## The Annals of Applied Probability

### Dynamic tree algorithms

#### 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

#### 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

#### References

• [1] Aldous, D. J. (1987). Ultimate instability of exponential back-off protocol for acknowledgement-based transmission control of random access communication channels. IEEE Trans. Inform. Theory 33 219–223.
• [2] Berger, T. (1981). The Poisson multiple-access conflict resolution problem. In Multi-User Communication Systems (G. Longo, ed.). CISM Courses and Lectures 265 1–28. Springer, New York.
• [3] Bertoin, J. (2001). Homogeneous fragmentation processes. Probab. Theory Related Fields 121 301–318.
• [4] Bertsekas, D. and Gallager, R. (1991). Data Networks, 2nd ed. Prentice-Hall, Upper Saddle River, NJ.
• [5] Boxma, O. J., Denteneer, D. and Resing, J. (2003). Delay models for contention trees in closed populations. Performance Evaluation 53 169–185.
• [6] Capetanakis, J. I. (1979). Tree algorithms for packet broadcast channels. IEEE Trans. Inform. Theory 25 505–515.
• [7] Ephremides, A. and Hajek, B. (1998). Information theory and communication networks: An unconsummated union. IEEE Trans. Inform. Theory 44 2416–2434.
• [8] Fayolle, G., Flajolet, P. and Hofri, M. (1986). On a functional equation arising in the analysis of a protocol for a multi-access broadcast channel. Adv. in Appl. Probab. 18 441–472.
• [9] Fayolle, G., Flajolet, P., Hofri, M. and Jacquet, P. (1985). Analysis of a stack algorithm for random multiple-access communication. IEEE Trans. Inform. Theory 31 244–254.
• [10] Flajolet, P., Gourdon, X. and Dumas, P. (1995). Mellin transforms and asymptotics: Harmonic sums. Theoret. Comput. Sci. 144 3–58.
• [11] Janssen, A. J. E. M. and De Jong, M. J. M. (2000). Analysis of contention tree algorithms. IEEE Trans. Inform. Theory 46 217–??2.
• [12] Massey, J. L. (1981). Collision-resolution algorithms and random access communication. In Multi-User Communication Systems (G. Longo, ed.). CISM Courses and Lectures 265 73–140. Springer, New York.
• [13] Mathys, P. and Flajolet, P. (1985). Q-ary collision resolution algorithms in random-access systems with free or blocked channel access. IEEE Trans. Inform. Theory 31 217–243.
• [14] Mohamed, H. and Robert, P. (2005). A probabilistic analysis of some tree algorithms. Ann. Appl. Probab. 15 2445–2471.
• [15] Tsybakov, B. S. and Mikhaĭlov, V. A. (1978). Free synchronous packet access in a broadcast channel with feedback. Probl. Inf. Transm. 14 32–59.
• [16] Tsybakov, B. S. and Vvedenskaya, N. D. (1981). Random multiple-access stack algorithm. Probl. Inf. Transm. 16 230–243.
• [17] Van Velthoven, J., Van Houdt, B. and Blondia, C. (2006). On the probability of abandonment in queues with limited sojourn and waiting times. Oper. Res. Lett. 34 333–338.