### Abstract

This talk will examine discrete and continuous time queueing systems in the context of recognising the so-called busy period as the first-passage time of a random walk process. As well as identifying the queue duration (busy-period) distribution, consideration is also given to the distribution of the maximum (extreme) queue length during a busy period and, much harder, the distribution of the total waiting time (area under the curve) during a busy period. Physical examples of interest include traffic jams, Abelian sandpile (avalanche) models in the compact directed percolation universality class, and the statistics of lattice polygon models. Throughout, the emphasis is on providing exact solutions.