Queueing with neighbours

In this paper we study asymptotic behaviour of a growth process generated by a semi-deterministic variant of cooperative sequential adsorption model (CSA). This model can also be viewed as a particular queueing system with local interactions. We show that quite limited randomness of the model still generates a rich collection of possible limiting behaviours

Stability of a growth process generated by monomer filling with nearest-neighbour cooperative effects

We study stability of a growth process generated by sequential adsorption of particles on a one-dimensional lattice torus, that is, the process formed by the numbers of adsorbed particles at lattice sites, called heights. Here the stability of process, loosely speaking, means that its components grow at approximately the same rate. To assess stability quantitatively, we investigate the stochastic process formed by differences of heights. The model can be regarded as a variant of a Polya urn scheme with local geometric interaction

Probabilistic models motivated by cooperative sequential adsorption

This survey concerns probabilistic models motivated by cooperative sequential adsorption (CSA) models. CSA models are widely used in physics and chemistry for modelling adsorption processes in which adsorption rates depend on the spatial configuration of already adsorbed particles. Corresponding probabilistic models describe random sequential allocation of particles either in a subset of Euclidean space, or at vertices of a graph. Depending on a technical setup these probabilistic models are stated in terms of spatial or integer-valued interacting birth-and-death processes. In this survey we consider several such models that have been studied in recent years

On a model of sequential point patterns

A sequential point process motivated by the cooperative sequential adsorption is studied. Analytical properties of the process are considered in details. It is shown that the point process is useful for modelling cluster point patterns. The test on the real life data is carried ou

Gaussian fluctuations of random point measures generated by cooperative sequential adsorption

A finite number of points are sequentially allocated in a finite domain of d-dimensional space. The probability distribution of a point depends on all previously allocated points. We consider a situation when this dependence vanishes as the domain is saturated by points. The law of large numbers and the central limit theorem are proved for the generated sequence of random point measures as the number of points goes to infinit

Long Term Behaviour of Locally Interacting Birth-and-Death Processes

In this paper we study long-term evolution of a finite system of locally interacting birth-and-death processes labelled by vertices of a finite connected graph. A detailed description of the asymptotic behaviour is obtained in the case of both constant vertex degree graphs and star graphs. The model is motivated by modelling interactions between populations and is related to interacting particle systems, Gibbs models with unbounded spins, as well as urn models with interaction.Comment: 26 pages, 2 figure

Spread of infection on homogeneous tree

This paper concerns a probabilistic model of spread of infection on a homogeneous tree. The model is related to stochastic SIR model on graphs. We derive an integral equation for the distribution of the waiting time for an individual to get infected. In a special case the integral equation is equivalent to the Bernoulli differential equation and implies the classic SIR model under an appropriate scaling