The Annals of Applied Probability

Dynamic tree algorithms

Hanène Mohamed and Philippe Robert

Full-text: Open access

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 λc so that if the arrival rate is smaller than λc, 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 λc 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.

Article information

Source
Ann. Appl. Probab., Volume 20, Number 1 (2010), 26-51.

Dates
First available in Project Euclid: 8 January 2010

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1262962317

Digital Object Identifier
doi:10.1214/09-AAP617

Mathematical Reviews number (MathSciNet)
MR2582641

Zentralblatt MATH identifier
1183.68764

Subjects
Primary: 68W40: Analysis of algorithms [See also 68Q25] 60K20: Applications of Markov renewal processes (reliability, queueing networks, etc.) [See also 90Bxx]
Secondary: 90B15: Network models, stochastic

Keywords
Tree structures renewal theorems auto-regressive processes random access networks Capetanakis–Tsybakov–Mikhailov algorithm

Citation

Mohamed, Hanène; Robert, Philippe. Dynamic tree algorithms. Ann. Appl. Probab. 20 (2010), no. 1, 26--51. doi:10.1214/09-AAP617. https://projecteuclid.org/euclid.aoap/1262962317


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