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February, 1991 On the Properties of a Tree-Structured Server Process
J. Komlos, A. Odlyzko, L. Ozarow, L. A. Shepp
Ann. Appl. Probab. 1(1): 118-125 (February, 1991). DOI: 10.1214/aoap/1177005984

Abstract

Let $X_0$ be a nonnegative integer-valued random variable and let an independent copy of $X_0$ be assigned to each leaf of a binary tree of depth $k$. If $X_0$ and $X'_0$ are adjacent leaves, let $X_1 = (X_0 - 1)^+ + (X'_0 - 1)^+$ be assigned to the parent node. In general, if $X_j$ and $X'_j$ are assigned to adjacent nodes at level $j = 0, \cdots, k - 1$, then $X_j$ and $X'_j$ are, in turn, independent and the value assigned to their parent node is then $X_{j+1} = (X_j - 1)^+ + (X'_j - 1)^+$. We ask what is the behavior of $X_k$ as $k \rightarrow \infty$. We give sufficient conditions for $X_k \rightarrow \infty$ and for $X_k \rightarrow 0$ and ask whether these are the only nontrivial possibilities. The problem is of interest because it asks for the asymptotics of a nonlinear transform which has an expansive term (the + in the sense of addition) and a contractive term (the + in the sense of positive part).

Citation

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J. Komlos. A. Odlyzko. L. Ozarow. L. A. Shepp. "On the Properties of a Tree-Structured Server Process." Ann. Appl. Probab. 1 (1) 118 - 125, February, 1991. https://doi.org/10.1214/aoap/1177005984

Information

Published: February, 1991
First available in Project Euclid: 19 April 2007

zbMATH: 0719.60127
MathSciNet: MR1097467
Digital Object Identifier: 10.1214/aoap/1177005984

Subjects:
Primary: 60K99

Keywords: Aloha , nonlinear recurrence , Poisson tree

Rights: Copyright © 1991 Institute of Mathematical Statistics

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Vol.1 • No. 1 • February, 1991
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