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arXiv · 2607.14954

A Queueing-Stability Criterion for Causal IPD-QIM Network Flow Watermarking

Jiuxiang Cao·Guang Cheng·Guangjie Liu
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Abstract

On multi-hop encrypted links such as Tor and cascaded VPNs, tunneling flattens packet lengths and protocol fields, leaving inter-packet delay (IPD) as the main carrier for active flow attribution. Causality lets the embedder delay packets but never advance them, so each quantization-index-modulation (QIM) alignment injects nonnegative dwell into a delay buffer; unbounded dwell breaks lattice alignment and delays the host connection unacceptably. Whether a causal QIM watermark embeds stably on bursty traffic has largely been left to empirical configuration rather than analysis. We model the embedder as a reflected dwell queue under the fixed dual-lattice, equiprobable-bit rule, where injection is state-dependent -- set by the current interval and bit -- rather than exogenous. The substitution $Y_i=\delta_i-r_i$ gives only an algebraic Lindley-form identity; stability is governed by the busy-state drift at large dwell, where the effective interval collapses to zero and the mean injection becomes $\Delta/4$. Away from the critical boundary, the buffer is stable iff $\mu_d>\Delta/4$ (i.e. $\Delta<4\mu_d$) for i.i.d. backgrounds, and, under stationary-ergodic and finite-state Markov-modulated traffic with instantaneous overload, iff the time-average intensity $\bar\rho<1$. With the exogenous decoding floor $\Delta\ge c\sigma_\xi$ ($c=4Q^{-1}(\epsilon/2)$), this yields the operating window $\Delta\in[c\sigma_\xi,4\bar\mu_d)$. Simulations confirm a sharp transition at $\rho=1$ set only by the mean; on four real IPD traces, with each simulated chain confined to a single flow, the criterion gives the correct stability direction under flow-local correlation and burstiness, while pooled cross-flow means overestimate the margin. These results give a testable stable-embeddability criterion and a quantization-step configuration baseline for causal QIM network flow watermarking.

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Jiuxiang Cao, Guang Cheng, Guangjie Liu. 2026-07-16. A Queueing-Stability Criterion for Causal IPD-QIM Network Flow Watermarking. https://arxiv.org/abs/2607.14954

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