Convergence rates for M/G/1 queues and ruin problems with heavy tails

The time-dependent virtual waiting time in a M/G/1 queue converges to a proper limit when the traffic intensity is less than one. In this paper we give precise rates on the speed of this convergence when the service time distribution has a heavy regularly varying tail. The result also applies to the classical ruin problem. We obtain the exact rate of convergence for the ruin probability after time t for the case where claims arrive according to a Poisson process and claim sizes are heavy tailed. Our result supplements similar theorems on exponential convergence rates for relaxation times in queueing theory and ruin probabilities in risk theory.