论文标题
带有不对称反馈和反射的bins模型
Balls-in-bins models with asymmetric feedback and reflection
论文作者
论文摘要
球中的球模型描述了无限多个球的随机顺序分配到有限数量的垃圾箱中。在这些模型中，将球放入垃圾箱中，概率与给定功能成正比（反馈功能），这取决于垃圾箱中现有球的数量。通常，所有垃圾箱的反馈函数（对称反馈）都是相同的，并且箱中的球数没有任何约束。在本文中，我们研究了带有两个垃圾箱的BB模型的版本，其中违反了上述假设。在第一个感兴趣的模型中，反馈功能可以取决于垃圾箱（具有非对称反馈的BB模型）。在两个反馈函数都是幂律和超级线性的情况下，一个垃圾箱几乎可以肯定地收到几乎有限的很多球，我们研究了给定垃圾箱发生这种情况的可能性。特别是，在某些初始条件下，我们得出了此概率的正常近似值，这概括了在对称反馈的情况下获得的[5]中的结果。本文的主要部分涉及BB模型，其不对称反馈会受到对分配球数量的某些限制。该模型可以解释为反映曲线楔中随机行走的瞬态，我们可以对其长期行为进行完整的分类。
Balls-in-bins models describe a random sequential allocation of infinitely many balls into a finite number of bins. In these models a ball is placed into a bin with probability proportional to a given function (feedback function), which depends on the number of existing balls in the bin. Typically, the feedback function is the same for all bins (symmetric feedback), and there are no constraints on the number of balls in the bins. In this paper we study versions of BB models with two bins, in which the above assumptions are violated. In the first model of interest the feedback functions can depend on a bin (BB model with asymmetric feedback). In the case when both feedback functions are power law and superlinear, a single bin receives all but finitely many balls almost surely, and we study the probability that this happens for a given bin. In particular, under certain initial conditions we derive the normal approximation for this probability, which generalizes the result in [5] obtained in the case of the symmetric feedback. The main part of the paper concerns the BB model with asymmetric feedback evolving subject to certain constraints on the numbers of allocated balls. The model can be interpreted as a transient reflecting random walk in a curvilinear wedge, and we obtain a complete classification of its long term behavior.
