Dynamic tree algorithms

TL;DR

This paper analyzes the stability and asymptotic behavior of dynamic tree algorithms, especially in communication networks, using probabilistic methods and fixed earlier proof gaps, with explicit results for Poisson arrivals.

Contribution

It provides a general stability criterion for dynamic tree algorithms and fixes gaps in previous proofs, with explicit characterization for Poisson arrivals.

Findings

Existence of a critical arrival rate for stability

Explicit stability threshold for Poisson arrivals

Asymptotic analysis of tree size under stability

Abstract

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.