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 /4. Away from the critical boundary, the buffer is stable iff mu d> /4 (i.e. <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 rho<1. With the exogenous decoding floor c sigma xi (c=4Q-1(epsilon/2)), this yields the operating window in[c sigma xi,4 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.