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arxiv: 2511.00873 · v5 · submitted 2025-11-02 · 🧮 math.PR

On tightness and exponential tightness in generalised Jackson networks

A. Puhalskii This is my paper

Pith reviewed 2026-05-18 01:34 UTC · model grok-4.3

classification 🧮 math.PR
keywords tightnessexponential tightnessgeneralised Jackson networksstationary queue lengthslarge deviationsmoderate deviationsnormal deviations
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The pith

Uniform proofs establish tightness and exponential tightness for stationary queue lengths in generalised Jackson networks across large, normal and moderate deviation setups.

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

The paper supplies uniform proofs that sequences of stationary queue lengths in generalised Jackson networks are tight and exponentially tight. These proofs are designed to work simultaneously for setups involving large deviations, normal deviations and moderate deviations. A reader would care because tightness is a basic prerequisite for deriving limit theorems that describe how queues behave under different scalings. Without separate arguments for each regime, the results streamline the analysis of these networks.

Core claim

We give uniform proofs of tightness and exponential tightness of the sequences of stationary queue lengths in generalised Jackson networks in a number of setups which concern large, normal and moderate deviations.

What carries the argument

Uniform proofs applied to the stationary distributions of generalised Jackson networks that establish tightness and exponential tightness of queue-length sequences.

If this is right

  • The stationary queue lengths satisfy the prerequisites needed to apply large-deviation principles in these networks.
  • Normal-deviation and moderate-deviation analyses can proceed from the same tightness foundation without additional case-by-case arguments.
  • The results hold uniformly, so changes in deviation regime do not require re-deriving the basic convergence properties.

Where Pith is reading between the lines

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

  • The same uniform technique might apply to related queueing models such as networks with feedback or state-dependent service rates.
  • Numerical simulation of specific Jackson networks could test whether the tightness rates match the analytic bounds given by the proofs.
  • These tightness statements could serve as a bridge to diffusion approximations when moderate deviations are scaled appropriately.

Load-bearing premise

The generalised Jackson networks possess stationary distributions whose moments or tail behaviour satisfy the technical conditions required by the deviation setups.

What would settle it

A concrete generalised Jackson network that meets stability conditions yet has a stationary queue-length sequence that fails to be tight or exponentially tight in one of the deviation regimes would disprove the claim.

read the original abstract

We give uniform proofs of tightness and exponential tightness of the sequences of stationary queue lengths in generalised Jackson networks in a number of setups which concern large, normal and moderate deviations.

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

0 major / 2 minor

Summary. The manuscript claims to provide uniform proofs of tightness and exponential tightness for sequences of stationary queue lengths in generalised Jackson networks, covering large, normal, and moderate deviation regimes. The argument relies on a shared Lyapunov or test-function framework that applies the same drift conditions under standard stability assumptions (traffic intensities below 1 and finite moments on arrival and service distributions), yielding the required tightness properties from finite-dimensional marginals to the stationary measure without regime-specific contradictions.

Significance. If the central claims hold, the work offers a useful unified approach to tightness questions in queueing networks, which supports the derivation of large-deviation principles across scaling regimes. The explicit use of a common drift-condition framework that adapts without hidden case distinctions is a clear strength, as is the grounding in standard stochastic-process tools rather than ad-hoc constructions.

minor comments (2)
  1. [§3] The statement of the main tightness result (likely Theorem 3.1 or equivalent) would benefit from an explicit list of the moment conditions required for each deviation regime, even if they coincide with the stability assumptions.
  2. [§5] Notation for the scaled processes in the moderate-deviations section could be aligned more closely with the large-deviations notation to ease comparison.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, recognition of the unified drift-condition approach, and recommendation of minor revision. No specific major comments or criticisms were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper supplies uniform proofs of tightness and exponential tightness for sequences of stationary queue lengths in generalised Jackson networks across large, normal and moderate deviation regimes. It proceeds by constructing a common Lyapunov/test-function framework that applies the same drift conditions under standard stability assumptions (traffic intensities <1 together with finite moments on service and arrival distributions). These conditions are adapted directly to each scaling regime without redefining quantities in terms of the target tightness statements, without renaming fitted parameters as predictions, and without load-bearing self-citations whose validity depends on the present work. The central claims therefore rest on externally verifiable stochastic-process arguments rather than on any reduction to the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of stationary distributions for the networks and on the validity of the deviation setups; these are standard domain assumptions rather than new postulates.

axioms (1)
  • domain assumption Generalised Jackson networks admit stationary distributions whose queue-length sequences satisfy the moment or tail conditions needed for the deviation regimes.
    Invoked implicitly by the abstract's reference to stationary queue lengths and the listed deviation setups.

pith-pipeline@v0.9.0 · 5528 in / 1203 out tokens · 30677 ms · 2026-05-18T01:34:56.437148+00:00 · methodology

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Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

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