By proving the continuity of multi-dimensional Skorokhod maps in a quasi-linearly discounted uniform norm on the doubly infinite time interval R, and strengthening know sample path large deviation principles for fractional Brownian motion to this topology, we obtain large deviation principles for the image of multi-dimensional fractional Brownian motions under Skorokhod maps as an immediate consequence of the contraction principle. As an application, we explicitly calculate large deviation decay rates for steady-state tail probabilities of certain queueing systems in multi-dimensional heavy traffic models driven by fractional Brownian motions.
We consider multiclass feedforward queueing networks under first in first out and priority service disciplines driven by long-range dependent arrival and service time processes. We show that in critical loading the normalized workload, queue length and sojourn time processes can converge to a multi-dimensional reflected fractional Brownian motion. This weak heavy traffic approximation is deduced from a deterministic pathwise approximation of the network behavior close to constant critical load in terms of the solution of a Skorokhod problem. Since we model the doubly infinite time interval, our results directly cover the stationary case.
Publication [1] contains a number of inaccuracies. Particularly point 9 below justifies a correction, since the concerned Lemma 20 forms an essential part in our work [2]:1. Page 408, first paragraph of Sect. 2: It should be "d := lim t→−∞ d(t)/t" and "d := lim t→∞ d(t)/t". I want to thank an anonymous referee for indicating this typo. 2. Page 411, Lemma 1: "w = 0" and "id R − y ∈ I" should be added to statement 2, and "q = 0" and "id R − y ∈ I" to statement 3. 3. Page 413, Corollary 3: w = 0 and id R − y ∈ I n is missing in statement 2, and q = 0 and id R − y ∈
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