Packing a bin online to maximize the total number of items

A bin of capacity 1 and a nite sequence of items of\ud sizes a1; a2; : : : are considered, where the items are given one by one\ud without information about the future. An online algorithm A must\ud irrevocably decide whether or not to put an item into the bin whenever\ud it is presented. The goal is to maximize the number of items collected.\ud A is f-competitive for some function f if n() f(nA()) holds for all\ud sequences , where n is the (theoretical) optimum and nA the number\ud of items collected by A.\ud A necessary condition on f for the existence of an f-competitive\ud (possibly randomized) online algorithm is given. On the other hand,\ud this condition is seen to guarantee the existence of a deterministic online\ud algorithm that is "almost" f-competitive in a well-dened sense

Precise algorithms to compute surface correlation functions of two-phase heterogeneous media and their applications

The quantitative characterization of the microstructure of random heterogeneous media in $d$-dimensional Euclidean space $\mathbb{R}^d$ via a variety of $n$-point correlation functions is of great importance, since the respective infinite set determines the effective physical properties of the media. In particular, surface-surface $F_{ss}$ and surface-void $F_{sv}$ correlation functions (obtainable from radiation scattering experiments) contain crucial interfacial information that enables one to estimate transport properties of the media (e.g., the mean survival time and fluid permeability) and complements the information content of the conventional two-point correlation function. However, the current technical difficulty involved in sampling surface correlation functions has been a stumbling block in their widespread use. We first present a concise derivation of the small-$r$ behaviors of these functions, which are linked to the \textit{mean curvature} of the system. Then we demonstrate that one can reduce the computational complexity of the problem by extracting the necessary interfacial information from a cut of the system with an infinitely long line. Accordingly, we devise algorithms based on this idea and test them for two-phase media in continuous and discrete spaces. Specifically for the exact benchmark model of overlapping spheres, we find excellent agreement between numerical and exact results. We compute surface correlation functions and corresponding local surface-area variances for a variety of other model microstructures, including hard spheres in equilibrium, decorated "stealthy" patterns, as well as snapshots of evolving pattern formation processes (e.g., spinodal decomposition). It is demonstrated that the precise determination of surface correlation functions provides a powerful means to characterize a wide class of complex multiphase microstructures

Load Balancing in Large-Scale Systems with Multiple Dispatchers

Load balancing algorithms play a crucial role in delivering robust application performance in data centers and cloud networks. Recently, strong interest has emerged in Join-the-Idle-Queue (JIQ) algorithms, which rely on tokens issued by idle servers in dispatching tasks and outperform power-of-$d$ policies. Specifically, JIQ strategies involve minimal information exchange, and yet achieve zero blocking and wait in the many-server limit. The latter property prevails in a multiple-dispatcher scenario when the loads are strictly equal among dispatchers. For various reasons it is not uncommon however for skewed load patterns to occur. We leverage product-form representations and fluid limits to establish that the blocking and wait then no longer vanish, even for arbitrarily low overall load. Remarkably, it is the least-loaded dispatcher that throttles tokens and leaves idle servers stranded, thus acting as bottleneck. Motivated by the above issues, we introduce two enhancements of the ordinary JIQ scheme where tokens are either distributed non-uniformly or occasionally exchanged among the various dispatchers. We prove that these extensions can achieve zero blocking and wait in the many-server limit, for any subcritical overall load and arbitrarily skewed load profiles. Extensive simulation experiments demonstrate that the asymptotic results are highly accurate, even for moderately sized systems