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Fluctuation limit theorems for age-dependent critical binary branching systems

José Alfredo López-Mimbela, Antonio Murillo-Salas (2011)

ESAIM: Proceedings

We consider an age-dependent branching particle system in ℝd, where the particles are subject to α-stable migration (0 < α ≤ 2), critical binary branching, and general (non-arithmetic) lifetimes distribution. The population starts off from a Poisson random field in ℝd with Lebesgue intensity. We prove functional central limit theorems and strong laws of large numbers under two rescalings: high particle density, and a space-time rescaling...

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...

Fragmentation-Coagulation Models of Phytoplankton

Ryszard Rudnicki, Radosław Wieczorek (2006)

Bulletin of the Polish Academy of Sciences. Mathematics

We present two new models of the dynamics of phytoplankton aggregates. The first one is an individual-based model. Passing to infinity with the number of individuals, we obtain an Eulerian model. This model describes the evolution of the density of the spatial-mass distribution of aggregates. We show the existence and uniqueness of solutions of the evolution equation.

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 } × ℝ 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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