RECURRENCE OF TWO-DIMENSIONAL QUEUEING PROCESSES, AND RANDOM WALK EXIT TIMES FROM THE QUADRANT

• 成果类型: Article
• 署名作者: Peigne, Marc; Woess, Wolfgang
• 署名单位: Centre National de la Recherche Scientifique (CNRS); Universite de Orleans; Universite de Tours; Graz University of Technology
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/20-AAP1654
• 发表日期: 2021
• 页码: 2519-2537
• 关键词: stochastic dynamical-systems; weak contractivity; queues; RD

摘要

Let X = (X-1, X-2) be a two-dimensional random variable and X(n), n is an element of N, a sequence of i.i.d. copies of X. The associated random walk is S(n) = X(1)+ ... + X(n). The corresponding absorbed-reflected walk W(n), n is an element of N, in the first quadrant is given by W(0) = x is an element of R-+(2) and W(n) = max{0, W(n - 1) - X(n)}, where the maximum is taken coordinate-wise. This is often called the Lindley process and models the waiting times in a two-server queue. We characterize recurrence of this process, assuming suitable, rather mild moment conditions on X. It turns out that this is directly related with the tail asymptotics of the exit time of the random walk x + S(n) from the quadrant, so that the main part of this paper is devoted to an analysis of that exit time in relation with the drift vector, that is, the expectation of X.