Tail Asymptotics for the Busy Period in the GI/G/1 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 (de Meyer and Teugels 1980), who treated the M/G/1 queue with a regularly varying service time distribution. Our method of proof is, opposed to the one in de Meyer and Teugels (1980), probabilistic, and reveals an insightful relationship between the busy period and the cycle maximum.