  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 $1/4$.
