Tail asymptotics for the busy period in the G1/G1 queue

We characterise the tail behaviour of the busy period distribution in the GI/G/1 queue under the assumption that the tail of the service time distribution is of intermediate regular variation. This extends a result of De Meyer and Teugels [16] who treated the M/G/1 queue with a regularly varying service time distribution. Our method of proof is, opposed to the one in [16], probabilistic and reveals an insightful relationship between the busy period and the cycle maximum