On the M/G/1 queue with heavy-tailed service time distributions

In present teletraffic applications of queueing theory service time distributions $B(t)$ with a heavy tail occur, i.e. $1-B(t) sim t^{-nu$ for $t rightarrow infty$ with $nu > 1$. For such service time distributions not much explicit information is available concerning the tail probabilities of the inherent waiting time distribution $W (t)$. In the present study the waiting time distribution is studied for a stable $M/G/1$ model for a class of service time distributions with $1 < nu < 2$. For $nu =1, frac{1{2$ the explicit expression for $Q(t)$ is derived. For rational $nu$ with $1< nu < 2$, an asymptotic series for the tail probabilities of $W(t)$ is derived.