链入仓流程
Chains-into-Bins Processes
摘要：
对{ \em球入箱过程}或{ \em占用问题}的研究有很长的历史。这些过程可以用来以自然的方式将现实问题转化为数学问题。一般来说，balls-into-bins过程的目标是将一组独立的对象（任务，作业，球）分配给一组资源（服务器，垃圾箱，骨灰盒），从而最小化最大负载。在本文中，我们分析了{ \em chains-into-bins}问题的最大负载，其定义如下。存在要分配的$n$箱和$m$对象。每个对象由连接到长度为$\ell$的链中的球组成，因此总共有$m \ell$球。我们假设链不能被破坏，并且一个链中的球必须被分配到$\ell$连续的bin。我们允许每个链$d$独立和均匀随机的bin选择其起始位置。使用接收该链的球的任何箱的最大负载最小化的规则来分配链。我们表明，对于$d \ge 2$和$m\cdot\ell=O(n)$，最大负载是$((\ln \ln m)/\ln d) +O(1)$，概率为$1-\tilde O(1/m^{d-1})$。
英文摘要：
The study of {\em balls-into-bins processes} or {\em occupancy problems} has a long history. These processes can be used to translate realistic problems into mathematical ones in a natural way. In general, the goal of a balls-into-bins process is to allocate a set of independent objects (tasks, jobs, balls) to a set of resources (servers, bins, urns) and, thereby, to minimize the maximum load. In this paper, we analyze the maximum load for the {\em chains-into-bins} problem, which is defined as follows. There are $n$ bins, and $m$ objects to be allocated. Each object consists of balls connected into a chain of length $\ell$, so that there are $m \ell$ balls in total. We assume the chains cannot be broken, and that the balls in one chain have to be allocated to $\ell$ consecutive bins. We allow each chain $d$ independent and uniformly random bin choices for its starting position. The chain is allocated using the rule that the maximum load of any bin receiving a ball of that chain is minimized. We show that, for $d \ge 2$ and $m\cdot\ell=O(n)$, the maximum load is $((\ln \ln m)/\ln d) +O(1)$ with probability $1-\tilde O(1/m^{d-1})$.
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