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

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 $\mathrm{\Delta }\mathrm{/}4$. Away from the critical boundary, the buffer is stable iff ${\mu }_d>\mathrm{\Delta }\mathrm{/}4$ (i.e. $\mathrm{\Delta }<4{\mu }_d$) 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 $\mathrm{\Delta }\ge c{\sigma }_{\xi }$ ($c=4Q^{-1}(\epsilon \mathrm{/}2)$), this yields the operating window $\mathrm{\Delta }\in [c{\sigma }_{\xi },4{\bar{\mu }}_d)$. 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.