On cycle maxima, first passage problems, and extreme value theory for queues

The distribution of the maximum of the virtual waiting time during a cycle C is studied for a variety of queueing models. For the queue, the idea is a generalization of the ladder height representation of the steady-state limit , and the results are explicit in terms of the failure rate and the density. For queues with a general Markovian arrival process and phase-type service times, the basic idea is to represent the distribution of by means of a multivariate version of the failure rate which again is related to generalize ladder heights. The fundamental step in the evaluation of is the determination of a set of first passage probabilities, which can be done either by solving a set of linear equations, or by deriving a matrix Ricatti differential equation having an explicit matrix-exponential solution; both approaches require the steady-state characteristics.