We consider a single server queue that serves a finite population of n customers that will enter the queue (require service) only once, also known as the ∆ (i) /G/1 queue. This paper presents a method for analyzing heavy-traffic behavior by using uniform acceleration, which simultaneously lets n and the service rate grow large, while the initial resource utilization approaches one. A key feature of the model is that, as time progresses, more customers have joined the queue, and fewer customers can potentially join. This diminishing population gives rise to a class of reflected stochastic processes that vanish over time, and hence do not have a stationary distribution. We establish that, when the arrival times are exponentially distributed, by suitably rescaling space and time, the queue length process converges to a Brownian motion with parabolic drift, a stochastic-process limit that captures the effect of a diminishing population by a negative quadratic drift. When the arrival times are generally distributed, our techniques provide information on the typical queue length and the first busy period. arXiv:1412.5329v2 [math.PR] 30 Nov 2015
The interactions between and with nanostructures can only be fully understood when the functional group distribution on their surfaces can be quantified accurately. Here we apply a combination of direct stochastic optical reconstruction microscopy (dSTORM) imaging and probabilistic modelling to analyse molecular distributions on spherical nanoparticles. The properties of individual fluorophores are assessed and incorporated into a model for the dSTORM imaging process. Using this tailored model, overcounting artefacts are greatly reduced and the locations of dye labels can be accurately estimated, revealing their spatial distribution. We show that standard chemical protocols for dye attachment lead to inhomogeneous functionalization in the case of ubiquitous polystyrene nanoparticles. Moreover, we demonstrate that stochastic fluctuations result in large variability of the local group density between particles. These results cast doubt on the uniform surface coverage commonly assumed in the creation of amorphous functional nanoparticles and expose a striking difference between the average population and individual nanoparticle coverage.
We consider a class of weakly interacting particle systems of meanfield type. The interactions between the particles are encoded in a graph sequence, i.e., two particles are interacting if and only if they are connected in the underlying graph. We establish a Law of Large Numbers for the empirical measure of the system that holds whenever the graph sequence is convergent in the sense of graph limits theory. The limit is shown to be the solution to a non-linear Fokker-Planck equation weighted by the (possibly random) graph limit. No regularity assumptions are made on the graphon limit so that our analysis allows for very general graph sequences, such as exchangeable random graphs. For these, we also prove a propagation of chaos result. Finally, we fully characterize the graph sequences for which the associated empirical measure converges to the mean-field limit, i.e., to the solution of the classical McKean-Vlasov equation.
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