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Finite time asymptotics of fluid and ruin models: multiplexed fractional Brownian motions case

Krzysztof Dębicki, Grzegorz Sikora (2011)

Applicationes Mathematicae

Motivated by applications in queueing fluid models and ruin theory, we analyze the asymptotics of ( s u p t [ 0 , T ] ( i = 1 n λ i B H i ( t ) - c t ) > u ) , where B H i ( t ) : t 0 , i = 1,...,n, are independent fractional Brownian motions with Hurst parameters H i ( 0 , 1 ] and λ₁,...,λₙ > 0. The asymptotics takes one of three different qualitative forms, depending on the value of m i n i = 1 , . . . , n H i .

Fluid limits for the queue length of jobs in multiserver open queueing networks

Saulius Minkevičius (2014)

RAIRO - Operations Research - Recherche Opérationnelle

The object of this research in the queueing theory is a theorem about the Strong-Law-of-Large-Numbers (SLLN) under the conditions of heavy traffic in a multiserver open queueing network. SLLN is known as a fluid limit or fluid approximation. In this work, we prove that the long-term average rate of growth of the queue length process of a multiserver open queueing network under heavy traffic strongly converges to a particular vector of rates. SLLN is proved for the values of an important probabilistic...

From a kinetic equation to a diffusion under an anomalous scaling

Giada Basile (2014)

Annales de l'I.H.P. Probabilités et statistiques

A linear Boltzmann equation is interpreted as the forward equation for the probability density of a Markov process ( K ( t ) , i ( t ) , Y ( t ) ) on ( 𝕋 2 × { 1 , 2 } × 2 ) , where 𝕋 2 is the two-dimensional torus. Here ( K ( t ) , i ( t ) ) is an autonomous reversible jump process, with waiting times between two jumps with finite expectation value but infinite variance. Y ( t ) is an additive functional of K , defined as 0 t v ( K ( s ) ) d s , where | v | 1 for small k . We prove that the rescaled process ( N ln N ) - 1 / 2 Y ( N t ) converges in distribution to a two-dimensional Brownian motion. As a consequence, the appropriately...

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