The critical greedy server on the integers is recurrent

TL;DR

This paper proves that in the critical case where arrival and service rates are equal, a greedy server on the integers visits every site infinitely often and characterizes its position with precise asymptotics.

Contribution

It establishes recurrence of the critical greedy server on the integers and provides a sharp iterated logarithm description of its position.

Findings

The server is recurrent in the critical case $\lambda = \mu$.

The times between emptying queues grow doubly exponentially.

The probability of the server changing direction approaches 1/4.

Abstract

Each site of $\mathbb{Z}$ hosts a queue with arrival rate $\lambda$. A single server, starting at the origin, serves its current queue at rate $\mu$ until that queue is empty, and then moves to the longest neighbouring queue. In the critical case $\lambda = \mu$, we show that the server returns to every site infinitely often. We also give a sharp iterated logarithm result for the server's position. Important ingredients in the proofs are that the times between successive queues being emptied exhibit doubly exponential growth, and that the probability that the server changes its direction is asymptotically equal to $1/4$.