Sojourn time tails in the M/D/1 processor sharing queue

We consider the sojourn time V in the M/D/1 processor sharing (PS) queue and show that P(V > x) is of the form Ce-[gamma]x as x becomes large. The proof involves a geometric random sum representation of V and a connection with Yule processes, which also enables us to simplify Ott's [21] derivation of the Laplace transform of V. Numerical experiments show that the approximation P(V > x) [approximate] Ce-[gamma]x is excellent even for moderate values of x