer flows in generalized Jackson networks, where the external arrival are characterized by independent renewal processes and the service times are independent and identically distributed random variables, independent of the external arrival processes. We show how these limiting theorems can be exploited to establish a set of linear equations, whose solution serves as approximations of the index of dispersion for counts of the flows in an open queueing network. We prove that this set of equations is asymptotically exact in approximating the index of dispersion for counts of the stationary flows. With the index of dispersion for counts available, the network is decomposed into single-server queues and the Robust Queueing algorithm can be applied to obtain performance approximation. This algorithm is referred to as the Robust Queueing Network Analyzer. We perform extensive simulation study to validate the effectiveness of our algorithm. We show that our algorithm can be applied not only to models with non-exponential distirbutions but also to models with more complex arrival processes than renewal processes, including those with Markovian arrival processes.

研究动机与目标

• 解决现有分解方法与重负载近似在具有依赖到达与反馈的非马氏排队网络中的局限性。
• 提升外部到达非泊松分布且服务时间一般时，开放排队网络中性能近似的准确性。
• 开发一种可扩展算法，利用计数分散指数（IDC）对客户流量中的时变变异性进行建模。
• 引入反馈消除程序，以提升在高负载、易产生反馈的网络中的准确性。
• 为云计算、医疗保健和联络中心等实际服务系统提供一种实用且计算高效的性能预测算法。

提出的方法

• 将客户流量建模为连续时间过程，其特征由到达率和计数分散指数（IDC）描述，即缩放的方差-时间曲线。
• 从模型原始参数或数据中推导并求解线性方程组，以计算内部到达过程的IDC值。
• 基于重负载极限应用鲁棒排队近似，使用一维反射布朗运动（RBM）来估计平均稳态性能。
• 集成反馈消除：在应用主RQNA算法前，识别并移除近似即时的反馈流量，以减少瓶颈队列中的误差。
• 结合总外部与内部到达过程的IDC及服务时间分布，近似平均停留时间与队列长度。
• 为树状结构网络开发简化版本的算法，以提升计算效率。

实验结果

研究问题

• RQ1 如何利用计数分散指数（IDC）对到达过程中的时间依赖性进行建模与利用，以改进排队网络近似？
• RQ2 客户反馈与相关到达对基于分解的近似方法（如QNA与QNET）的准确性有何影响？
• RQ3 反馈消除程序是否能显著提升鲁棒排队近似在高负载、反馈密集网络中的准确性？
• RQ4 在不同网络拓扑结构下，RQNA-IDC算法在平均停留时间估计方面与仿真及其他近似方法（如QNA、QNET、SBD、RQ）相比表现如何？
• RQ5 对一维反射布朗运动的依赖在紧密耦合、高负载队列中在多大程度上限制了RQNA-IDC的准确性？

主要发现

• 在未使用反馈消除的情况下，RQNA-IDC在瓶颈节点5处误差最高达211%，表明在高反馈场景下存在严重不准确。
• 反馈消除后，RQNA-IDC在节点4和5的误差降低至15–18%，使其在性能上与SBD和QNET等成熟方法具有竞争力。
• 在引入反馈消除后，RQNA-IDC的总系统停留时间近似误差为9.9%，而RQ为18%，RQNA无消除时高达102%。
• 在节点10，RQNA-IDC在反馈消除后仅产生-8.7%的误差，优于RQ（3.9%）和SBD（1.7%）。
• 反馈消除程序有效缓解了高负载、反馈密集网络中IDC分解的失效问题。
• 该方法在包括反馈环路在内的多种复杂网络结构中表现出鲁棒性，经大量仿真研究与重负载极限验证。