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

2026-07-16

AI Summary

This study addresses the instability of inter-packet delay (IPD)-based QIM network flow steganography under causality constraints in multi-hop encrypted links, where bursty traffic induces unbounded buffer growth, disrupting watermark alignment and degrading performance, with no existing theoretical stability criterion. The authors model the causal QIM embedder as a state-dependent injection-controlled reflected waiting queue and, by integrating queueing theory, Lindley’s recursion, and Markov-modulated traffic models, establish the first verifiable stability criterion. They prove that stability depends solely on the average injection rate and background traffic intensity, and derive the effective step size interval for Δ as [cσ_ξ, 4μ̄_d). Experiments on four real-world IPD traces confirm the criterion accurately predicts stability direction, outperforming cross-flow aggregate mean methods, while simulations reveal a sharp phase transition at ρ = 1.

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 $Y_i=δ_i-r_i$ 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 $Δ/4$. Away from the critical boundary, the buffer is stable iff $μ_d>Δ/4$ (i.e. $Δ<4μ_d$) for i.i.d. backgrounds, and, under stationary-ergodic and finite-state Markov-modulated traffic with instantaneous overload, iff the time-average intensity $\barρ<1$. With the exogenous decoding floor $Δ\ge cσ_ξ$ ($c=4Q^{-1}(ε/2)$), this yields the operating window $Δ\in[cσ_ξ,4\barμ_d)$. Simulations confirm a sharp transition at $ρ=1$ 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.

Problem

• causal watermarking
• queueing stability
• inter-packet delay
• QIM
• network flow

Innovation

• causal QIM watermarking
• queueing stability
• inter-packet delay (IPD)
• reflected dwell queue
• stable embeddability criterion