The M/G/1 queue with heavy-tailed service time distribution

In modern teletraffic applications of queueing theory, service time distributions B(t) with a heavy tail occur, i.e., 1-B(t)~Ct^-v for t→∞ with v>1. For such service time distributions, not much explicit information is available concerning the tail probabilities of the corresponding waiting time distribution W(t). In the present study, which is devoted to the M/G/1 queue, a class of heavy-tailed service time distributions is introduced that does allow a rather detailed analysis of the tail behavior of the waiting time distribution. For v=1½, an explicit expression for W(t) is derived. For rational v with 1<v<2, an asymptotic series for the tail probabilities of W(t) is derived. In addition, we present an approximation for W(t), which is based on a heavy-traffic limit theorem for the M/G/1 queue with heavy-tailed service time distribution (with infinite variance); this approximation is shown to yield excellent results for values of t which are not too small, even when the load is not heavy