## The Annals of Applied Probability

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

### Dynamic tree algorithms

Hanène Mohamed and Philippe Robert

#### 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 *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.

#### 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