  Consider a setting where Willie generates a Poisson stream of jobs and routes
them to a single server that follows the first-in first-out discipline. Suppose
there is an adversary Alice, who desires to receive service without being
detected. We ask the question: what is the number of jobs that she can receive
covertly, i.e. without being detected by Willie? In the case where both Willie
and Alice jobs have exponential service times with respective rates $\mu_1$ and
$\mu_2$, we demonstrate a phase-transition when Alice adopts the strategy of
inserting a single job probabilistically when the server idles : over $n$ busy
periods, she can achieve a covert throughput, measured by the expected number
of jobs covertly inserted, of $\mathcal{O}(\sqrt{n})$ when $\mu_1 < 2\mu_2$,
$\mathcal{O}(\sqrt{n/\log n})$ when $\mu_1 = 2\mu_2$, and
$\mathcal{O}(n^{\mu_2/\mu_1})$ when $\mu_1 > 2\mu_2$. When both Willie and
Alice jobs have general service times we establish an upper bound for the
number of jobs Alice can execute covertly. This bound is related to the Fisher
information. More general insertion policies are also discussed.
