The impact of a heavy-tailed service-time distribution upon the M/GI/s waiting-time distribution

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 disciplin e are bounded below by tails of Poisson distributions. As a consequence, th e tail 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-exc ess 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 establish additional results th at enable us to nearly capture the full asymptotics in both light and heavy traffic. The difference between the asymptotic behavior in these two regio ns shows that the actual asymptotic form must be quite complicated.