Extreme value theory for queues via cycle maxima

The present state of extreme value theory for queues is surveyed. The exposition focuses on the regenerative properties of queueing systems, which reduces the problem to the study of the tail of the maximum $overline X( au)$ of the queueing process ${X(t)}$ during a regenerative cycle $ au$. For simple queues, methods for obtaining the distribution of $overline X( au)$ both explicitly and asymptotically are reviewed. In greater generality, the study leads to Wiener-Hopf problems. Extensions to queues in a Markov regime, for example, to those governed by Markov-modulated Poisson arrivals, are also considered.