Dynamic tree algorithms

Details Dynamic tree algorithms Dynamic tree algorithms

2008-09-21

arxiv.org/abs/0809.3577v2

Annals of Applied Probability 2010, Vol. 20, No. 1, 26-51

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/09-AAP617 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Inst

Scientific paper

10.1214/09-AAP617

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 $\lambda_c$ so that if the arrival rate is smaller than $\lambda_c$, 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 $\lambda_c$ 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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