Gaussian queues in light and heavy traffic

In this paper we investigate Gaussian queues in the light-traffic and in the heavy-traffic regime. The setting considered is that of a centered Gaussian process $X\equiv\{X(t):t\in\mathbb R\}$ with stationary increments and variance function $\sigma^2_X(\cdot)$, equipped with a deterministic drift $c>0$, reflected at 0: $$Q_X^{(c)}(t)=\sup_{-\infty<s\le t}(X(t)-X(s)-c(t-s)).$$ We study the resulting stationary workload process $Q^{(c)}_X\equiv\{Q_X^{(c)}(t):t\ge0\}$ in the limiting regimes $c\to 0$ (heavy traffic) and $c\to\infty$ (light traffic). The primary contribution is that we show for both limiting regimes that, under mild regularity conditions on the variance function, there exists a normalizing function $\delta(c)$ such that $Q^{(c)}_X(\delta(c)\cdot)/\sigma_X(\delta(c))$ converges to a non-trivial limit in $C[0,\infty)$

A multi-ethnic genome-wide association study implicates collagen matrix integrity and cell differentiation pathways in keratoconus

Keratoconus is characterised by reduced rigidity of the cornea with distortion and focal thinning that causes blurred vision, however, the pathogenetic mechanisms are unknown. It can lead to severe visual morbidity in children and young adults and is a common indication for corneal transplantation worldwide. Here we report the first large scale genome-wide association study of keratoconus including 4,669 cases and 116,547 controls. We have identified significant association with 36 genomic loci that, for the first time, implicate both dysregulation of corneal collagen matrix integrity and cell differentiation pathways as primary disease-causing mechanisms. The results also suggest pleiotropy, with some disease mechanisms shared with other corneal diseases, such as Fuchs endothelial corneal dystrophy. The common variants associated with keratoconus explain 12.5% of the genetic variance, which shows potential for the future development of a diagnostic test to detect susceptibility to disease