Tail Behaviour of the Area Under the Queue Length Process of the Single Server Queue with Regularly Varying Service Times

This paper considers a stable GI|GI|1 queue with a regularly varying service time distribution. We derive the tail behaviour of the integral of the queue length process Q(t) over the busy period. We show that an occurrence of a large integral is related to an occurrence of a large maximum of the queueing process over the busy period and we exploit asymptotic result for this variable. We also prove the functional central limit theorem for ∫ t 0 Q(s) ds.