On recurrence of the multidimensional Lindley process

TL;DR

This paper investigates the recurrence properties of multidimensional Lindley processes, extending classical queueing theory to higher dimensions and analyzing their behavior through random walk relations and fluctuation theory.

Contribution

It introduces new criteria for recurrence of multidimensional Lindley processes using tail behavior assumptions and fluctuation analysis of associated random walks.

Findings

Recurrence depends on tail behavior of the underlying random walk.

Established links between Lindley processes and random walk fluctuations.

Provided conditions under which multidimensional Lindley processes are recurrent.

Abstract

A Lindley process arises from classical studies in queueing theory and it usually reflects waiting times of customers in single server models. In this note we study recurrence of its higher dimensional counterpart under some mild assumptions on the tail behaviour of the underlying random walk. There are several links between the Lindley process and the associated random walk and we build upon such relations. We apply a method related to discrete subordination for random walks on the integer lattice together with various facts from the theory of fluctuations of random walks.