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May 2014 On the tail asymptotics of the area swept under the Brownian storage graph
Marek Arendarczyk, Krzysztof Dȩbicki, Michel Mandjes
Bernoulli 20(2): 395-415 (May 2014). DOI: 10.3150/12-BEJ491

Abstract

In this paper, the area swept under the workload graph is analyzed: with $\{Q(t)\colon\ t\ge0\}$ denoting the stationary workload process, the asymptotic behavior of

\[\pi_{T(u)}(u):=\mathbb{P}\biggl(\int_{0}^{T(u)}Q(r)\,\mathrm{d}r>u\biggr)\]

is analyzed. Focusing on regulated Brownian motion, first the exact asymptotics of $\pi_{T(u)}(u)$ are given for the case that $T(u)$ grows slower than $\sqrt{u}$, and then logarithmic asymptotics for (i) $T(u)=T\sqrt{u}$ (relying on sample-path large deviations), and (ii) $\sqrt{u}=\mathrm{o}(T(u))$ but $T(u)=\mathrm{o}(u)$. Finally, the Laplace transform of the residual busy period are given in terms of the Airy function.

Citation

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Marek Arendarczyk. Krzysztof Dȩbicki. Michel Mandjes. "On the tail asymptotics of the area swept under the Brownian storage graph." Bernoulli 20 (2) 395 - 415, May 2014. https://doi.org/10.3150/12-BEJ491

Information

Published: May 2014
First available in Project Euclid: 28 February 2014

zbMATH: 1314.60148
MathSciNet: MR3178504
Digital Object Identifier: 10.3150/12-BEJ491

Rights: Copyright © 2014 Bernoulli Society for Mathematical Statistics and Probability

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Vol.20 • No. 2 • May 2014
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