Queue-Reactive Model in Stochastic Systems

• Queue-Reactive Model is a framework where event intensities are dynamically adjusted based on current queue states, enhancing system responsiveness.
• It underpins applications in order-book modeling, multi-agent allocation, traffic assignment, and formal computational theory with state-dependent dynamics.
• The approach utilizes precise mathematical formulations, calibration methods, and simulation techniques to analyze ergodicity, price diffusion, and order-flow clustering.

A Queue-Reactive Model is a state-dependent stochastic framework in which the arrival and service dynamics at a set of queues are governed by functions of the instantaneous queue configuration. The paradigm appears across order-book modeling in quantitative finance, resource allocation in distributed robotics, traffic assignment, and formal computational theory. The hallmark of queue-reactivity is that event intensities, costs, or capacities are explicitly coupled to the current or recent state of one or more queues, enabling endogenous responses to congestion, order imbalance, or computational process state. This article surveys foundational instances, mathematical formulations, tractable extensions, parameter estimation methodologies, and key empirical findings.

1. Core Mathematical Structures and Definitions

The canonical queue-reactive model (QR) in the context of order books is formulated as a continuous-time Markov jump process on a state space $\mathbb{N}^{2K}$, where each component encodes the length or volume at a price-specific limit-order queue. For each queue $i$, arrivals (limit orders) and departures (cancels, market orders) are modeled as Poisson processes with rate functions

$$\lambda^{L}_i(q_i),\quad \lambda^{C}_i(q_i),\quad \lambda^{M}_i(q_i)$$

that depend on the current queue size $q_i$. The infinitesimal generator $\mathcal{L}$ acts according to

$$(\mathcal{L} f)(q) = \sum_{i} \left[ \lambda^L_i(q_i) (f(q+e_i){-}f(q)) + (\lambda^C_i(q_i){+}\lambda^M_i(q_i))\,\mathbb{1}_{q_i>0}(f(q-e_i){-}f(q)) \right]$$

where $e_i$ is the $i$-th unit vector. Under mild stability conditions, the process is ergodic with an invariant distribution admitting closed-form product-of-ratios expression per queue in specialized, independence-assuming cases (Huang et al., 2013, Wu et al., 2019, Espana et al., 19 Nov 2025).

Queue-reactivity also arises in centralized task allocation for multi-agent systems, where the cost of assigning an agent to a task incorporates explicit penalties based on current queue lengths $|\mathcal{Q}_i|$ and individual task waiting times $w_j$, producing a global cost of

$$C_{k,j} = c_{k,j} + \alpha |\mathcal{Q}_{i(j)}| + \beta w_j$$

and yielding a linear integer program for optimal allocation under mapping and capacity constraints (Dahlquist et al., 2023).

In computational theory, "queue-reactive" characterizes computational models (queue automata, reactive Turing machines) whose transition rules and computational state evolve interactively with queue contents, yielding maximal expressiveness in the Chomsky–Turing hierarchy (Baeten et al., 12 Feb 2025).

2. Extensions and Generalizations

Order-Book Models:

Queue-reactive models have been extended to:

• Hawkes-augmented QR (QRH-I/QRH-II): Adds self- and cross-excitation via Hawkes processes to the baseline state-dependent rates, leading to stochastic intensities of the form

$$\lambda^\ell(t) = \mu^\ell(q(t^-)) + \sum_{m,u} \alpha^{\ell m}_u g^m_u(t)$$

where $g^m_u$ encode exponential Hawkes memory (Wu et al., 2019).

• Size-Aware QR (SAQR): Models not just event type and arrival rate but incorporates explicit order size dependency, expanding event types to $(\eta,s)$ and calibrating size-conditional intensities $\lambda^{(\eta,s)}_i(n)$ (Bodor et al., 28 May 2024).
• Deep QR (MDQR): Utilizes neural architectures to relax queue-independence, learn cross-queue and feature-enriched state dependencies, and generate realistic event and size distributions, operating within a point-process likelihood framework (Bodor et al., 15 Jan 2025).

Traffic Assignment:

Queue-reactive principles have been applied to model equilibrium link flows and queuing delays in congested networks, where the link cost function includes both conventional flow-performance terms and a queue-dependent delay, and the effective capacity is a decreasing function of the queue, enforcing equilibrium flows below nominal capacity in oversaturated scenarios (Fu et al., 11 Jan 2025).

Multi-Asset and Signal-Driven Models:

In multivariate extensions, queue-reactive models are coupled to latent "efficient prices" (Brownian motion) or observable signals, so that event intensities and reference price mechanisms reconcile tick-level microstructure with macroscopic price diffusion, while enabling cross-asset correlation and market-impact response (Sfendourakis, 13 Jun 2025).

3. Parameter Estimation and Calibration

All realizations of the queue-reactive paradigm admit data-driven calibration:

• Closed-form MLE: For Poisson queue-event models, maximum likelihood estimation of $\lambda^\eta(n)$ is reduced to frequency and duration statistics over event types and queue-size bins

$$\hat\lambda^\eta(n) = \frac{\#\{\text{events of type }\eta \text{ at } n\}}{\sum_{\text{occurrences at}\;n} \Delta t_k}$$

(Huang et al., 2013, Mariotti et al., 2022, Bodor et al., 28 May 2024).

• Hawkes variants: MLE or least-squares procedure over parametric kernels (typically exponentials), using analytic recursions and grid-based state aggregation (Wu et al., 2019).
• Deep queue-reactive: Stochastic gradient descent on joint log-likelihood and cross-entropy losses for event type/timing and order size (categorical) via neural network models (Bodor et al., 15 Jan 2025).
• Signal-driven/efficient price models: Likelihood evaluated through backward solution of a sequence of conditional (linear parabolic) PDEs on efficient-reference price gap, tractably implemented with parametric intensity expansions and system-of-ODE reductions (Sfendourakis, 13 Jun 2025).

4. Queue-Reactive Properties and Empirical Performance

All queue-reactive models exhibit strong empirical tractability:

• Ergodicity: Under negative-drift and boundedness conditions, there exists a unique V-uniformly ergodic law, admitting exponential convergence for queue-size and event statistics (Huang et al., 2013, Wu et al., 2019, Sfendourakis, 13 Jun 2025).
• Price diffusion: The mid-price or reference price generated by queue-reactive dynamics exhibits the correct macroscopic diffusion and mean-reversion when parameters governing price moves and re-initialization are calibrated to market data (Huang et al., 2013, Mariotti et al., 2022).
• Order-flow clustering: Hawkes-augmented models produce event clustering and inter-arrival statistics (heavy tails, self-excitation) matching empirical order flow (Wu et al., 2019).
• Cross-queue dependencies: Non-independence and feature learning (MDQR) enable correct reproduction of bid/ask volume correlations, square-root market impact scaling, non-trivial excitation patterns, and intraday seasonality (Bodor et al., 15 Jan 2025).
• Traffic assignment: The queue-reactive static assignment model uniquely identifies sub-capacity equilibrium flows under congestion, matching detector data and supporting robust alternating-minimization algorithms (Fu et al., 11 Jan 2025).

5. Algorithmic Solutions and Computational Aspects

Auction-based task allocation: Reactive models admit tractable linear integer programming (LIP) formulations, with explicit objective and constraints optimizing joint travel, queue-length, and individual waiting-time penalties. The LIP is solved efficiently using standard solvers, with empirical guidelines for tuning queue versus waiting penalties (Dahlquist et al., 2023).

Simulation and RL: Queue-reactive simulators are computationally efficient for both high-frequency simulation (orders of 1M+ events/day under 1 ms/event inference) and reinforcement learning orchestration, providing realistic environments for optimal execution strategy testing (Espana et al., 19 Nov 2025, Bodor et al., 15 Jan 2025).

Traffic assignment: Alternating minimization (gradient-projection) algorithms are effective for large-scale static assignment under queue-reactive constraints, converging in reasonable time and producing interpretable solutions for link flow and congestion (Fu et al., 11 Jan 2025).

6. Theoretical and Hierarchical Implications

In formal models of computation, the queue-reactive framework achieves maximal expressiveness within the Chomsky–Turing hierarchy: non-deterministic queue automata and reactive Turing machines (RTM) are computationally equivalent, and any executable process is equivalently representable via queue automata, pushing queue-reactivity to the top class of effective process descriptions (Baeten et al., 12 Feb 2025). Closure properties, simulation constructions, and equivalence theorems are established up to strong bisimulation.

7. Applications, Limitations, and Future Directions

Applications include:

• Market microstructure simulation, strategy backtesting, and regulatory stress testing (Huang et al., 2013, Mariotti et al., 2022, Bodor et al., 15 Jan 2025).
• Balanced multi-agent task allocation with robust queue management (Dahlquist et al., 2023).
• Congestion modeling and capacity planning in transportation networks (Fu et al., 11 Jan 2025).
• Theoretical paper of executable process semantics (Baeten et al., 12 Feb 2025).
• Multi-asset market impact and cross-observable signal integration (Sfendourakis, 13 Jun 2025).

Limitations: Basic queue-reactive models are memoryless (do not exhibit excitation), may ignore order size granularity, or assume queue independence. Advanced variants address these via Hawkes memory, deep feature sets, or explicit size modeling. Open avenues include integrating periodicity, exogenous seasonality, and multi-period/dynamic extensions while preserving tractability and ergodicity (Bodor et al., 28 May 2024, Fu et al., 11 Jan 2025, Wu et al., 2019).

Empirical findings indicate that queue-reactive models and their extensions consistently outperform zero-intelligence baselines and pure Hawkes models in reproducing queue-size distributions, inter-event statistics, volatility, and cost variance relevant for real-world system design and analysis.

Key References: (Huang et al., 2013, Wu et al., 2019, Dahlquist et al., 2023, Bodor et al., 28 May 2024, Bodor et al., 15 Jan 2025, Fu et al., 11 Jan 2025, Baeten et al., 12 Feb 2025, Sfendourakis, 13 Jun 2025, Espana et al., 19 Nov 2025, Mariotti et al., 2022).