We investigate the functional limits of generalized Jackson networks in a multi-scale heavy traffic regime where stations approach full utilization at distinct, separated rates. Our main result shows that the appropriately scaled queue length processes converge weakly to a limit process whose coordinates are mutually independent. This finding provides the fundamental dynamic mechanism that explains the asymptotic independence previously observed only in stationary distributions. The specific form of the limit processes is shown to depend on the initial conditions. Moreover, we introduce and analyze a blockwise multi-scale heavy traffic regime. In this regime, the network's stations are partitioned into blocks, where stations in different blocks approach the heavy traffic at different rates, while stations within the same block share a common rate. We obtain the functional limits in this regime as well, showing that the limit process exhibits blockwise independence.

Inspired by Dai et al. [2023], we propose a novel multi-scaling asymptotic regime for semimartingale reflected Brownian motion (SRBM). We first prove that the stationary distribution of SRBM converges to a product-form limit with each component following an exponential distribution under the multi-scaling regime when the reflection matrix is an M-matrix. Then, we extend the results to a few examples of SRBM where the reflection matrix is a P-matrix by verifying a certain uniform moment bound condition. Our proof approach is rooted in the basic adjoint relationship (BAR) for SRBM proposed by Harrison and Williams [1987].

This paper examines a continuous-time routing system with general interarrival and service time distributions, operating under the join-the-shortest-queue and power-of-two-choices policies. Under a weaker set of assumptions than those commonly found in the literature, we prove that the scaled steady-state queue length at each station converges weakly to an identical exponential random variable in heavy traffic. Specifically, our results hold under the assumption of the $(2+\delta_0)$th moment for the interarrival and service distributions with some $\delta_0>0$. The proof leverages the Palm version of the basic adjoint relationship (BAR) as a key technique.

We establish uniform moment bounds for steady-state queue lengths of generalized Jackson networks (GJNs) in multi-scale heavy traffic as recently proposed by Dai et al. [2023]. Uniform moment bounds lay the foundation for further analysis of the limit stationary distribution. Our result can be used to verify the crucial moment state space collapse (SSC) assumption in Dai et al. [2023] to establish a product-form limit of GJN in the multi-scale heavy traffic regime. Our proof critically utilizes the Palm version of the basic adjoint relationship (BAR) as developed in Braverman et al. [2023].

We collaborate with a large teaching hospital in Shenzhen, China and build a high-fidelity simulation model for its ultrasound center to predict key performance metrics, including the distributions of queue length, waiting time and sojourn time, with high accuracy. The key challenge to build an accurate simulation model is to understand the complicated patient routing at the ultrasound center. To address the issue, we propose a novel two-level routing component to the queueing network model and use machine learning tools to calibrate the routing components from data. Our empirical results show that the calibrated model is of high fidelity and yields accurate prediction results for the performance metrics.