Large deviations for multi-dimensional reflected fractional Brownian motion

Abstract

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.

Large deviation properties of constant rate data streams sharing a buffer with long-range dependent traffic in critical loading

“…To some extend these two observations (which also partially apply to a single queue and short-range dependent traffic) have motivated a large number of publications which address the analysis of queueing systems by investigating large deviations of reflected Gaussian processes [1], [20], [26], [28], [31], [32]. In such models the Gaussian input approximates statistical properties of the traffic processes, and the Skorokhod map captures the response of the queueing system to this input.…”

“…However, again, only a few explicit results about the behavior of multi-dimensional reflected fractional Brownian motion have been obtained so far. Most of them can be classified as logarithmic asymptotics for certain tail probabilities of their distribution [20], [26].…”

Asymptotic Expansion of Stationary Distribution for Reflected Brownian Motion in the Quarter Plane via Analytic Approach

“…Two-dimensional semimartingale reflecting Brownian motion (SRBM) in the quarter plane received a lot of attention from the mathematical community. Problems such as SRBM existence [39,40], stationary distribution conditions [19,22], explicit forms of stationary distribution in special cases [7,8,19,23,30], large deviations [1,7,33,34] construction of Lyapunov functions [10], and queueing networks approximations [19,21,31,32,43] have been intensively studied in the literature. References cited above are non-exhaustive, see also [42] for a survey of some of these topics.…”

Reflected Brownian motion in the quadrant: tail behavior of the stationary distribution

“…Thus we can assume without loss of generality that R has the form (A.1), and must consider the following three cases: (a) r 1 , r 2 ≤ 0 and r 1 r 2 < 1; (b) r 1 , r 2 ≥ 0 and r 1 r 2 < 1; and (c) r 1 and r 2 have opposite signs. In case (a) the reflection matrix R is what Berman and Plemmons [2] call an M -matrix; Majewski [9] proved that the LDP holds under the stability condition (1.3) in this case. (Majewski uses the term "K-matrix" instead of the more standard term "M -matrix.")…”