Before revision:  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, or equivalently an expected number of jobs inserted covertly , of O(n) when \mu_1 < 2\mu_2, O(n/\log n) when \mu_1 = 2\mu_2, and O(n^{\mu_2/\mu_1}) when \mu_1 > 2\mu_2.
After revision:  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, or equivalently an expected number of jobs covertly inserted , of O(n) when \mu_1 < 2\mu_2, O(n/\log n) when \mu_1 = 2\mu_2, and O(n^{\mu_2/\mu_1}) when \mu_1 > 2\mu_2.