By exploiting an infinite-server-model lower bound, we show that the tails of the steady-state and transient waiting-time distributions in the M/GI/s queue with unlimited waiting room and the first-come first-served discipline are bounded below by tails of Poisson distributions. As a consequence, the trial of the steady-state waiting-time distribution is bounded below by a constant times the sth power of the tail of the service-time stationary-excess distribution. We apply that bound to show that the steady-state waiting-time distribution has a heavy tail (with appropriate definition) whenever the service-time distribution does. We also established additional results that enable us to nearly capture the full asymptotics in both light and heavy traffic. The difference between the asymptotic behavior in these two regions shows that the actual asymptotic form must be quite complicated.
