On the Properties of a Tree-Structured Server Process

Loading...
Thumbnail Image
Penn collection
Statistics Papers
Degree type
Discipline
Subject
aloha
poisson tree
nonlinear recurrence
Statistics and Probability
Funder
Grant number
License
Copyright date
Distributor
Related resources
Author
Komlos, Janos
Odlyzko, Andrew
Ozarow, L.
Shepp, Larry A
Contributor
Abstract

Let X0 be a nonnegative integer-valued random variable and let an independent copy of X0 be assigned to each leaf of a binary tree of depth k. If X0 and X0′ are adjacent leaves, let X1=(X0−1)++(X0′−1)+ be assigned to the parent node. In general, if Xj and Xj′ are assigned to adjacent nodes at level j = 0,⋯, k − 1, then Xj and Xj′ are, in turn, independent and the value assigned to their parent node is then Xj+1=(Xj−1)++(Xj′−1)+. We ask what is the behavior of Xk as k→∞. We give sufficient conditions for Xk→∞ and for Xk→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).

Advisor
Date Range for Data Collection (Start Date)
Date Range for Data Collection (End Date)
Digital Object Identifier
Series name and number
Publication date
1991
Journal title
The Annals of Applied Probability
Volume number
Issue number
Publisher
Publisher DOI
Journal Issue
Comments
Recommended citation
Collection