The efficient management of large-scale queueing networks is critical for a variety of sectors, including healthcare, logistics, and customer service, where system performance has profound implications for operational effectiveness and cost management. To address this key challenge, our paper introduces simulation techniques tailored for complex, large-scale Markovian queueing networks. We develop two simulation schemes based on Euler approximation, namely the backward and forward schemes. These schemes can accommodate time-varying dynamics and are optimized for efficient implementation using vectorization. Assuming a feedforward queueing network structure, we establish that the two schemes provide stochastic upper and lower bounds for the system state, while the approximation error remains bounded over the simulation horizon. With the recommended choice of time step, we show that our approximation schemes exhibit diminishing asymptotic relative error as the system scales up, while maintaining much lower computational complexity compared to traditional discrete-event simulation and achieving speedups up to tens of thousands times. This study highlights the substantial potential of Euler approximation in simulating large-scale discrete systems.

翻译：大规模排队网络的高效管理对医疗保健、物流和客户服务等多个领域至关重要，系统性能直接影响运营效率与成本管理。为应对这一关键挑战，本文提出了针对复杂大规模马尔可夫排队网络的仿真技术。我们基于欧拉近似开发了两种仿真方案，即后向方案和前向方案。这两种方案能够适应时变动态特性，并通过向量化技术实现高效实施。在假设前馈排队网络结构的前提下，我们证明这两种方案可为系统状态提供随机上界与下界，且仿真跨度内的近似误差保持有界。在推荐的时间步长选择下，我们证明随着系统规模扩大，近似方案的渐近相对误差呈现递减趋势，同时其计算复杂度远低于传统离散事件仿真，可实现数万倍的加速效果。本研究凸显了欧拉近似在仿真大规模离散系统中的巨大潜力。