# Recurrence of 2-dimensional queueing processes, and random walk exit times from the quadrant

Abstract : Let $X = (X_1, X_2)$ be a 2-dimensional random variable and $X(n), n \in \mathbb{N}$ a sequence of i.i.d. copies of $X$. The associated random walk is $S(n)= X(1) + \cdots +X(n)$. The corresponding absorbed-reflected walk $W(n), n \in \mathbb{N}$ in the first quadrant is given by $W(0) = x \in \mathbb{R}_+^2$ 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, i.e., the expectation of $X$.
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Cited literature [25 references]

https://hal.archives-ouvertes.fr/hal-02281986
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Submitted on : Monday, September 9, 2019 - 4:07:20 PM
Last modification on : Friday, February 19, 2021 - 4:10:03 PM
Long-term archiving on: : Friday, February 7, 2020 - 11:38:12 PM

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### Identifiers

• HAL Id : hal-02281986, version 1
• ARXIV : 1909.00616

### Citation

Marc Peigné, Wolfgang Woess. Recurrence of 2-dimensional queueing processes, and random walk exit times from the quadrant. 2019. ⟨hal-02281986⟩

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