On the Asymptotic Insensitivity of the Supermarket Model in Processor Sharing Systems

The supermarket model is a popular load balancing model where each incoming job is assigned to a server with the least number of jobs among d randomly selected servers. Several authors have shown that the large scale limit in case of processor sharing servers has a unique insensitive fixed point, which naturally leads to the belief that the queue length distribution in such a system is insensitive to the job size distribution as the number of servers tends to infinity. Simulation results that support this belief have also been reported. However, global attraction of the unique fixed point of the large scale limit was not proven except for exponential job sizes, which is needed to formally prove asymptotic insensitivity. The difficulty lies in the fact that with processor sharing servers, the limiting system is in general not monotone. In this paper we focus on the class of hyperexponential distributions of order 2 and demonstrate that for this class of distributions global attraction of the unique fixed point can still be established using monotonicity by picking a suitable state space and partial order. This allows us to formally show that we have asymptotic insensitivity within this class of job size distributions. We further demonstrate that our result can be leveraged to prove asymptotic insensitivity within this class of distributions for other load balancing systems.

The geometric average of curl-free fields in periodic geometries

In periodic homogenization problems, one considers a sequence ( u η ) η {(u^{\eta})_{\eta}} of solutions to periodic problems and derives a homogenized equation for an effective quantity u ^ {\hat{u}} . In many applications, u ^ {\hat{u}} is the weak limit of ( u η ) η {(u^{\eta})_{\eta}} , but in some applications u ^ {\hat{u}} must be defined differently. In the homogenization of Maxwell’s equations in periodic media, the effective magnetic field is given by the geometric average of the two-scale limit. The notion of a geometric average has been introduced in [G. Bouchitté, C. Bourel and D. Felbacq, Homogenization of the 3D Maxwell system near resonances and artificial magnetism, C. R. Math. Acad. Sci. Paris 347 2009, 9–10, 571–576]; it associates to a curl-free field Y ∖ Σ ¯ → ℝ 3 {Y\setminus\overline{\Sigma}\to\mathbb{R}^{3}} , where Y is the periodicity cell and Σ an inclusion, a vector in ℝ 3 {\mathbb{R}^{3}} . In this article, we extend previous definitions to more general inclusions, in particular inclusions that are not compactly supported in the periodicity cell. The physical relevance of the geometric average is demonstrated by various results, e.g., a continuity property of limits of tangential traces.

The Maximum Entropy of a Metric Space

We define a one-parameter family of entropies, each assigning a real number to any probability measure on a compact metric space (or, more generally, a compact Hausdorff space with a notion of similarity between points). These generalize the Shannon and Rényi entropies of information theory. We prove that on any space X, there is a single probability measure maximizing all these entropies simultaneously. Moreover, all the entropies have the same maximum value: the maximum entropy of X. As X is scaled up, the maximum entropy grows, and its asymptotics determine geometric information about X, including the volume and dimension. And the large-scale limit of the maximizing measure itself provides an answer to the question: what is the canonical measure on a metric space? Primarily, we work not with entropy itself but its exponential, which in its finite form is already in use as a measure of biodiversity. Our main theorem was first proved in the finite case by Leinster and Meckes.

Euler-scale dynamical correlations in integrable systems with fluid motion

We devise an iterative scheme for numerically calculating dynamical two-point correlation functions in integrable many-body systems, in the Eulerian scaling limit. Expressions for these were originally derived in Ref. [1] by combining the fluctuation-dissipation principle with generalized hydrodynamics. Crucially, the scheme is able to address non-stationary, inhomogeneous situations, when motion occurs at the Euler-scale of hydrodynamics. In such situations, in interacting systems, the simple correlations due to fluid modes propagating with the flow receive subtle corrections, which we test. Using our scheme, we study the spreading of correlations in several integrable models from inhomogeneous initial states. For the classical hard-rod model we compare our results with Monte-Carlo simulations and observe excellent agreement at long time-scales, thus providing the first demonstration of validity for the expressions derived in Ref. [1]. We also observe the onset of the Euler-scale limit for the dynamical correlations.

The Nontraditional Coriolis Terms and Tropical Convective Clouds

The full, three-dimensional Coriolis force includes the familiar sine-of-latitude terms as well as frequently dropped cosine-of-latitude terms [nontraditional Coriolis terms (NCT)]. The latter are often ignored because they couple the zonal and vertical momentum equations that in the large-scale limit of weak vertical velocity are considered insignificant almost everywhere. Here, we ask whether equatorial mesoscale clouds that fall outside the large-scale limit are affected by the NCT. A simple scaling indicates that a Lagrangian parcel convecting at 10 m s−1 through the depth of the troposphere should be deflected over 2 km to the west. To understand the real impact of NCT, we develop a mathematical framework that describes an azimuthally symmetric convective circulation with an analytical expression for an incompressible poloidal flow. Because the model incorporates the full three-dimensional flow associated with convection, it uniquely predicts not only the westward tilt of clouds but also a meridional diffluence of western cloud outflow. To test these predictions, we perform a set of cloud-resolving simulations whose results show preferential lifting of surface parcels with positive zonal momentum and zonal asymmetry in convective strength. RCE simulations show changes to the organization of coherent precipitation regions and a decrease in mean convective intensity of approximately 2 m s−1 above the freezing level. An additional pair of dry cloud-resolving simulations designed to mimic the steady-state flow of the model show maximum perturbations to the upper-level zonal flow of 8 m s−1. Together, the numerical and analytic results suggest the NCT consequentially alter equatorial mesoscale convective circulations and should be considered in conceptual models.

Testing tidal alignment models for anisotropic correlations of halo ellipticities with N-body simulations

ABSTRACT There is a growing interest of using the intrinsic alignment (IA) of galaxy images as a tool to extract cosmological information complimentary to galaxy clustering analysis. Recently, Okumura & Taruya derived useful formulas for the intrinsic ellipticity–ellipticity correlation, the gravitational shear–intrinsic ellipticity correlation, and the velocity–intrinsic ellipticity correlation functions based on the linear alignment (LA) model. In this paper, using large-volume N-body simulations, we measure these alignment statistics for dark-matter haloes in real and redshift space and compare them to the LA and non-linear alignment model predictions. We find that anisotropic features of baryon acoustic oscillations in the IA statistics can be accurately predicted by our models. The anisotropy due to redshift-space distortions (RSDs) is also well described in the large-scale limit. Our results indicate that one can extract the cosmological information encoded in the IA through the Alcock–Paczynski and RSD effects.