Blockwise Multi-Scale Heavy Traffic Regime

• The paper demonstrates that blockwise multi-scale heavy traffic regimes partition systems into distinct blocks with specific scaling parameters, leading to independent SRBM limits.
• It leverages block-specific scaling techniques to derive product-form stationary distributions and establish state space collapse in complex queueing networks.
• The regime’s framework facilitates accurate performance evaluation and optimized resource allocation in heterogeneous service systems through refined diffusion approximations.

A blockwise multi-scale heavy traffic regime formalizes queueing and network models in which multiple subsystems ("blocks") or stations are simultaneously driven into heavy traffic, but at rates that may differ greatly by block or even by station. Rather than assuming a single scaling for all components, blockwise multi-scale frameworks accommodate the distinct time and space scales imposed when resource utilization approaches capacity along inhomogeneous, often strictly separated, trajectories. These regimes have become central in advancing diffusion approximations, product-form stationary characterizations, and decomposition principles for complex queueing systems, generalized Jackson networks, and matching models.

1. Definition and Conceptual Structure

The blockwise multi-scale heavy traffic regime partitions the state space (e.g., stations in a network or classes in a multiclass priority system) into $K$ "blocks" $\{A_1,\ldots,A_K\}$. Each block $A_k$ is indexed by a block-specific scaling parameter $\gamma_k(r)$, which approaches zero as $r\to0$ but with strict scale separation. For instance, one may set $\gamma_1(r) = r$, $\gamma_2(r) = r^2$, ..., $\gamma_K(r) = r^K$, capturing the fact that blocks are pushed to saturation at rates that may differ by orders of magnitude. Within each block, the criticality parameter for station $j\in A_k$ is

$$\mu_j - \lambda_j = \gamma_k(r)\, b_j, \qquad b_j > 0,$$

so traffic intensities satisfy $1 - \rho_j \asymp \gamma_k(r)$ for $j\in A_k$ (Chen et al., 23 Aug 2025).

The key property is that stations (or queues) in different blocks experience heavy-traffic scaling on different functional time and space scales, inducing a separation in fluctuation magnitudes and mixing times. Inter-block couplings—e.g., routing—may exist but become asymptotically negligible due to this separation.

2. Mathematical Formulation and Limit Processes

In blockwise multi-scale heavy traffic, queue length or workload processes are scaled differently by block. For a block $A_k$,

$$\{\gamma_k(r)\,Z_{A_k}(t/\gamma_k^2(r)) : t \geq 0\}$$

is the considered process, with $Z_{A_k}$ denoting the joint queue length vector for block $A_k$ (Chen et al., 23 Aug 2025).

Under appropriate initial conditions—typically "matching-rate initial conditions" ensuring separation across blocks—the main functional limit theorem is: $$\big\{\big(\gamma_1(r)\,Z_{A_1}(t/\gamma_1^2(r)),\,\ldots,\,\gamma_K(r)\,Z_{A_K}(t/\gamma_K^2(r))\big);\, t \geq 0\big\} \;\Rightarrow\; Z^*,$$ where $Z^*$ is a $J$-dimensional diffusion process whose blockwise projections $Z^*_{A_k}$ are mutually independent. Each block's limit is characterized as a semimartingale reflected Brownian motion (SRBM) with data inherited from the "local" drift, covariance, and reflection structure of the original system, typically as follows: $$Z^*_{A_k}\,=\, {\rm SRBM}(\,\xi_{A_k}, -G_{A_k,A_k}^{(m_k)}\,b^{(k)},\, (E^{(m_k)} \Gamma (E^{(m_k)})')_{A_k,A_k},\, G_{A_k,A_k}^{(m_k)}\,)$$ where $G^{(m_k)}$ is obtained by an inductive blockwise Gaussian elimination of the reflection matrix, $E^{(m_k)}$ similarly from the block structure, and $\Gamma$ encodes the input noise (Chen et al., 23 Aug 2025). The independence across blocks is a consequence of the time scale disparity, which ensures that asymptotic inter-block influence vanishes.

The stationary distributions of the scaled queue lengths also exhibit product-form limits: $$\bigl(\gamma_1(r)Z_{A_1},\, \ldots,\, \gamma_K(r) Z_{A_K}\bigr)\;\Rightarrow\; (d_1 E_1,\, \ldots,\, d_K E_K)$$ where each $E_k$ is a standard exponential random variable and each $d_k$ is a constant computable from local data (arrival rates, variances, routing) (Dai et al., 2023, Dai et al., 6 Mar 2024).

3. Dynamic Mechanism of Blockwise Asymptotic Independence

A defining feature of this regime is dynamic decoupling: as the scaling parameter $r\to0$, the stochastic evolution of each block (on its respective time and space scales) becomes asymptotically independent of other blocks. This arises for two reasons:

1. Scale Separation: For blocks $A_k$ and $A_l$ with $k<l$, $\gamma_k(r)/\gamma_l(r)\to\infty$. The processes on block $A_l$ evolve so "slowly" that, on the block $A_k$ time scale, the $A_l$ block is essentially frozen.
2. Initial Condition Matching: If, for $k\geq2$, $\gamma_{k-1}(r)Z_{A_k}(0)\to\infty$, then for each block the initial state is overwhelming relative to finer scales, ensuring no inter-block interaction in the fluid or diffusion limits (Chen et al., 23 Aug 2025).

Technically, strong approximations (using the Law of the Iterated Logarithm and Skorokhod reflection properties) show that cross-block noise and drift terms vanish in the limit; diffusion approximations thus "lift" asymptotic independence from input processes to state processes.

4. Applications in Queueing Networks and Service Systems

Blockwise multi-scale heavy traffic models have broad implications:

• In generalized Jackson networks, they explain why product-form steady-state distributions emerge in the scaling limit, even when prelimit networks are highly coupled and nonreversible (Dai et al., 2023, Guang et al., 26 Jan 2024, Dai et al., 6 Mar 2024).
• In multiclass and priority queueing systems with static buffer priorities, blockwise scaling produces exponential product-form asymptotics for the noncollapsing (e.g., low-priority) classes, thus enabling tractable performance evaluation under critical loading (Dai et al., 6 Mar 2024).
• In matching systems (e.g., assemble-to-order, blood banks), heavy traffic limits of coupled queues with perishable and jointly matched items lead to diffusion equations where the regulator term is a blockwise minimum, tightly coupling and decoupling componentwise behavior depending on scaling (Xie, 2022).
• For large-scale service systems such as call centers or cloud computing clusters, subsystem partitions (geographical, by customer segment, or by priority) may naturally suggest blockwise scalings to yield accurate delay and congestion approximations (Janssen et al., 2014, Leeuwaarden et al., 2017).

Blockwise multi-scale regimes also support operational insights, e.g., allowing capacity sizing and resource allocation to be optimized per-block using explicit steady-state approximations.

5. Uniform Moment Bounds and State Space Collapse

Theoretical analysis of these regimes requires establishing uniform in $r$ moment bounds for scaled queue lengths. These "uniform moment bounds" (e.g.,

$$\sup_{r\in(0,r_0)} \mathbb{E}[ (\gamma_k(r) Z_k)^{M} ] < \infty$$

for all blocks/classes $k$) underpin the tightness of the limiting distributions and justify interchanging limits for steady-state and scaling (Guang et al., 26 Jan 2024). They are proved via advanced test function and generator (BAR) methods, often invoking properties of a reflection matrix (which should be a $P$-matrix for uniqueness and nondegeneracy in some multiclass models) (Dai et al., 6 Mar 2024).

Such bounds also verify the state space collapse property: fluctuations in the full system can be projected to a low-dimensional (blockwise) subspace, simplifying the functional limit analysis and facilitating Lyapunov drift arguments for performance guarantees.

6. Connections to "Product-form" and Independence Principles

A key outcome of the blockwise multi-scale heavy traffic regime is the emergence of product-form stationary and dynamic limits, under blockwise scaling:

• Each block's scaled queue length converges in law to an exponential (in stationarity) or to an SRBM (dynamically), and blocks are independent in the limit.
• For generalized Jackson networks, this rigorous product-form property resolves longstanding questions about the dynamic origin of such results (moving beyond the steady-state algebra of Jackson-type networks under critical loading) (Dai et al., 2023, Chen et al., 23 Aug 2025).
• In functional limit theorems, blockwise independence emerges as the fundamental "mechanism" (in contrast with systems that only admit weak, one-dimensional approximations or that remain strongly coupled).

Blockwise decompositions thus convert complex, high-dimensional, and heterogeneously scaled queueing problems into analysis on tractable, independent lower-dimensional components, providing a robust foundation for simulation, control, and analytic performance computation.

7. Extensions and Ongoing Research Directions

Recent work has expanded the blockwise multi-scale paradigm:

• To general multiclass queueing networks (with or without priorities) and to settings with nontrivial routing and feedback, via block Gaussian elimination and advanced stochastic comparison (Chen et al., 23 Aug 2025, Dai et al., 2023).
• To process-level (functional) limit theorems, not just stationary distributions, thereby explaining how dynamic (asymptotic) independence emerges over time and not solely in equilibrium (Chen et al., 23 Aug 2025).
• To queueing networks with Gaussian or Lévy input, Markovian or non-Markovian arrivals, highlighting the universality of these regimes across model classes (Kriukov et al., 20 Mar 2025, Koops et al., 2015).
• To uncertainty-aware control policies, where adversarial disturbance at the diffusion scale may force differentiated blockwise limits, leading to discontinuous SDE models (Atar et al., 2022).

A plausible implication is that the blockwise multi-scale regime will play a central role in the next-generation analysis of large, heterogeneous networks (e.g., cloud infrastructure, supply chains, decentralized service platforms) where heterogeneity in bottlenecks and resources is the norm.

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In summary, the blockwise multi-scale heavy traffic regime provides the rigorous asymptotic theory and algorithmic toolkit for analyzing, controlling, and simulating complex stochastic service systems approaching capacity on multiple separated scales. Its core phenomena—decoupling, product-form limits, and state-space collapse—are now foundational in both probability theory and operations research.