动态树算法
Dynamic tree algorithms
摘要：
本文分析了一种处理随机波达流的通用树算法。在具有随机接入的通信网络的上下文中的Capetanakis-Tsybakov-Mikhailov协议是这种算法的一个示例。在计算机科学中，这对应于具有动态输入的trie结构。从数学上讲，它与外来到达（移民）的停止分支过程有关。在相当一般的假设下，对到达数的分布和分支过程，它表明，存在一个正的常数$λ_c$，使如果到达率小于$λ_c$，那么该算法是稳定的请求流下，也就是说，相关联的树的总大小是可积的。同时，在文献中的稳定性的早期证明的差距是固定的。当到达为泊松分布时，给出了$λ_c$的一个显式特征.在稳定性条件下，分析了从大量个体开始的树的平均大小的渐近行为。得到的结果与概率重写的函数方程描述系统的动态的帮助。证明广泛使用这种随机背景的整个文件。在这个分析中，两个基本的极限定理起着关键作用：更新定理和收敛到平衡的自回归过程与移动平均。
英文摘要：
In this paper, a general tree algorithm processing a random flow of arrivals is analyzed. Capetanakis--Tsybakov--Mikhailov's protocol in the context of communication networks with random access is an example of such an algorithm. In computer science, this corresponds to a trie structure with a dynamic input. Mathematically, it is related to a stopped branching process with exogeneous arrivals (immigration). Under quite general assumptions on the distribution of the number of arrivals and on the branching procedure, it is shown that there exists a positive constant $λ_c$ so that if the arrival rate is smaller than $λ_c$, then the algorithm is stable under the flow of requests, that is, that the total size of an associated tree is integrable. At the same time, a gap in the earlier proofs of stability in the literature is fixed. When the arrivals are Poisson, an explicit characterization of $λ_c$ is given. Under the stability condition, the asymptotic behavior of the average size of a tree starting with a large number of individuals is analyzed. The results are obtained with the help of a probabilistic rewriting of the functional equations describing the dynamics of the system. The proofs use extensively this stochastic background throughout the paper. In this analysis, two basic limit theorems play a key role: the renewal theorem and the convergence to equilibrium of an auto-regressive process with a moving average.
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