In the framework of quantum probability, we present a simple geometrical mechanism which gives rise to binomial distributions, Gaussian distributions, Poisson distributions, and their interrelation. More specifically, by virtue of coherent states and a toy analogue of the Bargmann transform, we calculate the probability distributions of the position observable and the Hamiltonian arising in the representation of the classic group SU(2). This representation may be viewed as a constrained harmonic oscillator with a two-dimensional sphere as the phase space. It turns out that both the position observable and the Hamiltonian have binomial distributions, but with different asymptotic behaviours: with large radius and high spin limit, the former tends to the Gaussian while the latter tends to the Poisson.

We derive some asymptotic results for the rate of convergence to equilibrium for the number of busy servers in an M/M/N/N queue with input rate λN and service rate 1 for N → ∞ in the ‘subcritical’ case λ ∈]0, 1[. These results improve recent contributions of Fricker, Robert and Tibi.

Fast stable methods for inverting multidimensional Laplace transforms have been developed in recent years by Abate, Whitt and others. We use these methods here to compute numerically the first-passage-time distribution for a spectrally one-sided Lévy process; the basic algorithm is not easy to apply, and we have to develop a variant of it. The numerical performance is as good as the original algorithm.

A probabilistic proof of a Stein factor bound is given, extending the Poisson case in Xia (1999).