Sojourn times in a random queue with and without preemption

We consider an equilibrium M/M/1 queue in which customers arrive at a random position, both for the preemptive and non-preemptive cases. The Laplace-Stieltjes transforms of their sojourn time distributions are derived and in the latter case it is verified that the result is that of an M/M/1 queue where the server chooses the next task to serve at random.