How long does a Gaussian queue remember?

A fluid queue is fed by fractional Brownian motion and drained at rate c: the stationary workload is Q(t) = sup_u≥0 (W(t) − W(t−u)) with W(t) = B_H(t) − ct. The marginal law of Q has been understood for decades; its temporal dependence — an open problem listed by Dębicki & Mandjes — is what this paper settles. The lookback time τ, how far into the past the supremum reaches, is the key: for H ≠ ½, Cov(Q(0), Q(r)) ~ H(2H−1) (E τ)² r^2H−2 — the queue inherits the dependence range of its input, positive and non-integrable for H > ½, eventually negative for H < ½. Brownian input (H = ½) is the Markovian exception, with exponential decay r^−3/2 e^−c²r/2 (Abate–Whitt). From Dębicki, Hashorva & Novikov, Covariance decay for Gaussian queues.

live: the net input W(t) = B_H(t) − ct (dark), its running minimum (dashed), the workload Q(t) as the gap between them (blue), and the lookback τ(t) — the bracket from now back to the time of the minimum

a stretch of the workload Q(t) (blue area) and, below it, the lookback τ(t): the age of the increment that realises the supremum

empirical covariance Cov(Q(0), Q(r)) against lag r on log–log axes (dots; hollow = negative), with the theoretical asymptote (dashed); its fBm constant uses the simulated mean lookback

E τ (from the simulation)

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mean lookback over time

Var Q

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Asymptotic constant

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Hurst index H = 0.75 Drain rate c = 1.0

Fractional Gaussian noise by circulant embedding (Davies–Harte, FFT), time step 0.02, eight independent runs of length 5 243 each, discarding the first 15% as burn-in. The running-minimum construction starts at time zero, so this is a finite-history approximation to the stationary queue. Differences between independent runs estimate covariance without the downward bias from subtracting each run’s own sample mean. The discrete supremum underestimates the continuous one; finite history and long memory can cause further bias, especially for high H and small c. Large-lag dots are noisy, and the dashed fBm constant is itself estimated.