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arxiv: 2605.16868 · v1 · pith:D5RZCSR6new · submitted 2026-05-16 · 🧮 math.PR

Growing open Markovian Jackson networks: Fluid limit and infinite-dimensional Skorokhod problem

Louis T. Clarke , Guodong Pang , Ruoyu Wu This is my paper

Pith reviewed 2026-05-19 19:46 UTC · model grok-4.3

classification 🧮 math.PR MSC 60K2590B22
keywords Jackson networksfluid limitsinfinite-dimensional Skorokhod problemqueueing networksMarkovian networksempirical measure convergencespectral radius condition
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The pith

Under suitable growth conditions, open Jackson networks converge in the fluid scale to the unique solution of an infinite-dimensional Skorokhod problem with a kernel reflection operator.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies open Jackson networks in which each station is a single-server queue with Poisson arrivals and exponential service, and the network itself is growing. After fluid scaling of the queue-length process, the system is shown to converge to a deterministic infinite-dimensional limit in which a kernel function replaces the usual transition matrix. This limit satisfies a new infinite-dimensional Skorokhod reflection problem whose existence, uniqueness, and Lipschitz continuity are proved once the spectral radius of the reflection operator is strictly less than one. The proof proceeds by introducing an intermediate process that removes the compensated Poisson noise, lifting it to infinite dimensions, applying the Lipschitz property, and controlling the original-to-intermediate difference via martingale estimates. Convergence of the empirical measure of the queues and of associated performance functionals then follows directly.

Core claim

The fluid-scaled queue-length process of a growing open Markovian Jackson network converges to the unique solution of an infinite-dimensional Skorokhod problem driven by a kernel function in place of the routing matrix; existence, uniqueness, and Lipschitz continuity of the mapping hold when the spectral radius of the reflecting operator is less than one.

What carries the argument

The infinite-dimensional Skorokhod mapping, which applies a reflection operator with spectral radius less than 1 to an infinite-dimensional driving process whose kernel encodes the network's routing and growth rates.

If this is right

  • Large growing networks can be approximated by a deterministic continuous-path process that tracks the measure of queue lengths.
  • Performance measures such as average queue length or waiting time converge to the corresponding functionals of the infinite-dimensional fluid limit.
  • The new theory applies to a broader class of reflection operators and infinite-dimensional processes than previously treated.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same kernel-based reflection structure may apply to other growing mean-field systems whose state can be represented by a measure on the nodes.
  • Numerical solution of the infinite-dimensional Skorokhod problem could serve as a fast surrogate for direct simulation of very large networks.
  • Relaxing the spectral-radius condition might be possible by allowing time-dependent or state-dependent operators while retaining pathwise uniqueness.

Load-bearing premise

The network growth rates and service parameters must be chosen so that the reflection operator has spectral radius strictly less than one.

What would settle it

A concrete counter-example network in which the spectral radius of the reflection operator equals or exceeds one, for which the fluid-scaled processes fail to converge or the Skorokhod mapping loses uniqueness or Lipschitz continuity.

read the original abstract

We study growing open Jackson networks where each station is a single-server queue that follows the first-come first-served discipline with Poisson arrivals and exponentially distributed service times, characterized by node-specific rates. In applying a fluid scaling to the queue-length process, we show that under certain conditions the queueing system can be approximated by an infinite-dimensional fluid limit with a kernel function in place of the transition matrix. This limiting process can be characterized by an infinite-dimensional Skorokhod problem, for which we develop a new theory by considering a broader class of reflection operators and general infinite-dimensional processes. We establish existence and uniqueness of a solution along with Lipschitz continuity provided the reflecting operator has a spectral radius less than 1.By introducing an intermediate process in which the compensated Poisson components are removed, and then lifting this to an infinite-dimensional process, we exploit the new Lipschitz property of the infinite-dimensional Skorokhod mapping to prove convergence of the intermediate process. We then prove the necessary estimates for the difference between the original and intermediate processes by using martingale properties. Finally, we consider the empirical measure of the queueing processes, for which we show convergence to the measure associated with the path of the infinite-dimensional fluid limit, extending to the convergence of specific performance-related functionals.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper studies fluid limits for growing open Markovian Jackson networks with node-specific Poisson arrivals and exponential services. Under growth and rate conditions, the scaled queue-length process is shown to converge to an infinite-dimensional fluid limit whose reflection operator is given by a kernel in place of a finite transition matrix. This limit is characterized as the solution to an infinite-dimensional Skorokhod problem; the authors develop existence, uniqueness, and Lipschitz continuity of the solution map when the spectral radius of the reflection operator is strictly less than 1. Convergence is established by removing compensated Poisson terms to obtain an intermediate process, lifting it to the infinite-dimensional setting, invoking the new Lipschitz property, and controlling the difference via martingale estimates. The argument is extended to convergence of the empirical measure of the queue lengths and of associated performance functionals.

Significance. If the central claims hold, the work provides a substantial extension of fluid-limit theory to growing networks whose dimension increases with the scaling parameter, replacing the usual finite matrix with a kernel operator. The new theory for infinite-dimensional Skorokhod problems, including the Lipschitz continuity result under the spectral-radius condition, is a technical contribution that may be useful beyond queueing. The intermediate-process-plus-martingale approach offers a reusable template, and the empirical-measure convergence adds direct applicability to performance analysis.

major comments (2)
  1. [Introduction and the section defining the reflection operator / spectral-radius condition] The abstract and introduction assert that the stated growth and rate conditions on the Jackson network deliver a reflection operator whose spectral radius is strictly less than 1, which is required for the Lipschitz property used in the lifting step. No explicit operator-norm estimate or uniform bound confirming that the kernel induced by the growing routing probabilities satisfies this inequality in the infinite-dimensional limit is indicated; if the norm approaches 1, the Lipschitz constant becomes unbounded and the passage from intermediate-process convergence to the original process fails. This is load-bearing for the entire convergence argument.
  2. [Section on the intermediate process and lifting to the infinite-dimensional Skorokhod problem] The lifting argument in the convergence proof invokes the new Lipschitz property of the infinite-dimensional Skorokhod map. A concrete estimate showing that the growth conditions imply spectral radius <1 uniformly (for example, via a bound on the kernel norm that remains strictly below 1 after passage to the limit) is needed to close the argument.
minor comments (2)
  1. [Model description] Clarify the precise definition of the kernel function that replaces the transition matrix and how it arises from the growing routing probabilities.
  2. [Literature review] Add a short discussion of how the new infinite-dimensional theory relates to existing finite-dimensional Skorokhod mappings and to prior work on measure-valued processes.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for the thorough review and insightful comments on our paper. The major comments correctly identify a need for more explicit verification of the spectral radius condition under our growth assumptions. We will revise the manuscript to include the necessary estimates and bounds. Our point-by-point responses follow.

read point-by-point responses

  1. Referee: [Introduction and the section defining the reflection operator / spectral-radius condition] The abstract and introduction assert that the stated growth and rate conditions on the Jackson network deliver a reflection operator whose spectral radius is strictly less than 1, which is required for the Lipschitz property used in the lifting step. No explicit operator-norm estimate or uniform bound confirming that the kernel induced by the growing routing probabilities satisfies this inequality in the infinite-dimensional limit is indicated; if the norm approaches 1, the Lipschitz constant becomes unbounded and the passage from intermediate-process convergence to the original process fails. This is load-bearing for the entire convergence argument.

    Authors: Thank you for pointing this out. The growth and rate conditions in the paper are intended to ensure that the induced kernel operator has spectral radius strictly less than 1. However, we acknowledge that an explicit operator-norm estimate is not provided in the current version. We will add a new proposition or lemma immediately following the definition of the reflection operator. This lemma will use the assumptions on the node-specific arrival and service rates together with the growth conditions to derive a uniform upper bound on the operator norm of the kernel that is strictly less than 1. This bound will be independent of the scaling parameter and will ensure the Lipschitz constant of the Skorokhod map remains finite. We believe this addition will fully address the concern and strengthen the convergence argument. revision: yes

  2. Referee: [Section on the intermediate process and lifting to the infinite-dimensional Skorokhod problem] The lifting argument in the convergence proof invokes the new Lipschitz property of the infinite-dimensional Skorokhod map. A concrete estimate showing that the growth conditions imply spectral radius <1 uniformly (for example, via a bound on the kernel norm that remains strictly below 1 after passage to the limit) is needed to close the argument.

    Authors: We agree that the lifting step requires an explicit verification. In the revised manuscript, we will reference the new bound from the added lemma when applying the Lipschitz property in the convergence proof. Specifically, we will insert a paragraph in the lifting argument section that invokes the uniform spectral radius bound to confirm that the conditions for the Lipschitz continuity hold uniformly. This will close the gap in the argument from the intermediate process to the original scaled queue-length process. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on independent theory and external conditions

full rationale

The paper develops a new existence/uniqueness/Lipschitz theory for the infinite-dimensional Skorokhod problem under the spectral-radius-<1 assumption on the reflection operator, then applies it to the fluid-scaled Jackson network via an intermediate process (with compensated Poisson terms removed) and standard martingale estimates for the original-to-intermediate difference. The growth and rate conditions are stated to ensure the spectral radius bound holds for the kernel-induced operator, but this is an external verification step rather than a self-referential definition or fit. No equation reduces to another by construction, no parameter is fitted to a subset and renamed a prediction, and no load-bearing uniqueness or ansatz is smuggled via self-citation. The chain is self-contained against the stated assumptions and classical probabilistic tools.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard assumptions of Poisson arrivals and exponential services together with the spectral radius condition on the reflection operator; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption The reflecting operator has spectral radius strictly less than 1.
    Invoked to obtain Lipschitz continuity of the infinite-dimensional Skorokhod mapping.
  • domain assumption The network growth satisfies conditions allowing the fluid limit to exist with a kernel function replacing the transition matrix.
    Required for the approximation statement and convergence arguments.

pith-pipeline@v0.9.0 · 5759 in / 1445 out tokens · 35189 ms · 2026-05-19T19:46:44.297971+00:00 · methodology

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Reference graph

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