Open problems in Gaussian fluid queueing theory

Dȩbicki, Krzysztof; Mandjes, Michel

doi:10.1007/s11134-011-9237-y

Cited by 9 publications

(8 citation statements)

References 26 publications

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“…Hereafter, we assume that all considered Gaussian random fields (or processes) have almost surely continuous sample paths. We need to introduce some more notation, starting with the well-known Pickands constant H α given by Pickands (1969), Albin (1990), Piterbarg (1996), Dȩbicki (2002), , Mandjes (2007), Dȩbicki and Mandjes (2011), and Dieker and Yakir (2014) for various properties of Pickands' constant and its generalizations. Next, we introduce another constant, usually referred to as the Piterbarg constant, given by…”

Section: Further Results and The Proof Of Theorem 11mentioning

confidence: 99%

Approximation of Passage Times of γ-Reflected Processes with FBM Input

Hashorva¹,

Ji²

2014

J. Appl. Probab.

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Define a γ -reflected process W γ (t) = Y H (t) − γ inf s∈ [0,t] Y H (s), t ≥ 0, with input process {Y H (t), t ≥ 0}, which is a fractional Brownian motion with Hurst index H ∈ (0, 1) and a negative linear trend. In risk theory R γ (u) = u − W γ (t), t ≥ 0, is referred to as the risk process with tax payments of a loss-carry-forward type. For various risk processes, numerous results are known for the approximation of the first and last passage times to 0 (ruin times) when the initial reserve u goes to ∞. In this paper we show that, for the γ -reflected process, the conditional (standardized) first and last passage times are jointly asymptotically Gaussian and completely dependent. An important contribution of this paper is that it links ruin problems with extremes of nonhomogeneous Gaussian random fields defined by Y H , which we also investigate.

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Section: Further Results and The Proof Of Theorem 11mentioning

confidence: 99%

Approximation of Passage Times of γ-Reflected Processes with FBM Input

Hashorva¹,

Ji²

2014

J. Appl. Probab.

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“…with {B α (t), t ≥ 0} an FBM with Hurst index α/2 ∈ (0, 1]. It is known that H 1 = 1 and H 2 = 1/ √ π ; see Pickands (1969), Albin (1990), Piterbarg (1996), Dȩbicki (2002), , Mandjes (2007), Dȩbicki andMandjes (2011), andDieker andYakir (2014) for various properties of Pickands' constant and its generalizations. Next, we introduce another constant, usually referred to as the Piterbarg constant, given by…”

Section: Further Results and The Proof Of Theorem 11mentioning

confidence: 99%

Approximation of Passage Times of γ-Reflected Processes with FBM Input

Hashorva

2014

Journal of Applied Probability

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Define a γ-reflected processWγ(t) =YH(t) - γinfs∈[0,t]YH(s),t≥ 0, with input process {YH(t),t≥ 0}, which is a fractional Brownian motion with Hurst indexH∈ (0, 1) and a negative linear trend. In risk theoryRγ(u) =u-Wγ(t),t≥ 0, is referred to as the risk process with tax payments of a loss-carry-forward type. For various risk processes, numerous results are known for the approximation of the first and last passage times to 0 (ruin times) when the initial reserveugoes to ∞. In this paper we show that, for the γ-reflected process, the conditional (standardized) first and last passage times are jointly asymptotically Gaussian and completely dependent. An important contribution of this paper is that it links ruin problems with extremes of nonhomogeneous Gaussian random fields defined byYH, which we also investigate.

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“…These constants have remained so elusive that devising an estimation algorithm with certain performance guarantees has remained outside the scope of current methodology (Dȩbicki and Mandjes [18]). The current paper resolves this open problem for the classical Pickands' constants.…”

Section: Introductionmentioning

confidence: 99%

On asymptotic constants in the theory of extremes for Gaussian processes

Dieker

Yakir

2014

Bernoulli

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This paper gives a new representation of Pickands' constants, which arise in the study of extremes for a variety of Gaussian processes. Using this representation, we resolve the long-standing problem of devising a reliable algorithm for estimating these constants. A detailed error analysis illustrates the strength of our approach.Comment: Published in at http://dx.doi.org/10.3150/13-BEJ534 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

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Open problems in Gaussian fluid queueing theory

Cited by 9 publications

References 26 publications

Approximation of Passage Times of γ-Reflected Processes with FBM Input

Approximation of Passage Times of γ-Reflected Processes with FBM Input

Approximation of Passage Times of γ-Reflected Processes with FBM Input

On asymptotic constants in the theory of extremes for Gaussian processes

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