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Renewal processes, population dynamics, and unimodular trees

Part of: Graph theory Markov processes Special processes Other groups of matrices

Published online by Cambridge University Press:  30 July 2019

François Baccelli  and
Antonio Sodre
François Baccelli*
Affiliation:
The University of Texas at Austin
Antonio Sodre*
Affiliation:
The University of Texas at Austin
*
*Postal address: Department of Mathematics, The University of Texas at Austin, 2515 Speedway, RLM 8.100, Austin, TX 78712-1202, USA.
*Postal address: Department of Mathematics, The University of Texas at Austin, 2515 Speedway, RLM 8.100, Austin, TX 78712-1202, USA.
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Abstract

Based on a simple object, an i.i.d. sequence of positive integer-valued random variables {an}n∊ℤ, we introduce and study two random structures and their connections. First, a population dynamics, in which each individual is born at time n and dies at time n + an. This dynamics is that of a D/GI/∞ queue, with arrivals at integer times and service times given by {an}n∊ℤ. Second, the directed random graph Tf on ℤ generated by the random map f(n) = n + an. Assuming only that E [a0] < ∞ and P [a0 = 1] > 0, we show that, in steady state, the population dynamics is regenerative, with one individual alive at each regeneration epoch. We identify a unimodular structure in this dynamics. More precisely, Tf is a unimodular directed tree, in which f(n) is the parent of n. This tree has a unique bi-infinite path. Moreover, Tf splits the integers into two categories: ephemeral integers, with a finite number of descendants of all degrees, and successful integers, with an infinite number. Each regeneration epoch is a successful individual such that all integers less than it are its descendants of some order. Ephemeral, successful, and regeneration integers form stationary and mixing point processes on ℤ.

Keywords

Population dynamicsqueueing systemunimodular random grapheternal family treebranching process

MSC classification

Primary: 92D25: Population dynamics (general) 60K25: Queueing theory 05C80: Random graphs
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 05C63: Infinite graphs 20H05: Unimodular groups, congruence subgroups
Type
Research Papers
Information
Journal of Applied Probability , Volume 56 , Issue 2 , June 2019 , pp. 339 - 357
Copyright
© Applied Probability Trust 2019 

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