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

Jiuxiang Cai; Jiuxiang Cao; Guang Cheng; Guangjie Liu

computer security

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

arXiv:2607.14954

summary

The paper derives a queueing‑theoretic stability condition for embedding causal QIM watermarks in inter‑packet delays of encrypted multi‑hop traffic, showing when the induced delay buffer remains bounded and validating the criterion on real IPD traces.

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 $Y_{i} = I^´_{i} - r_{i}$ 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 $I^/4$ . Away from the critical boundary, the buffer is stable iff $I^¼_{d} > I^/4$ (i.e. $I^ < 4 I^¼_{d}$ ) for i.i.d. backgrounds, and, under stationary-ergodic and finite-state Markov-modulated traffic with instantaneous overload, iff the time-average intensity $I ¨ ˉ < 1$ . With the exogenous decoding floor $I^ \geq c I ¨ _{I^} ¾$ ( $c = 4 Q^{- 1} (I^µ/2)$ ), this yields the operating window $I^ \in [c I ¨ _{I^} ¾, 4 I^ˉ ¼_{d})$ . Simulations confirm a sharp transition at $I ¨ = 1$ 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.

Topics & keywords

#network flow watermarking #queueing theory #inter-packet delay #causal quantization-index-modulation #traffic burstiness #stability analysisreflected dwell queueLindley equationMarkov-modulated trafficquantization stepembedding stabilityIPD watermark