  We study a model for email communication due to Gabrielli and Caldarelli,
where someone receives and answers emails at the times of independent Poisson
processes with intensities $\lambda_{\rm in}>\lambda_{\rm out}$. The receiver
assigns i.i.d. priorities to incoming emails according to some atomless law and
always answers the email in the mailbox with the highest priority. Since the
frequency of incoming emails is higher than the frequency of answering, below a
critical priority, the mailbox fills up ad infinitum. We prove a theorem about
the limiting shape of the mailbox just above the critical point, linking it to
the convex hull of Brownian motion. We conjecture that this limiting shape is
universal in a class of similar models, including a model for the evolution of
an order book due to Stigler and Luckock.
